Solving fuzzy fractional q -differential equations via fuzzy q -differential transform

Abstract

This paper proposes a method for solving fuzzy linear and nonlinear partial q-differential equations by the fuzzy q-differential transform. Further, we implemented the fuzzy fractional q-differential transform for solving some types of fuzzy fractional q-differential equations. The technique investigated is based on gH-differentiability, fuzzy q-derivative, and fuzzy q-fractional derivative. Various concrete problems have been tested by implementing the new method, and the results show great performance. The results also reveal that the method is a very effective and quite accurate mathematical tool for solving fuzzy fractional and integer q-differential equations. Finally, we have provided some examples illustrating our method.

Keywords

Fuzzy numbers fuzzy-valued functions

1 Introduction

As we all know, the differential (integral) theory of fuzzy number valued function based on the theory of fuzzy number space is one of the main research contents of fuzzy analysis, especially the fuzzy differential and integral equations, which are widely used in engineering technology and social science, have aroused great interest of scholars in different fields. The research of fuzzy differential equations is mainly based on the following three methods. The first is based on the H-derivative and the generalized derivative of Bede. The second is based on Zadeh’s extension principle. The third is based on the theory of differential inclusion, and the theory of fuzzy differential equations in these three meanings is different from each other.

In this work, on the based of the H-derivative and the generalized derivative of Bede, we present the solutions of fuzzy partial q-differential equations via two-dimensional fuzzy q-DTM. Moreover, we study one-dimensional fuzzy fractional q-DTM to obtain the solutions of some types of fuzzy fractional q-differential and q-integro-differential equations. Finally, we consider the two-dimensional fuzzy fractional q-DTM for solving fuzzy fractional partial q-differential equations systematically studied.

Fuzzy analysis and fuzzy differential equations have been proposed to deal with uncertainty due to incomplete information that appears in several mathematical or computer models of certain deterministic real-world phenomena. This theory has furthermore developed and a large number of applications of this theory have been considered. As a result, fuzzy fractional differential equations and fractional differential equations have emerged as important topics [53 –55]. Stefanini and Bede [50] have proposed the generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Also, Bede and Stefanini [17] introduced the generalized differentiability of fuzzy-valued functions. Gomes and Barros [21] discussed the generalized difference and the generalized differentiability. The concept of the fuzzy-type Riemann-Liouville differentiability based on Hukuhara differentiability was introduced in [8, 9] and using the Hausdorff measure of non-compactness, the researchers presented some fuzzy integral equations using appropriate compression-type conditions. Various approaches and techniques, based on Hukuhara differentiability or generalized Hukuhara differentiability [17], have been introduced in some of the works in the literature; see [6 , 78].

In recent years, the fuzzy partial differential equations (FPDEs) have attracted great interest because of their practical applications in many fields such as physics, social science, and other areas of science and engineering. The FPDEs have been studied by many authors using different methods. Keshavarza et al. [32] presented the fuzzy solution to the mathematical model of a cancer tumor under Caputo-generalized Hukuhara partial differentiability by using fuzzy integral transforms. Keshavarz and Allahviranloo [31] studied the fuzzy fundamental triangular solution of the fractional diffusion equation under Caputo generalized Hukuhara partial differentiability by using the fuzzy Laplace transform and the fuzzy Fourier transform. Furthermore; see [1–3 , 48].

Recently, q-calculus appears as a connection between mathematics and physics. Ernst [59] introduced a new notation for q-calculus a new q-Taylor’s formula. Jing and Fan [60] proposed q-Taylor’s formula with its q-remainder. Koekoek and Swarttow [29] presented askey-Scheme of hypergeometric orthogonal polynomials and its q-analogue. Koelink [30] considered the eight lectures on quantum groups and q-special functions. Koornwinder and Swarttow, [33] studied on q-analogues of the Fourier and Hankel transforms. Noeiaghdam et al. [40] studied the introduction of the numerical methods for solving the fuzzy q-differential equations that many real life problems can be modelized in the form of these equations. Noeiaghdam et al. [41] introduced the fuzzy q-derivative and fuzzy q-fractional derivative in Caputo sense by using generalized Hukuhara difference, and then provide the related theorems and properties in detail. More details, one can refer to [12 , 51].

Differential equations (DEs) that use q-difference calculus have become a powerful way to model many problems in engineering, physics, and mathematics [64 –66]. Different researchers [67 –70] have looked at DEs with fractional q-difference calculus. Fractional q-DEs have several interesting topics about the existence and stability of solutions. Recently, many researchers have looked into the existence, uniqueness, and different types of Ulam stability (US) of solutions to fractional q-DEs; see [71–73 , 77] for some of these studies. In addition, many scholars have been studying sequential fractional DEs, for example; see [75 –77].

The differential transform method (DTM) was originally discussed by Zhou [57] in 1986, this technique adopts an analytic solution in polynomial form, which is different from the traditional higher-order Taylor formula technique. Lately, many researchers have proposed this method to solve many problems. Allahviranloo et al. [4] used the DTM for solving fuzzy differential equations. Rivaz et al. [46] presented the generalized DTM for solving fuzzy fractional differential equations under Caputo fractional derivative. Keshavarz et al. [32] studied the fuzzy solution to the mathematical model of a cancer tumor under Caputo-generalized Hukuhara partial differentiability by using fuzzy integral transforms. Osman et al. [42] investigated the comparison of fuzzy DTM, with other techniques to obtain the solutions of fuzzy (1 + n)-dimensional Burgers’ equation.

This paper is structured as follows. In Section 2, we recall some basic definitions, remarks, and theorems that will be used are given. In Section 3, we consider the fuzzy linear and nonlinear partial q-differential equations via two-dimensional fuzzy q-differential transform. In Section 4, the applications of one-and two-dimensional fuzzy fractional q-differential transform for solving fuzzy fractional q-differential equations are discussed. Finally, a conclusion is drawn in Section 5.

2 Basic concepts

We denote by $E^{1}$ , the set of fuzzy numbers, that is, normal, fuzzy convex, upper semi-continuous and compactly supported fuzzy sets which defined over the real line. For 0 < ϱ ≤ 1, set [u] _ϱ = {x ∈ Ru (x) ≥ ϱ} , and [u] ₀ = cl {x ∈ R|u (x) >0} . We explain $[u]_{λ} = [{\underline{u}}_{λ}, {\bar{u}}_{λ}]$ , consequently if $u \in E^{1},$ the λ-level set [u] _λ is a closed interval for all λ ∈ [0, 1] (see in [5, 39]). Let $u, v \in E^{1}$ and k ∈ R, the addition and scalar multiplication are given by

[u + v] _λ = [u] _λ + [v] _λ,

[ku] _λ = k [u] _λ .

The triangular fuzzy number defined as a fuzzy set in

E^{1},

that determined by u = (a, b, c) ∈ R and a ≤ b ≤ c such that

{\underline{u}}_{λ} = a + (b - a) λ

and

{\bar{u}}_{λ} = c - (c - b) λ

are the endpoints of λ-level sets for all λ ∈ [0, 1] . A support of fuzzy number u is given as

sup p (u) = cl {x \in R | u (x) > 0},

where cl is closure of set {x ∈ R|u (x) >0} .

The Hausdorff distance between fuzzy numbers is defined by $D : E^{1} \times E^{1} ⟶ R^{+} \cup {0}$ as in [34] $\begin{matrix} D (u, v) & = sup_{λ \in [0, 1]} {d ([u]_{λ}, [v]_{λ})} \\ = sup_{λ \in [0, 1]} max {| {\underline{u}}_{λ} - {\underline{v}}_{λ} |, | {\bar{u}}_{λ} - {\bar{v}}_{λ} |}, \end{matrix}$ where d is the Hausdorff metric.

The metric space $(E^{1}, D)$ is complete, separable and locally compact and the following properties from [34] for metric D are valid

$D (u \oplus w, v \oplus w) = D (u, v), \forall u, v, w \in E^{1},$

$D (u \oplus v, w \oplus e) \leq D (u, w) + D (v, e), \forall u, v, w, e \in E^{1}$ ,

$D (k ⊙ u, k ⊙ v) = | k | D (u, v), \forall u, v \in E^{1}, k \in R,$

$D (k_{1} ⊙ u, k_{2} ⊙ u) = | k_{1} - k_{2} | D (u, 0), \forall u \in E^{1}, k_{1}, k_{2} \in R,$ with k₁ · k₂ ≥ 0.

$D (u ⊖ v, w ⊖ e) \leq D (u, w) + D (v, e), as long as u ⊖ v, and w ⊖ e \forall u, v, w, e \in E^{1}$ ,

where ⊖ the H-difference, it means that w ⊖ v = u if and only if u ⊕ v = w .

Definition 2.1 [17, 23]. The gH-difference between two fuzzy numbers $u, v \in E^{1}$ is defined as follows:

$u \underset{gH}{⊖} v = e \Leftrightarrow {\begin{matrix} (i) u = v \oplus e, \\ or (ii) v = u \oplus (- e) . \end{matrix}$ (2.1)

In terms of λ-levels, we get $[u \underset{gH}{⊖} v] = [min {{\underline{u}}_{λ} - {\underline{v}}_{λ}, {\bar{u}}_{λ} - {\bar{v}}_{λ}}, max {{\underline{u}}_{λ} - {\underline{v}}_{λ}, {\bar{u}}_{λ} - {\bar{v}}_{λ}}]$ and if the H-difference exists, then $u ⊖ v = u \underset{gH}{⊖} v$ ; the conditions for existence of $e = u \underset{gH}{⊖} v \in E^{1}$ are $Case (i) {\begin{matrix} {\underline{e}}_{λ} = {\underline{u}}_{λ} - {\underline{v}}_{λ} and {\bar{e}}_{λ} = {\bar{u}}_{λ} - {\bar{v}}_{λ}, \forall λ \in [0, 1], \\ with {\underline{e}}_{λ} increasing, {\bar{e}}_{λ} decreasing, {\underline{e}}_{λ} \leq {\bar{e}}_{λ} . \end{matrix}$ (2.2) $Case (ii) {\begin{matrix} {\underline{e}}_{λ} = {\bar{u}}_{λ} - {\bar{v}}_{λ} and {\bar{e}}_{λ} = {\underline{u}}_{λ} - {\underline{v}}_{λ}, \forall λ \in [0, 1], \\ with {\underline{e}}_{λ} increasing, {\bar{e}}_{λ} decreasing, {\underline{e}}_{λ} \leq {\bar{e}}_{λ} . \end{matrix}$ (2.3)

It is easy to show that (i) and (ii) are both valid if and only if e is a crisp number.

Definition 2.2 [22]. Let $f (x, t) : D \to E^{1}$ and (x₀, t) ∈ D. We say that f is strongly generalized differentiable on (x₀, t) if there exists an element $f^{'} (x_{0}, t) \in E^{1}$ such that

for all l^* > 0 sufficiently small, $\exists f (x_{0} + l^{*}, t) \underset{H}{⊖} f (x_{0}, t), f (x_{0}, t) \underset{H}{⊖} f (x_{0} - l^{*}, t)$ and the limits (in the metric D) $\begin{matrix} lim_{l^{*} \to 0 +} \frac{f (x_{0} + l^{*}, t) ⊖_{H} f (x_{0}, t)}{l^{*}} \\ = lim_{l^{*} \to 0 +} \frac{f (x_{0}, t) ⊖_{H} f (x_{0} - l^{*}, t)}{l^{*}} \\ = f^{'} (x_{0}, t), \end{matrix}$ or

for all l^* > 0 sufficiently small, $\exists_{H} f (x_{0}, t) \underset{H}{⊖} f (x_{0} + l^{*}, t), f (x_{0} - l^{*}, t) \underset{H}{⊖} f (x_{0}, t)$ and the limits $\begin{matrix} lim_{l^{*} \to 0 +} \frac{f (x_{0}, t) ⊖_{H} f (x_{0} + l^{*}, t)}{- l^{*}} \\ = lim_{l^{*} \to 0 +} \frac{f (x_{0} - l^{*}, t) ⊖_{H} f (x_{0}, t)}{- l^{*}} \\ = f^{'} (x_{0}, t), \end{matrix}$ or

for all l^* > 0 sufficiently small, $\exists f (x_{0} + l^{*}, t) \underset{H}{⊖} f (x_{0}, t), f (x_{0} - l^{*}, t) \underset{H}{⊖} f (x_{0}, t)$ and the limits $\begin{matrix} lim_{l^{*} \to 0 +} \frac{f (x_{0} + l^{*}, t) ⊖_{H} f (x_{0}, t)}{l^{*}} \\ = lim_{l^{*} \to 0 +} \frac{f (x_{0} - l^{*}, t) ⊖_{H} f (x_{0}, t)}{- l^{*}} \\ = f^{'} (x_{0}, t), \end{matrix}$ or

for all l^* > 0 sufficiently small, $\exists f (x_{0}, t) \underset{H}{⊖} f (x_{0} + l^{*}, t), f (x_{0}, t) \underset{H}{⊖} f (x_{0} - l^{*}, t)$ and the limits $\begin{matrix} lim_{l^{*} \to 0 +} \frac{f (x_{0}, t) ⊖_{H} f (x_{0} + l^{*}, t)}{- l^{*}} \\ = lim_{l^{*} \to 0 +} \frac{f (x_{0}, t) ⊖_{H} f (x_{0} - l^{*}, t)}{l^{*}} \\ = f^{'} (x_{0}, t) . \end{matrix}$

Definition 2.3 [4]. Let $f (x, t) : D \to E^{1}$ be a function and set $[f (x, t)]_{λ} = ((\underline{f} (x, t; λ), \bar{f} (x, t; λ))$ for each λ ∈ [0, 1]. Then (1) If f is gH-differentiable in the first form (i), then $(\underline{f} (x, t; λ)$ and $\bar{f} (x, t; λ)$ are differentiable functions and $[f^{'} (x, t)]_{λ} = [{\underline{f}}^{'} (x, t; λ), {\bar{f}}^{'} (x, t; λ)] .$ (2.4) (2) If f is gH-differentiable in the second form (ii), then $(\bar{f} (x, t; λ)$ and $\underline{f} (x, t; λ)$ are differentiable functions and $[f^{'} (x, t)]_{λ} = [{\bar{f}}^{'} (x, t; λ), {\underline{f}}^{'} (x, t; λ)] .$ (2.5)

Remark 2.4 For $f (x, t) : D \to E^{1},$ the following results hold. $[f^{''} (x, t)]_{λ} = [{\underline{f}}^{''} (x, t; λ), {\bar{f}}^{''} (x, t; λ)],$ (2.6) in cases for that (i, i) , (ii, ii)-f^′′ (x, t ; λ) exist; $[f^{''} (x, t)]_{λ} = [{\bar{f}}^{''} (x, t; λ), {\underline{f}}^{''} (x, t; λ)],$ (2.7) in cases for that (i, ii) , (ii, i)-f^′′ (x, t ; λ) exist.

Definition 2.5 [56]. A fuzzy-valued function $f : [a, b] \to E^{1}$ is said to be continuous at t₀ ∈ [a, b] if for each ɛ > 0 there is δ > 0 such that D (f (t) , f (t₀)) < ɛ whenever |t - t₀| < δ . If f is continuous for each t ∈ [a, b] then we say that f is fuzzy continuous on [a, b] .

Definition 2.6 [13]. A fuzzy-valued function $f : [a, b] \to E^{1}$ is said to bounded iff there is M ≥ 0 such that D (f (t) , 0) = ∥ f (u) ∥ ≤ M for all t ∈ [a, b] . Equivalently we obtain χ_-M ≤ f (x) ≤ χ_M, ∀ x ∈ [a, b] .

Definition 2.7 [5]. A fuzzy-valued function f of two variable is a rule that assigns to each ordered pair of real numbers, (x, t), in a set D, denoted by f (x, t), the unique fuzzy number. The domain of f denotes D, and its range is the set of values that f can take, so {f (x, t) | (x, t) ∈ D}.

Let us consider $f : D \to E^{1}$ is the parametric representation of the fuzzy-valued function. Then, for all (x, t) ∈ D and λ ∈ [0, 1] the following inequality holds: $f (x, t; λ) = [\underline{f} (x, t; λ), \bar{f} (x, t; λ)] .$

2.1 Quantum calculus

The quantum calculus emerges in the latter portion of the twentieth century as an attempt to bridge the gap between mathematics and physics [59 –63]. Numerous fields of mathematics, including number theory, combinatorics, orthogonal polynomials, fundamental hypergeometric functions, and other branches of science, including quantum theory, mechanics, and the theory of relativity, all make use of this concept to some extent.

In expression $\frac{f (x) - f (x_{0})}{x - x_{0}}$ as x approaches x₀, the limit if it exists, gives the familiar definition of the derivative $D f (x)$ (normal derivative) of a function f (x) at x = x₀. However, if we take x = qx₀ or x = x₀ + h, where q is a fixed number different from 1, and h is a fixed number different from 0 and do not take the limit, with these changes we obtain the so-called quantum calculus. To make this statement precisely, we need the following definitions: Let $μ \in ℝ$ [40, 41]. The time-scale $T_{μ}$ , for 0 < q < 1 is defined as

$T_{μ} = {x : x = μ q^{n}, n \in ℤ^{+}} \cup {0}$ (2.8) where $ℤ^{+}$ is the set of nonnegative integer numbers. The q-analogues of a positive integer number a is defined as follows

$[a]_{q} = \frac{1 - q^{a}}{1 - q} .$ (2.9) Consider an arbitrary function $f : T_{μ} \to R$ . The q-differential and q-derivative of f are respectively,

$\begin{matrix} d_{q} f (x) & = f (x) - f (qx), \\ D_{q} f (x) & = \frac{d_{q} f (x)}{d_{q} x} = \frac{f (x) - f (qx)}{(1 - q) x}, x \in T_{μ} - {0} . \end{matrix}$ (2.10) It is obvious that if x = 0 then $D_{q} f (0) = D f (0)$ , where $D f$ is normal derivative. Further we define

$D_{q}^{0} f = f, D_{q}^{n} f = D_{q} (D_{q}^{n - 1} f), n = 1, 2, 3, \dots$ (2.11) If the function f is differentiable at x, we have $lim_{q \to 1} D_{q} f (x) = D f (x)$ . The q-derivative operator is a special case of Hahn’s q-operator, which is defined in [23]. The q-antiderivative _qI_a for f is defined in the following form

$\begin{matrix} \int_{a}^{b} f (s) d_{q} s = b (1 - q) \sum_{i = 0}^{\infty} q^{i} f (b q^{i}) - a (1 - q) \\ \sum_{i = 0}^{\infty} q^{i} f (a q^{i}) \end{matrix}$ (2.12) that is called the Jackson integral of f (x). As for q-derivative, we can define an operator q antiderivative $_{q} I_{a}^{n}$ as follows

$\begin{matrix} _{q} I_{a}^{0} f (x) = f (x),_{q} I_{a}^{n} f (x) =_{q} I_{a} (_{q} I_{a}^{n - 1} f (x)), \\ n = 1, 2, 3, \dots \end{matrix}$ (2.13) Also, for the operators _qI_a and $D_{q}$ we have

$D_{q \cdot q} I_{a} f (x) = f (x),_{q} I_{a} \cdot D_{q} f (x) = f (x) - f (a) .$ (2.14) The q-fractional function is defined by

$(x - s)_{q}^{n} = {\begin{matrix} \prod_{i = 0}^{n - 1} (x - q^{i} s), & n \in ℕ, \\ x^{n} \prod_{i = 0}^{\infty} \frac{1 - \frac{s}{t} q^{i}}{1 - \frac{s}{x} q^{i + n}}, & (x \neq 0) O . W . \end{matrix}$ (2.15) The q-gamma function denoted by Γ_q (.) is defined as follows $\begin{matrix} Γ_{q} (α) = \frac{(1 - q)_{q}^{α - 1}}{(1 - q)^{α - 1}}, α \in ℝ / {0} \cup ℤ_{-}, 0 < q < 1, \\ Γ_{q} (α + 1) = \frac{1 - q^{α}}{1 - q} Γ_{q} (α) = [α]_{q} Γ_{q} (α), Γ_{q} (1) = 1, α > 0 . \end{matrix}$

2.2 Fuzzy quantum differentiability

Let us consider f is the fuzzy-valued function $(f : T_{q} \to$ $E^{1})$ ; firstly, we introduce fuzzy q-derivative by gH-difference, refs. [40, 41].

Definition 2.8 We consider an arbitrary fuzzy-valued function $f : T_{q} \to$ $E^{1}$ q-differential by gH-difference is

$^{gH} d_{q} f (x) = f (x) \underset{gH}{⊖} f (qx),$ if $f (x) \underset{gH}{⊖} f (qx)$ exists, the fuzzy generalized Hukuhara q-derivative (for short $_{q}^{F} [gH]$ -derivative) of f is defined by $_{q}^{F} D f (x) = \frac{^{g H} d_{q} f (x)}{d_{q} t} = \frac{f (x) ⊖_{g H} f (q x)}{(1 - q) x}, x \in T_{q} - {0}$

On the other hand, for the fuzzy-valued functions f (x) and f (qx), we have

$\begin{matrix} [f (x) {\underset{gH}{⊖} f (qx)]}_{λ} \\ = [min {\underline{f} (x; λ) - \underline{f} (qx; λ), \bar{f} (x; λ) - \bar{f} (qx; λ)}, \\ max {\underline{f} (x; λ) - \underline{f} (qx; λ), \bar{f} (x; λ) - \bar{f} (qx; λ)}] \end{matrix}$ If the H-difference exists, then [f (x) ⊖ f (qx)] _λ = [f (qx) ${\underset{gH}{⊖} f (x)]}_{λ}$ . The conditions for the existence of ^gHd_qf (x) = $f (x) \underset{gH}{⊖} f (qx) \in ℝ_{F}$ are

$case (i) {\begin{matrix} d_{q} \underline{f} (x; λ) = \underline{f} (x; λ) - \underline{f} (qx; λ), is increasing \\ d_{q} \bar{f} (x; λ) = \bar{f} (x; λ) - \bar{f} (qx; λ), is decreasing \\ d_{q} \underline{f} (x; λ) \leq d_{q} \bar{f} (x; λ) \end{matrix}$

$case (ii) {\begin{matrix} d_{q} \underline{f} (x; λ) = \bar{f} (x; λ) - \bar{f} (qx; λ), is increasing \\ d_{q} \bar{f} (x; λ) = \underline{f} (x; λ) - \underline{f} (qx; λ), is decreasing \\ d_{q} \underline{f} (x; λ) \leq d_{q} \bar{f} (x; λ) \end{matrix}$

Definition 2.9 Let $f : T_{q} \to E^{1}$ , and $f (x) \underset{gH}{⊖} f (qx)$ exists. We say that f is $_{q}^{F} [(i) - gH]$ -differentiable if ${_{q}^{F} Df (x)]}_{λ} = [D_{q} \underline{f} (x; λ), D_{q} \bar{f} (x; λ)], 0 \leq λ \leq 1$ and f is $_{q}^{F} [(ii) - gH]$ -differentiable if ${_{q}^{F} Df (x)]}_{λ} = [D_{q} \bar{f} (x; λ), D_{q} \underline{f} (x; λ)], 0 \leq λ \leq 1$ where $\begin{matrix} D_{q} \underline{f} (x; λ) = \frac{\underline{f} (x; λ) - \underline{f} (qx; λ)}{(1 - q) x}, \\ D_{q} \bar{f} (x; λ) = \frac{\bar{f} (x; λ) - \bar{f} (qx; λ)}{(1 - q) t} . \end{matrix}$

Definition 2.10 Let $f : T_{q} \to E^{1}$ be a fuzzy-valued function on $T_{q}$ . A point $ξ_{0} \in T_{q} - {0}$ is said to be a switching point for the $_{q}^{F} [gH]$ -differentiably of f, if in any neighborhood V of ξ₀ there exist points t₁ < ξ₀ < t₂ such that

(type I) f is $_{q}^{F} [(i) - gH]$ -differentiable at x₁ while f is not $_{q}^{F} [(ii) - gH]$ -differentiable at x₁, and f is $_{q}^{F} [(ii) -$ gH]-differentiable at x₂ while f is not $_{q}^{F} [(i) - gH]$ differentiable at x₂, or

(type II) f is $_{q}^{F} [(ii) - gH]$ -differentiable at x₁ while f is not $_{q}^{F} [(i) - gH]$ -differentiable at x₁, and f is $_{q}^{F} [(i) -$ gH]-differentiable at x₂ while f is not $_{q}^{F} [(ii) -$ gH]-differentiable at x₂.

Definition 2.11 Let us consider $f : T_{q} \to E^{1}$ . We say that f is $_{q}^{F} [gH]$ differentiable of the n^xh-order at t₀ whenever the function f is $_{q}^{F} [gH]$ -differentiable of the order m (m = 1, 2, …, n - 1) at x₀, moreover the $_{q}^{F} [gH]$ -differentiable type has no change (there is not any switching point on $T_{q}$ ), then there exist $_{q}^{F} D^{n} f (t_{0}) \in E^{1}$ such that $_{q}^{F} D^{n} f (x_{0}) = \frac{_{q}^{F} D^{n - 1} f (x_{0}) ⊖_{g H}_{q}^{F} D^{n - 1} f (q x_{0})}{(1 - q) x_{0}}$

Theorem 2.12 Let $[f (x)]_{λ} = [\underline{f} (x; λ), \bar{f} (x; λ)]$ , the function f is $_{q}^{F} [gH]$ -differentiable if and only if $\underline{f} (x; λ)$ and $\bar{f} (x; λ)$ are q-differentiable with respect to for all λ ∈ [0, 1], and $\begin{matrix} {[_{q}^{F} Df (x)]}_{λ} = [min {D_{q} \underline{f} (x; λ), D_{q} \bar{f} (x; λ)} \\ max {D_{q} \underline{f} (x; λ), D_{q} \bar{f} (x; λ)}] . \end{matrix}$

Definition 2.13 Let f is fuzzy-valued function, by Definition 2.8 of $_{q}^{F} [gH]$ -derivative and the operator ${\hat{M}}_{q}$ defined by ${\hat{M}}_{q} (F (x)) = F (qx)$ , we have $\frac{1}{(1 - q) x} (1 \underset{gH}{⊖} {\hat{M}}_{q}) F (x) = \frac{F (x) ⊖_{gH} F (qx)}{(1 - q) x} = f (x)$ then, the fuzzy q-antiderivative can be written formally as $\begin{matrix} F (x) = \frac{1}{1 ⊖_{gH} {\hat{M}}_{q}} ((1 - q) xf (x)) = (1 - q) \\ ⊙ \sum_{i = 0}^{\infty} {\hat{M}}_{q}^{i} (xf (x)) \end{matrix}$ by assuming that a = 0 and using the expansion in geometric series, fuzzy Jackson integral is defined $\begin{matrix} [F (x)]_{λ} & = {[\int_{0}^{x} f (s) d_{q} s]}_{λ} \\ = [\int_{0}^{x} \underline{f} (s; λ) d_{q} s, \int_{0}^{x} \bar{f} (s; λ) d_{q} s] \\ = [(1 - q) x \sum_{i = 0}^{\infty} q^{i} \underline{f} (q^{i} x; λ), \\ (1 - q) t \sum_{i = 0}^{\infty} q^{i} \bar{f} (q^{i} x; λ)] \end{matrix}$

$\begin{matrix} = (1 - q) x \sum_{i = 0}^{\infty} q^{i} {[f (q^{i} x)]}_{λ} \\ \Rightarrow F (x) = (1 - q) x ⊙ \sum_{i = 0}^{\infty} q^{i} ⊙ f (q^{i} x) \end{matrix}$

Definition 2.14 Let us consider f is the fuzzy-valued function and $F (x) \underset{gH}{⊖} F (qx)$ exists. Then the fuzzy Jackson integral is given by

$F (x) = (1 - q) x ⊙ \sum_{j = 0}^{\infty} q^{j} ⊙ f (q^{j} x)$

The notation $C_{f} (T_{μ}, E^{1})$ denotes the set of fuzzy-valued continuous functions defined on $T_{μ}$ and $C_{f}^{n} (T_{μ}, E^{1}), n \in N$ for the space of fuzzy-valued functions f on $T_{μ}$ such that itself and its first n, q-derivatives are all in $C_{f} (T_{μ}, E^{1})$ .

Theorem 2.15 Let $f : T_{μ} \to E^{1}$ , be $_{q}^{F} [gH]$ -differentiable function such that the type of $_{q}^{F} [gH]$ -differentiability of f in $T_{μ}$ does not change. Then for $x \in T_{μ}$ ,

If f (s) is $_{q}^{F} [(i) - gH]$ -differentiable then $_{q}^{F} D f_{i - gH} (s)$ is q-integrable over $T_{μ}$ and $f (x) = f (a) \oplus \int_{a}^{x}_{q}^{F} D f_{i - gH} (s) d_{q} s .$

If f (s) is $_{q}^{F} [(ii) - gH]$ -differentiable then $_{q}^{F} D f_{ii - gH} (s)$ is q-integrable over $T_{μ}$ and $f (x) = f (a) \oplus (- 1) \int_{a}^{x}_{q}^{F} D f_{ii - gH} (s) d_{q} s .$

Theorem 2.16 Let $_{q}^{F} D^{(j)} f : T_{μ} \to E^{1}$ and $_{q}^{F} D^{(j)} f \in C_{f} (T_{μ}, E^{1}), j = 1, \dots$ , n. For all $x \in T_{μ}$

Assume that $_{q}^{F} D^{(j)} f, j = 1, \dots, n$ are $_{q}^{F} [(i) - gH]$ -differentiable and there is no change in the type of $_{q}^{F} [gH]$ -differentiability on $T_{μ}$ . Then $\begin{matrix} _{q}^{F} D^{(j - 1)} f_{i - gH} (x) =_{q}^{F} D^{(j - 1)} f_{i - gH} (a) \\ \oplus \int_{a}^{x}_{q}^{F} D^{(j)} f_{i - gH} (s) d_{q} s . \end{matrix}$

If $_{q}^{F} D^{(j)} f, j = 1, \dots, n$ are $_{q}^{F} [(ii) - gH]$ -differentiable and the type of $_{q}^{F} [gH]$ -differentiability does not change on $T_{μ}$ , then $\begin{matrix} _{q}^{F} D^{(j - 1)} f_{ii - gH} (x) =_{q}^{F} D^{(j - 1)} f_{ii - gH} (a) \\ \oplus \int_{a}^{x}_{q}^{F} D^{(j)} f_{ii - gH} (s) d_{q} s . \end{matrix}$

Assume that $_{q}^{F} D^{(j)} f$ for j = 1, …, n exist and in each order of differentiation, the type of $_{q}^{F} [gH]$ -differentiability changes on $T_{μ}$ (i.e. $_{q}^{F} [gH]$ -differentiability changes from $_{q}^{F} [(i) - gH]$ to $_{q}^{F} [(ii) - gH]$ and vice versa), then ${\begin{matrix} _{q}^{F} D^{(j - 1)} f_{i - gH} (x) =_{q}^{F} D^{(j - 1)} f_{i - gH} (a) ⊖ (- 1) \int_{a}^{x}_{q}^{F} D^{(j)} f_{ii - gH} (s) d_{q} s, & j is even number, \\ _{q}^{F} D_{ii - gH}^{(j - 1)} f (x) =_{q}^{F} D^{(j - 1)} f_{ii - gH} (a) ⊖ (- 1) \int_{a}^{x}_{q}^{F} D^{(j)} f_{i - gH} (s) d_{q} s, & j is odd number \end{matrix}$

Theorem 2.17 (Fuzzy q-Taylor’s Theorem) Suppose that $_{q}^{F} D^{(n)} f \in C_{f} (T_{μ}, E^{1})$ and $x \in T_{μ}$ . For finding the fuzzy q-Taylor’s expansion of f around $a \in T_{μ}$ , we obtain

If $_{q}^{F} D^{(j)} f, j = 0, 1, \dots, n - 1$ are $_{q}^{F} [(i) - gH]$ -differentiable, provided that type of $_{q}^{F} [gH]$ -differentiability has no change on $T_{μ}$ . Then there exists $η \in T_{μ}$ such that $\begin{matrix} f (x) = f (a) \oplus \sum_{j = 1}^{n - 1} [\frac{(x - a)_{q}^{j}}{Γ_{q} (j + 1)} ⊙_{q}^{F} D^{(j)} f_{i - gH} (a)] \\ \oplus \frac{(t - a)_{q}^{n}}{Γ_{q} (n + 1)} ⊙_{q}^{F} D^{(n)} f_{i - gH} (η) . \end{matrix}$

If $_{q}^{F} D^{(j)} f, j = 0, 1, \dots, n - 1$ are $_{q}^{F} [(ii) - gH]$ -differentiable, provided that type of $_{q}^{F} [gH]$ -differentiability has no change on $T_{μ}$ . Then there exists $η \in T_{μ}$ such that $\begin{matrix} f (x) = f (a) ⊙ (- 1) \sum_{j = 1}^{n - 1} \\ [\frac{(t - a)_{q}^{j}}{Γ_{q} (j + 1)} ⊙_{q}^{F} D^{(j)} f_{ii - gH} (a)] \\ ⊙ (- 1) \frac{(x - a)_{q}^{n}}{Γ_{q} (n + 1)} ⊙_{q}^{F} D^{(n)} f_{ii - gH} (η) . \end{matrix}$

Suppose that $_{q}^{F} D^{(j)} f, j = 1, \dots$ , n exist and in each order the type of $_{q}^{F} [gH]$ -differentiability changes on $T_{μ}$ , then there exists $η \in T_{μ}$ such that $\begin{matrix} f (x) & = f (a) ⊙ (- 1) \sum_{j = 1 jis odd}^{n - 1} \\ [\frac{(t - a)_{q}^{j}}{Γ_{q} (j + 1)} ⊙_{q}^{F} D^{(j)} f_{ii - gH} (a)] \\ \oplus \sum_{j = 1 j is even}^{n - 1} [\frac{(x - a)_{q}^{j}}{Γ_{q} (j + 1)} \end{matrix}$ $\begin{matrix} ⊙_{q}^{F} D^{(j)} f_{i - gH} (a)] ⊙ (- 1) \frac{(x - a)_{q}^{n}}{Γ_{q} (n + 1)} ⊙_{q}^{F} D^{(n)} f_{ii - gH} (η) \end{matrix}$

If f has a switching point at $ξ \in T_{μ}$ of type II (i.e. $_{q}^{F} [gH]$ -differentiability changes from $_{q}^{F} [(ii) - gH]$ to $_{q}^{F} [(i) - gH])$ and suppose that the type of differentiability for $_{q}^{F} D^{(j)} f, j =$ 1, …, n - 1 are $_{q}^{F} [(i) - gH]$ -differentiable. Then there exists $η \in T_{μ}$ such that $\begin{matrix} f (x) = \\ {\begin{matrix} f (a) ⊙ (- 1) (x - a) ⊙_{q}^{F} D f_{ii - gH} (a) \\ \oplus \sum_{j = 2}^{n - 1} [\frac{(x - a)_{q}^{j}}{Γ_{q} (j + 1)} ⊙_{q}^{F} D^{(j)} f_{i - gH} (a)], \\ \oplus \frac{(x - a)_{q}^{n}}{Γ_{q} (n + 1)} ⊙_{q}^{F} D^{(n)} f_{i - gH} (η), & 0 < x \leq ξ, \\ f (a) \oplus \sum_{j = 1}^{n - 1} [\frac{(x - a)_{q}^{j}}{Γ_{q} (j + 1)} ⊙_{q}^{F} D^{(j)} f_{i - gH} (a)] \\ \oplus \frac{(x - a)_{q}^{n}}{Γ_{q} (n + 1)} ⊙_{q}^{F} D^{(n)} f_{i - gH} (η), & ξ \leq x . \end{matrix} \end{matrix}$

If f has a switching point at $ξ \in T_{μ}$ of type I (i.e. $_{q}^{F} [gH]$ -differentiability changes from $_{q}^{F} [(i) - gH]$ to $_{q}^{F} [(ii) - gH])$ and suppose that the type of differentiability for $_{q}^{F} D^{(j)} f, j =$ 1, …, n - 1 are $_{q}^{F} [($ ii) - gH]-differentiable. Then there exists $η \in T_{μ}$ such that $\begin{matrix} f (x) = \\ {\begin{matrix} f (a) \oplus (x - a) ⊙_{q}^{F} D_{i - gH} (a) \\ ⊖ (- 1) \sum_{j = 2}^{n - 1} [\frac{(x - a)_{q}^{j}}{Γ_{q} (j + 1)} ⊙_{q}^{F} D^{(j)} f_{ii - gH} (a)] \\ ⊖ (- 1) \frac{(x - a)_{q}^{n}}{Γ_{q} (n + 1)} ⊙_{q}^{F} D^{(n)} f_{ii - gH} (η) 0 < x \leq ξ, \\ f (a) ⊙ (- 1) \sum_{j = 1}^{n - 1} [\frac{(x - a)_{q}^{j}}{Γ_{q} (j + 1)} ⊙_{q}^{F} D^{(k)} f_{ii - gH} (a)] \\ ⊖ (- 1) \frac{(x - a)_{q}^{n}}{Γ_{q} (n + 1)} ⊙_{q}^{F} D^{(n)} f_{ii - gH} (η), ξ \leq x . \end{matrix} \end{matrix}$

3 Fuzzy linear and nonlinear partial q-differential equations

In this section, we consider the following fuzzy linear and nonlinear partial q-differential equations using two-dimensional fuzzy q-differential transform.

Let q be positive real number, q ≠ 1. A q-complex number [n] _q is $\begin{matrix} [n]_{q} = \frac{1 - q^{n}}{1 - q} = 1 + q + . . . + q^{n - 1}, [n]_{q}! \\ = [1]_{q} [2]_{q} [3]_{q} \dots [n]_{q}, and {[\begin{matrix} n \\ p \end{matrix}]}_{q} \\ = \frac{[n]_{q}!}{[p]_{q}! [(n - p)]_{q}!}, \end{matrix}$ (3.1) where $n, p \in ℕ$ and n ≥ p. Using induction on n, it following the q-Lebniz rule [26] $\begin{matrix} D_{q, x}^{n} {f (x, t) \cdot g (x, t)} = \sum_{l = 0}^{n} {[\begin{matrix} n \\ l \end{matrix}]}_{q} \\ D_{q}^{n - l} f (x q^{l}, t) D_{q}^{l} g (x, t), \end{matrix}$ (3.2) where $D_{q, x}^{n} = \frac{\partial_{q}^{n}}{\partial_{q} x^{n}}$ .

3.1 On q-partial derivative

Suppose that $f (\vec{x})$ where $\vec{x} = (x_{1}, x_{2}, x_{3}, \dots, x_{n})$ a multivariate continuous real-valued function. The q-partial derivative of a function $f (\vec{x})$ to a variable x_ς (see [45]), we have $D_{q, x_{ς}} f (\vec{x}) = \frac{f (\vec{x}) - (ɛ_{q, ς} f) (\vec{x})}{(1 - q) x_{ς}}, (x_{ς} \neq 0, q \in (0, 1)),$ (3.3) where

$\begin{matrix} (ɛ_{q, ς} f) (\vec{x}) & = f (x_{1}, x_{2}, x_{3}, . . ., {qx}_{ς}, . . ., x_{n}), \\ (ɛ_{q} f) (\vec{x}) & = f ({qx}_{1}, {qx}_{2}, {qx}_{3}, . . ., {qx}_{n}) = f (q \vec{x}) . \end{matrix}$

The high q-partial derivatives are as:

$\begin{matrix} D_{q}^{0} f (\vec{x}) = f (\vec{x}), D_{q, x_{ς}^{μ} x_{ι}^{n}}^{μ + n} f (\vec{x}) = D_{q, x_{ι}^{n} x_{ς}^{μ}}^{μ + n} f (\vec{x}) \\ (ς, ι = 1, 2, 3, . . ., n and μ, n = 0, 1, 2, 3, . . .) . \end{matrix}$

3.2 q-Taylor formula

The q-Taylor formula for a multivariable functions (refer to [47]).

Theorem 3.1 Let that there exist all q-differentials of $f (\vec{x})$ be in some neighborhood of $\vec{a}$ . Then $f (\vec{x}) = \sum_{\vec{l} = 0}^{\vec{\infty}} \frac{d_{\vec{q}}^{\vec{l}} f (\vec{x}, \vec{a})}{[\vec{l}]_{\vec{q}}!},$ (3.4) where

$\begin{matrix} d_{q}^{l} f (\vec{x}, \vec{a}) = ((x_{1} - a_{1}) D_{q, x_{1}} + (x_{2} - a_{2}) D_{q, x_{2}} \\ + (x_{3} - a_{3}) D_{q, x_{3}} + \dots + (x_{n} - a_{n}) D_{q, x_{n}})^{(l)} f (\vec{a}), \\ (z - c)^{(0)} = 1, (z - c)^{(l)} = \prod_{ς = 0}^{l - 1} (z - {cq}^{ς}) (l \in ℕ) . \end{matrix}$

3.3 Fuzzy two-dimensional q-differential transform

We present the two-dimensional fuzzy q-differential transform for solving fuzzy partial q-differential equations, the theory of two-dimensional q-DTM with uncertainty represented by using fuzzy concepts is explained as follows.

Definition 3.2 Let us consider x (ϑ, t) be a differentiable of order l + l^* over time domain T, then $\begin{matrix} {\underline{X}}_{ς ι} (l, l^{*}; λ) & = {[\frac{\partial_{q}^{l + l^{*}} \underline{x} (ϑ, t; λ)}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}]}_{ϑ = ϑ_{τ}, t = t_{ι}}, \forall l, l^{*} \in K = {0, 1, 2, 3, . . .}, \\ {\bar{X}}_{ς ι} (l, l^{*}; λ) & = {[\frac{\partial_{q}^{l + l^{*}} \bar{x} (ϑ, t; λ)}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}]}_{ϑ = ϑ_{τ}, t = t_{ι}}, \forall l, l^{*} \in K = {0, 1, 2, 3, . . .}, \end{matrix}}$ (3.5) when x (ϑ, t) is (i)-differentiable and $\begin{matrix} {\underline{X}}_{ς ι} (l, l^{*}; λ) & = \frac{\partial_{q}^{l + l^{*}}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}} \bar{x} (ϑ, t; λ), (l is odd, l^{*} is even) or inverse), \\ {\bar{X}}_{ς ι} (l, l^{*}; λ) & = \frac{\partial_{q}^{l + l^{*}}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}} \underline{x} (ϑ, t; λ), (l is odd, l^{*} is even) or inverse), \end{matrix}}$ (3.6) and $\begin{matrix} {\underline{X}}_{ς ι} (l, l^{*}; λ) = \frac{\partial_{q}^{l + l^{*}}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}} \underline{x} (ϑ, t; λ), \\ both of l, l^{*} are even or odd, \\ {\bar{X}}_{ς ι} (l, l^{*}; λ) = \frac{\partial_{q}^{l + l^{*}}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}} \bar{x} (ϑ, t; λ), \\ both of l, l^{*} are even or odd, \end{matrix}}$ (3.7) when x (ϑ, t) is (ii)-differentiable.

Notice that ${\underline{X}}_{ς ι} (l, l^{*}; λ)$ and ${\bar{X}}_{ς ι} (l, l^{*}; λ)$ are called the lower and upper spectrum of x (ϑ, t) at ϑ = ϑ_ς and t = t_ι, respectively.

Thus, if x (ϑ, t) be (i)-differentiable, then x (ϑ, t) can be represented as:

$\begin{matrix} \underline{x} (ϑ, t; λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{(ϑ - ϑ_{ς})^{l} (t - t_{ι})^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \underline{X} (l, l^{*}; λ), \\ l, l^{*} \in K, 0 \leq λ \leq 1, \end{matrix}$ (3.8)

$\begin{matrix} \bar{x} (ϑ, t; λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{(ϑ - ϑ_{ς})^{l} (t - t_{ι})^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \bar{X} (l, l^{*}; λ), \\ l, l^{*} \in K, 0 \leq λ \leq 1, \end{matrix}$ (3.9) and if x (ϑ, t) be (ii)-differentiable, then x (ϑ, t) can be represented as: $\begin{matrix} \underline{x} (ϑ, t, λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{l} {(t - t_{ι})}^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \bar{X} (l, l^{*}; λ) \\ (l is odd, l^{*} is even) or inverse + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \\ \frac{{(ϑ - ϑ_{ς})}^{l} {(t - t_{ι})}^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \underline{X} (l, l^{*}; λ), 0 \leq λ \leq 1, \\ both of l, l^{*} are even or odd, \end{matrix}$ (3.10) and $\begin{matrix} \bar{x} (ϑ, t, λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{l} {(t - t_{ι})}^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \underline{X} (l, l^{*}; λ) \\ (l is odd, l^{*} is even) or inverse + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \\ \frac{{(ϑ - ϑ_{ς})}^{l} {(t - t_{ι})}^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \bar{X} (l, l^{*}; λ), 0 \leq λ \leq 1 . \\ both of l, l^{*} are even or odd \end{matrix}$ (3.11) The above-mentioned equations are known as the inverse transformation of X (l, l^* ; λ). If X (l, l^* ; λ) is defined as follows: $\begin{matrix} \underline{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{l + l^{*}} \underline{x (ϑ, t; λ)}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}]}_{ϑ = ϑ_{0}, t = t_{0}}, \forall l, l^{*} \in K, \\ \bar{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{l + l^{*}} \bar{x (ϑ, t; λ)}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}]}_{ϑ = ϑ_{0}, t = t_{0}}, \forall l, l^{*} \in K, \end{matrix}}$ (3.12) when x (ϑ, t) (i)-differentiable and $\begin{matrix} \underline{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{l + l^{*}}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}} (\bar{x (ϑ, t; λ)})]}_{ϑ = ϑ_{0}, t = t_{0}}, (l is odd, l^{*} is even) or inverse) \\ \bar{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{l + l^{*}}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}} (\underline{x (ϑ, t; λ)})]}_{ϑ = ϑ_{0}, t = t_{0}}, (l is odd, l^{*} is even) or inverse) \end{matrix}}$ (3.13) and $\begin{matrix} \underline{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{l + l^{*}}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}} (\underline{x (ϑ, t; λ)})]}_{ϑ = ϑ_{0}, t = t_{0}}, both of l, l^{*} are even or odd \\ \bar{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{l + l^{*}}}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}} (\bar{x (ϑ, t; λ)})]}_{ϑ = ϑ_{0}, t = t_{0}}, both of l, l^{*} are even or odd \end{matrix}}$ (3.14) when x (ϑ, t) is (ii)-differentiable, then, the function x (ϑ, t) can be introduced as: $\underline{x} (ϑ, t; λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{(ϑ - ϑ_{ς})^{l} (t - t_{ι})^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\underline{X} (l, l^{*}; λ)}{P (l, l^{*})}, l, l^{*} \in K, 0 \leq λ \leq 1,$ (3.15) $\bar{x} (ϑ, t; λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{(ϑ - ϑ_{ς})^{l} (t - t_{ι})^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\bar{X} (l, l^{*}; λ)}{P (l, l^{*})}, l, l^{*} \in K, 0 \leq λ \leq 1,$ (3.16) when x (ϑ, t) are (i)-differentiable. Then, the function x (ϑ, t) can be (ii)-differentiable we obtain the following: $\begin{matrix} \underline{x} (ϑ, t, λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{l} {(t - t_{ι})}^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\bar{X} (l, l^{*}; λ)}{P (l, l^{*})} \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{l} {(t - t_{ι})}^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\underline{X} (l, l^{*}; λ)}{P (l, l^{*})}, 0 \leq λ \leq 1, \\ both of l, l^{*} are even or odd \end{matrix}$ (3.17) and $\begin{matrix} \bar{x} (ϑ, t, λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{l} {(t - t_{ι})}^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\underline{X} (l, l^{*}; λ)}{P (l, l^{*})} \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{l} {(t - t_{ι})}^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\bar{X} (l, l^{*}; λ)}{P (l, l^{*})}, 0 \leq λ \leq 1, \\ both of l, l^{*} are even or odd \end{matrix}$ (3.18) where P (l, l^*) >0, P (l, l^*) is called the weighting factor. In this work $P (l, l^{*}) = \frac{G^{l} H^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!}$ is applied where G and H are the time horizon on interest. Consequently, if x (ϑ, t) be (i)-differentiable, then $\underline{X} (l, l^{*}; λ) = \frac{G^{l} H^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\partial_{q}^{l + l^{*}} \underline{x} (ϑ, t; λ)}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}, l \in K, 0 \leq λ \leq 1,$ (3.19) $\bar{X} (l, l^{*}; λ) = \frac{G^{l} H^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\partial_{q}^{l + l^{*}} \bar{x} (ϑ, t; λ)}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}, l \in K, 0 \leq λ \leq 1,$ (3.20) and if x (ϑ, t) be (ii)-differentiable, then

$\begin{matrix} \underline{X} (l, l^{*}; λ) & = \frac{G^{l} H^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\partial_{q}^{l + l^{*}} \bar{x} (ϑ, t; λ)}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}, (l is odd, l^{*} is even) or inverse \\ \bar{X} (l, l^{*}; λ) & = \frac{G^{l} H^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\partial_{q}^{l + l^{*}} \underline{x} (ϑ, t; λ)}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}, (l is odd, l^{*} is even) or inverse \end{matrix}}$ (3.21) and

$\begin{matrix} \underline{X} (l, l^{*}; λ) & = \frac{G^{l} H^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\partial_{q}^{l + l^{*}} \underline{x} (ϑ, t; λ)}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}, both of l, l^{*} are even or odd \\ \bar{X} (l, l^{*}; λ) & = \frac{G^{l} H^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\partial_{q}^{l + l^{*}} \bar{x} (ϑ, t; λ)}{\partial_{q} ϑ^{l} \partial_{q} t^{l^{*}}}, both of l, l^{*} are even or odd . \end{matrix}}$ (3.22) Unitizing the fuzzy q-differential transform, a fuzzy partial q-differential equation in the domain of interest can be transformed to an algebraic equation in the domain K and x (ϑ, t) can be obtained as the finite-term q-Taylor series plus a reminder as: $\begin{matrix} \underline{x} (ϑ, t; λ) & = \sum_{l = 0}^{n} \sum_{l^{*} = 0}^{m} \frac{(ϑ - ϑ_{0})^{l} (t - t_{0})^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\underline{X} (l, l^{*}; λ)}{P (l, l^{*})} + R_{(n + 1, m + 1)} (ϑ, t) \\ = \sum_{l = 0}^{n} \sum_{l^{*} = 0}^{m} {(\frac{ϑ - ϑ_{0}}{G})}^{l} {(\frac{t - t_{0}}{H})}^{l^{*}} \underline{X} (l, l^{*}; α) + R_{(n + 1, m + 1)} (ϑ, t), l, l^{*} \in K, 0 \leq λ \leq 1, \end{matrix}$ (3.23) $\begin{matrix} \bar{x} (ϑ, t; λ) & = \sum_{l = 0}^{n} \sum_{l^{*} = 0}^{m} \frac{(ϑ - ϑ_{0})^{l} (t - t_{0})^{l^{*}}}{[l]_{q}! [l^{*}]_{q}!} \frac{\bar{X} (l, l^{*}; λ)}{P (l, l^{*})} + R_{(n + 1, m + 1)} (ϑ, t) \\ = \sum_{l = 0}^{n} \sum_{l^{*} = 0}^{m} {(\frac{ϑ - ϑ_{0}}{G})}^{l} {(\frac{t - t_{0}}{H})}^{l^{*}} \bar{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), l, l^{*} \in K, 0 \leq λ \leq 1, \end{matrix}$ (3.24) when x (ϑ, t) is (i)-differentiable and

$\begin{matrix} \underline{x} (ϑ, t, λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} {(\frac{ϑ - ϑ_{0}}{G})}^{l} {(\frac{t - t_{0}}{H})}^{l^{*}} \bar{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} {(\frac{ϑ - ϑ_{0}}{G})}^{l} {(\frac{t - t_{0}}{H})}^{l^{*}} \underline{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), \\ both of l, l^{*} are even or odd \end{matrix}}$ (3.25) and $\begin{matrix} \bar{x} (ϑ, t, λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} {(\frac{ϑ - ϑ_{0}}{G})}^{l} {(\frac{t - t_{0}}{H})}^{l^{*}} \underline{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} {(\frac{ϑ - ϑ_{0}}{G})}^{l} {(\frac{t - t_{0}}{H})}^{l^{*}} \bar{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), \\ both of l, l^{*} are even or odd \end{matrix}}$ (3.26) when x (ϑ, t) is (ii)-differentiable.

The objective of this section is to find the solution of fuzzy partial q-differential equations at the equally spaced grid points [t₀, t₁, . . . , t_n] and $[ϑ_{0}, ϑ_{1}, . . ., ϑ_{n}^{'}]$ where ϑ_ς = a + ςl^* and t_ι = a + ιl^*′ for each (ς = 0, 1, 2, . . . n) and (ι = 0, 1, 2, . . . n′), and $l^{*} = \frac{b - a}{n}$ and $l^{*'} = \frac{b^{'} - a^{'}}{n^{'}}$ . That is, the domain of interest are divided to [a, b] × [a′, b′] is divided to n × n′ sub-domain, and the fuzzy approximation functions in each sub-domain are x_ς (ϑ, t ; λ) for ς = 0, 1, 2, . . , n - 1, and x_ι (ϑ, t ; λ) for ι = 0, 1, 2, . . , n′ - 1, respectively.

According to the initial conditions, we obtain

$\underline{X} (0, 0; λ) = \underline{x} (0, 0; λ), \bar{X} (0, 0; λ) = \bar{x} (0, 0; λ), 0 \leq λ \leq 1 .$

In the first sub-domain, $\underline{x} (ϑ, t, λ)$ and $\bar{x} (ϑ, t, λ)$ can be described by $\underline{x} (0, 0; λ) = {\underline{x}}_{0, 0} (λ)$ and $\bar{x} (0, 0; λ) = {\bar{x}}_{0, 0} (λ)$ , respectively. They can be represented in terms of their (n, n′) - t (l^*, l^*′) order bivariate q-Taylor series with respect to ϑ₀, t₀. That is

$\begin{matrix} \underline{x} (ϑ_{0}, t_{0}; λ) & = {\underline{X}}_{0} (0, 0; λ) + {\underline{X}}_{0} (0, 1; λ) (t - t_{0}) + {\underline{X}}_{0} (0, 2; λ) (t - t_{0})^{2} + . . . + {\underline{X}}_{0} (0, n^{'}; λ) (t - t_{0})^{n^{'}}, \\ + {\underline{X}}_{0} (1, 0; λ) (ϑ - ϑ_{0}) + {\underline{X}}_{0} (1, 1; λ) (ϑ - ϑ_{0}) (t - t_{0}) + {\underline{X}}_{0} (1, 2; λ) (ϑ - ϑ_{0}) (t - t_{0})^{2} \\ + . . . + {\underline{X}}_{0} (1, n^{'} - 1; λ) (ϑ - ϑ_{0}) (t - t_{0})^{n^{'}} + {\underline{X}}_{0} (n, n^{'} - 1; λ) (ϑ - ϑ_{0})^{n} (t - t_{0})^{n^{'} - 1} \\ + {\underline{X}}_{0} (n, n^{'}; λ) (ϑ - ϑ_{0})^{n} (t - t_{0})^{n^{'}}, \end{matrix}$ and

$\begin{matrix} \bar{x} (ϑ_{0}, t_{0}; λ) & = {\bar{X}}_{0} (0, 0; λ) + {\bar{X}}_{0} (0, 1; λ) (t - t_{0}) + {\bar{X}}_{0} (0, 2; λ) (t - t_{0})^{2} + . . . + {\bar{X}}_{0} (0, n^{'}; λ) (t - t_{0})^{n^{'}}, \\ + {\bar{X}}_{0} (1, 0; λ) (ϑ - ϑ_{0}) + {\bar{X}}_{0} (1, 1; λ) (ϑ - ϑ_{0}) (t - t_{0}) + {\bar{X}}_{0} (1, 2; λ) (ϑ - ϑ_{0}) (t - t_{0})^{2} \\ + . . . + {\bar{X}}_{0} (1, n^{'} - 1; λ) (ϑ - ϑ_{0}) (t - t_{0})^{n^{'}} + {\bar{X}}_{0} (n, n^{'} - 1; λ) (ϑ - ϑ_{0})^{n} (t - t_{0})^{n^{'} - 1} \\ + {\bar{X}}_{0} (n, n^{'}; λ) (ϑ - ϑ_{0})^{n} (t - t_{0})^{n^{'}} . \end{matrix}$ Additionally, using q-Taylor series for x (ϑ_ς, t_ι ; λ), the solution on the grid points ϑ_ς+1 and t_ι+1 can be obtained as:

$\begin{matrix} \underline{x} (ϑ_{ς + 1}, t_{ι + 1}; λ) & = {\underline{X}}_{ς ι} (ϑ_{ς + 1}, t_{ι + 1}; λ) = {\underline{X}}_{ς ι} (0, 0; λ), \\ + {\underline{X}}_{ς ι} (0, 1; λ) (t_{ς + 1} - t_{ς}) + {\underline{X}}_{ς ι} (0, 2; λ) (t_{ς + 1} - t_{ς})^{2} + . . . + {\underline{X}}_{ς ι} (0, n^{'}; λ) (t_{ς + 1} - t_{ς})^{n^{'}}, \\ + {\underline{X}}_{ς ι} (1, 0; λ) (ϑ_{ς + 1} - ϑ_{ς}) + {\underline{X}}_{ς ι} (1, 1; λ) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς}), \\ + {\underline{X}}_{ς ι} (1, 2; λ) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς})^{2} + . . . + {\underline{X}}_{ς ι} (1, n^{'} - 1; λ) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς})^{n^{'}} \\ + {\underline{X}}_{ς ι} (n, n^{'} - 1; λ) (ϑ_{ς + 1} - ϑ_{ς})^{n} (t_{ς + 1} - t_{ς})^{n^{'} - 1} \\ + {\underline{X}}_{ς ι} (n, n^{'}; λ) (ϑ_{ς + 1} - ϑ_{ς})^{n} (t_{ς + 1} - t_{ς})^{n^{'}}, \\ = \sum_{μ = 0}^{n} \sum_{i = 0}^{n^{'}} {\underline{X}}_{ς ι} (μ . n . λ) ξ^{μ} ξ^{' i}, \end{matrix}$ and

$\begin{matrix} \bar{x} (ϑ_{ς + 1}, t_{ι + 1}; λ) & = {\bar{X}}_{ς ι} (ϑ_{ς + 1}, t_{ι + 1}; λ) = {\bar{X}}_{ς ι} (0, 0; λ), \\ + {\bar{X}}_{ς ι} (0, 1; λ) (t_{ς + 1} - t_{ς}) + {\bar{X}}_{ς ι} (0, 2; λ) (t_{ς + 1} - t_{ς})^{2} + . . . + {\bar{X}}_{ς ι} (0, n^{'}; λ) (t_{ς + 1} - t_{ς})^{n^{'}}, \\ + {\bar{X}}_{ς ι} (1, 0; λ) (ϑ_{ς + 1} - ϑ_{ς}) + {\bar{X}}_{ς ι} (1, 1; λ) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς}), \\ + {\bar{X}}_{ς ι} (1, 2; λ) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς})^{2} + . . . + {\bar{X}}_{ς ι} (1, n^{'} - 1; λ) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς})^{n^{'}} \\ + {\bar{X}}_{ς ι} (n, n^{'} - 1; λ) (ϑ_{ς + 1} - ϑ_{ς})^{n} (t_{ς + 1} - t_{ς})^{n^{'} - 1} \\ + {\bar{X}}_{ς ι} (n, n^{'}; λ) (ϑ_{ς + 1} - ϑ_{ς})^{n} (t_{ς + 1} - t_{ς})^{n^{'}}, \\ = \sum_{μ = 0}^{n} \sum_{i = 0}^{n^{'}} {\bar{X}}_{ς ι} (μ . n . λ) ξ^{μ} ξ^{' i} . \end{matrix}$

3.3.1 The properties of fuzzy q-differential transformation

In this part, we propose the very important properties of fuzzy two-dimensional q-differential transform method.

Theorem 3.3 Suppose that u (x, t) and v (x, t) are fuzzy-valued functions and their fuzzy q-differential transformations denoted by U_q (l, l^*) and V_q (l, l^*), respectively. Then

If w (x, t) = u (x, t) ⊕ v (x, t) , then W_q (l, l^*) = U_q (l, l^*) ⊕ V_q (l, l^*) , l, l^* ∈ K

If $w (x, t) = u (x, t) \underset{gH}{⊖} v (x, t),$ then $W_{q} (l, l^{*}) = U_{q} (l, l^{*}) \underset{gH}{⊖} V_{q} (l, l^{*}), l, l^{*} \in K$

If w (x, t) = λ ⊙ u (x, t) , then W_q (l, l^*) = λ ⊙ U_q (l, l^*) , l, l^* ∈ K,

provided the generalized Hukuhara difference (gH-difference) exists.

Proof. By using definition 3.2, the proof is obvious.

Theorem 3.4 Let us consider the fuzzy-valued function $u \in E^{1}$ and, if $w (x, t) = D_{q, x} u (x, t) = \frac{\partial q}{\partial_{q} x} u (x, t)$ , then we obtain W_q (l, l^*) = [l + 1] _qU_q (l + 1, l^* ; λ), where W_q (l, l^*) and U_q (l, l^*) are the q-differential transformations of w and u, respectively.

Proof. Using definition 3.2, we obtain for λ ∈ [0, 1] $w (x, t; λ) = \frac{\partial q}{\partial_{q} x} u (x, t; λ) = {[\frac{\partial q}{\partial_{q} x} \underline{u} (x, t; λ), \frac{\partial q}{\partial_{q} x} \bar{u} (x, t; λ)]}_{x = x_{0}, t = t_{0}},$ then

$\begin{matrix} W_{q} (l, l^{*}; λ) & = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} y^{l^{*}}} [\frac{\partial q}{\partial_{q} x} \underline{u} (x, t; λ), \frac{\partial q}{\partial_{q} x} \bar{u} (x, t; λ)]]}_{x = x_{0}, t = t_{0}} \\ = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l + 1 + l^{*}}}{\partial_{q} x^{l + 1} \partial_{q} t^{l^{*}}} \underline{u} (x, t; λ), \frac{\partial_{q}^{l + 1 + l^{*}}}{\partial_{q} x} \bar{u} (x, t; λ)]}_{x = x_{0}, t = t_{0}} \\ = [l + 1]_{q} {[\frac{1}{[l + 1]_{q}! [l^{*}]_{q}!} [\frac{\partial_{q}^{l + 1 + l^{*}}}{\partial_{q} x^{l + 1} \partial_{q} t^{l^{*}}} \underline{u} (x, t; λ), \frac{\partial_{q}^{l + 1 + l^{*}}}{\partial_{q} x} \bar{u} (x, t; λ)]]}_{x = x_{0}, t = t_{0}} . \end{matrix}$ Thus, we can be obtained the proof as follows:

$W_{q} (l, l^{*}; λ) = [[l + 1]_{q} U_{q} (l + 1, l^{*}; λ)],$ the proof is completed.□

Theorem 3.5 Let us consider $u \in E^{1}$ and, if $w (x, t) = D_{q, x^{θ}, t^{η}} u (x, t) = \frac{\partial_{q}^{ζ + η}}{\partial_{q} x^{ζ} \partial_{q} t^{η}} u (x, t)$ , then we obtain W_q (l, l^*) = [l + 1] _q · [l + 2] _q ⋯ [l + ζ] _q [l^* + 1] _q ⋯ [l^* + η] _qU_q (l + ζ, l^* + η ; λ), where W_q (l, l^*) and U_q (l, l^*) are the q-differential transformations of w and u, respectively.

Proof. Using definition 3.2, we obtain $w (x, t; λ) = \frac{\partial_{q}^{ζ + η}}{\partial_{q}^{ζ + η} x^{ζ} \partial_{q} t^{η}} u (x, t; λ) = [\frac{\partial_{q}^{ζ + η}}{\partial_{q} x^{ζ} \partial_{q} t^{η}} \underline{u} (x, t; λ), \frac{\partial_{q}^{ζ + η}}{\partial_{q} x^{ζ} \partial_{q} t^{η}} \bar{u} (x, t; λ)], 0 \leq λ \leq 1$ then

$\begin{matrix} W_{q} (l, l^{*}; λ) & = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} [\frac{\partial^{ζ + η} q}{\partial_{q} x^{ζ} \partial_{q} t^{η}} \underline{u} (x, t; λ), \frac{\partial^{ζ + η} q}{\partial_{q} x^{ζ} \partial_{q} t^{η}} \bar{u} (x, t; λ)]]}_{x = x_{0}, t = t_{0}} \\ = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l + ζ + l^{*} + η}}{\partial_{q} x^{l + ζ} \partial_{q} t^{l^{*} + η}} \underline{u} (x, t; λ), \frac{\partial_{q}^{l + ζ + l^{*} + η}}{\partial_{q} x^{l + ζ} \partial_{q} t^{l^{*} + η}} \bar{u} (x, t; λ)]}_{x = x_{0}, t = t_{0}} \\ = ([l + 1]_{q} \cdot [l + 2]_{q} \cdot \cdot \cdot [l + ζ]_{q} [l^{*} + 1]_{q} \cdot \cdot \cdot [l^{*} + υ]_{q}) \\ \times {[\frac{1}{[l + η]_{q}! [l^{*} + υ]_{q}!} [\frac{\partial_{q}^{l + ζ + l^{*} + η}}{\partial_{q} x^{l + ζ} \partial_{q} t^{l^{*} + ζ}} \underline{u} (x, t; λ), \frac{\partial_{q}^{l + ζ + l^{*} + η}}{\partial_{q} x^{l + ζ} \partial_{q} t^{l^{*} + η}} \bar{u} (x, t; λ)]]}_{x = x_{0}, t = t_{0}} \end{matrix}$ Thus, we obtained the proof as follows:

$W_{q} (l, l^{*}; λ) = [l + 1]_{q} \cdot [l + 2]_{q} \dots [l + ζ]_{q} [l^{*} + 1]_{q} \dots [l^{*} + η]_{q} U_{q} (l + ζ, l^{*} + η; λ), 0 \leq λ \leq 1,$ the proof is completed.□

Lemma 3.6 Let us consider the fuzzy-valued functions $w \in E^{1}$ for the following relation

${[D_{q, x}^{μ} w ({xq}^{η}, t)]}_{x = 0} = a_{η, μ} (q) {[D_{q, x}^{μ} w (x, t)]}_{x = 0},$ (3.27) then we have

$a_{η, μ} (q) = \sum_{ς = 0}^{η} \sum_{ι = 0}^{μ} (- 1)^{ς} (1 - q)^{ς} [ς]_{q}! {[\begin{matrix} η \\ ς \end{matrix}]}_{q} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2} + ς ι} .$ (3.28)

Proof. For any λ ∈ [0, 1] , the λ-level set of $w ({xq}^{η}, t; λ) = [\underline{w} ({xq}^{η}, t; λ), \bar{w} ({xq}^{η}, t; λ)]$ . Then $\underline{w} ({xq}^{η}, t; λ) = \sum_{ς = 0}^{η} (- 1)^{ς} (1 - q)^{ς} {[\begin{matrix} η \\ ς \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2}} x^{ς} D_{q, x}^{ς} \underline{w} (x, t; λ)$ (3.29) $\bar{w} ({xq}^{η}, t; λ) = \sum_{ς = 0}^{s} (- 1)^{ς} (1 - q)^{ς} {[\begin{matrix} η \\ ς \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2}} x^{ς} D_{q, x}^{ς} \bar{w} (x, t; λ)$ (3.30) with $D_{q, x}^{μ} \underline{w} ({xq}^{η}, t; λ) = \sum_{ς = 0}^{η} (- 1)^{ς} (1 - q)^{ς} {[\begin{matrix} η \\ ς \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2}} D_{q, x}^{μ} {x^{ς} D_{q, x}^{ς} \underline{w} (x, t; λ)},$ (3.31) $D_{q, x}^{μ} \bar{w} ({xq}^{η}, t; λ) = \sum_{ς = 0}^{s} (- 1)^{ς} (1 - q)^{ς} {[\begin{matrix} η \\ ς \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2}} D_{q, x}^{μ} {x^{ς} D_{q, x}^{ς} \bar{w} (x, t; λ)} .$ (3.32) Consequently, we obtain

$\begin{matrix} D_{q}^{μ} {x^{ς} D_{q}^{ς} \underline{w} (x, t; λ)} & = \sum_{ι = 0}^{μ} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} D_{qx}^{μ - ι} ({xq}^{ι})^{ς} D_{x, q}^{ι} (D_{q, x}^{ς} \underline{w} (x, t; λ)) \\ = \sum_{ι = 0}^{μ} q^{ς ι} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} D_{qx}^{μ - ι} x^{ς} D_{q, x}^{ς + ι} \underline{w} (x, t; λ) \\ = \sum_{ι = 0}^{μ} q^{ς ι} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} [ς]_{q} [ς - 1]_{q} \cdot \cdot \cdot [ς + ι - μ + 1]_{q} x^{ς + ι - μ} D_{q, x}^{ς + ι} \underline{w} (x, t; λ), \end{matrix}$ (3.33) and

$\begin{matrix} D_{q}^{μ} {x^{ς} D_{q}^{ς} \bar{w} (x, t; λ)} & = \sum_{ι = 0}^{μ} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} D_{qx}^{μ - ι} ({xq}^{ι})^{ς} D_{x, q}^{ι} (D_{q, x}^{ς} \bar{w} (x, t; λ)) \\ = \sum_{ι = 0}^{μ} q^{ς ι} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} D_{qx}^{μ - ι} x^{ς} D_{q, x}^{ς + ι} \bar{w} (x, t; λ) \\ = \sum_{ι = 0}^{μ} q^{ς ι} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} [ς]_{q} [ς - 1]_{q} \cdot \cdot \cdot [ς + ι - μ + 1]_{q} x^{ς + ι - μ} D_{q, x}^{ς + ι} \bar{w} (x, t; λ) . \end{matrix}$ (3.34) From (3.34) into (3.29), we obtain

$\begin{matrix} D_{q, x}^{μ} \underline{w} ({xq}^{η}, t; λ) & = \sum_{ς = 0}^{η} (- 1)^{ς} (1 - q)^{ς} {[\begin{matrix} η \\ ς \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2}} \\ \times \sum_{ι = 0}^{μ} q^{ς ι} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} [ς]_{q} [ς - 1]_{q} \cdot \cdot \cdot [ς + ι - λ + 1]_{q} x^{ς + ι - μ} D_{q, x}^{ς + ι} \underline{w} (x, t; λ), \end{matrix}$ (3.35) and

$\begin{matrix} D_{q, x}^{μ} \bar{w} ({xq}^{η}, t; λ) & = \sum_{ς = 0}^{η} (- 1)^{ς} (1 - q)^{ς} {[\begin{matrix} η \\ ς \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2}} \\ \times \sum_{ι = 0}^{μ} q^{ς ι} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} [ς]_{q} [ς - 1]_{q} \cdot \cdot \cdot [ς + ι - μ + 1]_{q} x^{ς + ι - μ} D_{q, x}^{ς + ι} \bar{f} (x, t; λ) . \end{matrix}$ (3.36) At x = 0, all terms of the second series are disappear except at ς + ι - μ = 0, then

$\begin{matrix} {[D_{q, x}^{μ} \underline{w} ({xq}^{s}, t; λ)]}_{x = 0} & = \sum_{ς = 0}^{η} (- 1)^{ς} (1 - q)^{ς} {[\begin{matrix} η \\ ς \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2}} \sum_{ι = 0}^{μ} q^{ς ι} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} [ς]_{q}! {[D_{q, x}^{μ} \underline{w} (x, t; λ)]}_{x = 0} \\ = [\sum_{ς = 0}^{η} \sum_{ι = 0}^{μ} (- 1)^{ς} (1 - q)^{ς} [ς]_{q}! {[\begin{matrix} η \\ ς \end{matrix}]}_{q} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2} + ς ι}] {[D_{q, x}^{μ} \underline{w} (x, t; λ)]}_{x = 0}, \end{matrix}$ and

$\begin{matrix} {[D_{q, x}^{μ} \bar{w} ({xq}^{η}, t; λ)]}_{x = 0} & = \sum_{ς = 0}^{η} (- 1)^{ς} (1 - q)^{ς} {[\begin{matrix} η \\ ς \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2}} \sum_{ι = 0}^{μ} q^{ς ι} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} [ς]_{q}! {[D_{q, x}^{μ} \bar{w} (x, t; λ)]}_{x = 0} \\ = [\sum_{ς = 0}^{η} \sum_{ι = 0}^{μ} (- 1)^{ς} (1 - q)^{ς} [ς]_{q}! {[\begin{matrix} η \\ ς \end{matrix}]}_{q} {[\begin{matrix} μ \\ ι \end{matrix}]}_{q} q^{\frac{ς (ς - 1)}{2} + ς ι}] {[D_{q, x}^{μ} \bar{w} (x, t; λ)]}_{x = 0} . \end{matrix}$ Thus, we obtain

${[D_{q, x}^{μ} w ({xq}^{η}, t; λ)]}_{x = 0} = a_{η, μ} (q) {[D_{q, x}^{μ} w (x, t; λ)]}_{x = 0}, 0 \leq λ \leq 1,$ the proof is completed.□

Remark 3.7 The real-valued function a_η,μ (q) defined in (3.28) has the property

$lim_{q \to 1} a_{η, μ} (q) = 1 .$ (3.37) For q → 1, all terms in the series a_η,μ (q) are vanished except at ς = 0 . subsequently, ι = μ .

Theorem 3.8 Let us consider U_α,q (l, l^*) and V_α,q (l, l^*) are the two-dimensional q-differential transformations of u (x, t) and v (x, t) are positive real-valued functions, respectively. If f (x, t) = u (x, t) · v (x, t) , where (x₀, t₀) = (0, 0) then under (i)-differentiability of f, we have ${\underline{F}}_{α, q} (l, l^{*}; λ) = \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {\underline{U}}_{q} (l^{*} - ζ; l - ς; λ) {\underline{V}}_{q} (ζ; ς; λ), 0 \leq λ \leq 1,$ (3.38) ${\bar{F}}_{α, q} (l, l^{*}; λ) = \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {\bar{U}}_{q} (l^{*} - ζ; l - ς; λ) {\bar{V}}_{q} (ζ; ς; λ), 0 \leq λ \leq 1,$ (3.39) and (ii)-differentiability of f as:

$\begin{matrix} {\underline{F}}_{α, q} (l, l^{*}; λ) & = \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {\underline{U}}_{q} (l^{*} - ζ; l - ς; λ) {\underline{V}}_{q} (ζ; ς; λ) \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {\bar{U}}_{q} (l^{*} - ζ; l - ς; λ) {\bar{V}}_{q} (ζ; ς; λ), 0 \leq λ \leq 1, \\ both of l, l^{*} are even or odd, \end{matrix}$ (3.40) and

$\begin{matrix} {\bar{F}}_{α, q} (l, l^{*}; λ) & = \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {\bar{U}}_{q} (l^{*} - ζ; l - ς; λ) {\bar{V}}_{q} (ζ; ς; λ) \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {\underline{U}}_{q} (l^{*} - ζ; l - ς; λ) {\underline{V}}_{q} (ζ; ς; λ), 0 \leq λ \leq 1, \\ both of l, l^{*} are even or odd . \end{matrix}$ (3.41)

Proof. Using definition of two-dimensional fuzzy q-DTM and under (i)-differentiability of f, we have $\underline{F} (l, l^{*}; λ) = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l}} [\underline{u} (x, t; λ) \cdot \underline{v} (x, t; λ)]]}_{x = 0, t = 0}$ (3.42) $\bar{F} (l, l^{*}; λ) = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l}} [\bar{u} (x, t; λ) \cdot \bar{v} (x, t; λ)]]}_{x = 0, t = 0}$ (3.43) by the property (3.2.), one obtain $\underline{F} (l, l^{*}; λ) = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l^{*}}}{\partial_{q} t^{l^{*}}} \sum_{ς = 0}^{l} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} (\frac{\partial_{q}^{l - ς}}{\partial_{q} x^{l - ζ}} \underline{u} ({xq}^{ς}, t; λ)) (\frac{\partial_{q}^{ς}}{\partial_{q} x^{ζ}} \underline{v} (x, t; λ))]}_{x = 0, t = 0}$ (3.44) $\bar{F} (l, l^{*}; λ) = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l^{*}}}{\partial_{q} t^{l^{*}}} \sum_{ς = 0}^{l} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} (\frac{\partial_{q}^{l - ζ}}{\partial_{q} x^{l - ζ}} \bar{u} ({xq}^{ς}, t; λ)) (\frac{\partial_{q}^{ς}}{\partial_{q} x^{ς}} \bar{v} (x, t; λ))]}_{x = 0, t = 0}$ (3.45) taking the relation (3.27), we get

$\begin{matrix} \underline{F} (l, l^{*}; λ) & = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\sum_{ς = 0}^{l} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} a_{ς, l - ς} (q) D_{q, t}^{l^{*}} (D_{q, x}^{l - ς} \underline{u} (x, t; λ) \cdot D_{q, x}^{ς} \underline{v} (x, t; λ))]}_{x = 0, t = 0} \\ = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\sum_{ς = 0}^{l} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} a_{ς, l - ς} (q) D_{q, t}^{l^{*}} (\underline{ξ} (x, t; λ) \cdot \underline{φ} (x, t; λ))]}_{x = 0, t = 0} \end{matrix}$ (3.46) and

$\begin{matrix} \bar{F} (l, l^{*}; λ) & = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\sum_{ς = 0}^{l} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} a_{ς, l - ζ} (q) D_{q, t}^{l^{*}} (D_{q, x}^{l - ς} \bar{u} (x, t; λ) \cdot D_{q, x}^{ς} \bar{v} (x, t; λ))]}_{x = 0, t = 0} \\ = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\sum_{ς = 0}^{l} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} a_{ς, l - ς} (q) D_{q, t}^{l^{*}} (\bar{ξ} (x, t; λ) \cdot \bar{φ} (x, t; λ))]}_{x = 0, t = 0} \end{matrix}$ (3.47) where

$\begin{matrix} \underline{ξ} (x, t; λ) & = D_{q, x}^{l - ζ} \underline{u} (x, t; λ), \underline{φ} (x, t; λ) = D_{q, x}^{ζ} \underline{v} (x, t; λ), \\ \bar{ξ} (x, t; λ) & = D_{q, x}^{l - ζ} \bar{u} (x, t; λ), \bar{φ} (x, t; λ) = D_{q, x}^{ζ} \bar{v} (x, t; λ) . \end{matrix}$ Using the q-Leibniz rule, we obtain

$\begin{matrix} D_{q, t}^{l^{*}} (\underline{ξ} (x, t; λ) \cdot \underline{φ} (x, t; λ)) = \sum_{ζ = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ζ \end{matrix}]}_{q} D_{q, t}^{l^{*} - ζ} \underline{ξ} (x, {tq}^{ζ}; λ) D_{q, t}^{ζ} \underline{φ} (x, t; λ), \\ D_{q, t}^{l^{*}} (\bar{ξ} (x, t; λ) \cdot \bar{φ} (x, t; λ)) = \sum_{ζ = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ζ \end{matrix}]}_{q} D_{q, t}^{l^{*} - ζ} \bar{ξ} (x, {tq}^{ζ}; λ) D_{q, t}^{ζ} \bar{φ} (x, t; λ) . \end{matrix}$ Taking lemma (3.6) and derivative f . ζ . to t, we get $D_{q, t}^{l^{*}} (\underline{ξ} (x, t; λ) \cdot \underline{φ} (x, t; λ)) = {[\sum_{ζ = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ζ \end{matrix}]}_{q} a_{ζ, l^{*} - ζ} (q) D_{q, t}^{l^{*} - ζ} \underline{ξ} (x, t; λ) D_{q, t}^{ζ} \underline{φ} (x, t; λ)]}_{x = 0, t = 0}$ (3.48) $D_{q, t}^{l^{*}} (\bar{ξ} (x, t; λ) \cdot \bar{φ} (x, t; λ)) = {[\sum_{ζ = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ζ \end{matrix}]}_{q} a_{ζ, l^{*} - ζ} (q) D_{q, t}^{l^{*} - ζ} \bar{ξ} (x, t; λ) D_{q, t}^{ζ} \bar{φ} (x, t; λ)]}_{x = 0, t = 0}$ (3.49) From (3.49) into (3.46), we obtain

$\begin{matrix} \underline{F} (l, l^{*}; λ) & = \frac{1}{[l]_{q}! [l^{*}]_{q}!} \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ζ \end{matrix}]}_{q} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {[D_{q, t}^{l^{*} - ζ} \underline{ξ} (x, t; λ) D_{q, t}^{ζ} \underline{φ} (x, t; λ)]}_{x = 0, t = 0} \\ = \frac{1}{[l]_{q}! [l^{*}]_{q}!} \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ζ \end{matrix}]}_{q} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {[D_{q, x^{l - ς}, t^{l^{*} - ζ}}^{(l^{*} - ζ) + (l - ς)} \underline{u} (x, t; λ) D_{q, x^{ς}, t^{ζ}}^{ζ + ς} \underline{v} (x, t; λ)]}_{x = 0, t = 0} \\ = \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {[\frac{D_{q, x^{l - ς}, t^{l^{*} - ζ}}^{(l^{*} - ζ) + (l - ς)} \underline{u} (x, t; λ)}{[l^{*} - ζ]_{q}! [l - ς]_{q}!}]}_{x = 0, t = 0} \cdot {[\frac{D_{q, x^{ς}, t^{ζ}}^{ζ + ς} \underline{v} (x, t; λ)}{[ζ]_{q}! [ς]_{q}!}]}_{x = 0, t = 0}, \end{matrix}$ and

$\begin{matrix} \bar{F} (l, l^{*}; λ) & = \frac{1}{[l]_{q}! [l^{*}]_{q}!} \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ζ \end{matrix}]}_{q} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {[D_{q, t}^{l^{*} - ζ} \bar{ξ} (x, t; λ) D_{q, t}^{ζ} \bar{φ} (x, t; λ)]}_{x = 0, t = 0} \\ = \frac{1}{[l]_{q}! [l^{*}]_{q}!} \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ζ \end{matrix}]}_{q} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {[D_{q, x^{l - ς}, t^{l^{*} - ζ}}^{(l^{*} - ζ) + (l - ς)} \bar{u} (x, t; λ) D_{q, x^{ς}, t^{ζ}}^{ζ + ς} \bar{v} (x, t; λ)]}_{x = 0, t = 0} \\ = \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {[\frac{D_{q, x^{l - ς}, t^{l^{*} - ζ}}^{(l^{*} - ζ) + (l - ς)} \bar{u} (x, t; λ)}{[l^{*} - ζ]_{q}! [l - ς]_{q}!}]}_{x = 0, t = 0} \cdot {[\frac{D_{q, x^{ς}, t^{ζ}}^{ζ + ς} \bar{v} (x, t; λ)}{[ζ]_{q}! [ς]_{q}!}]}_{x = 0, t = 0} . \end{matrix}$ Consequently, taking the definition of fuzzy q-DTM, we obtain

$\begin{matrix} {\underline{F}}_{α, q} (l, l^{*}; λ) & = \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {\underline{U}}_{q} (l^{*} - ζ; l - ς; λ) {\underline{V}}_{q} (ζ; ς; λ), 0 \leq λ \leq 1, \\ {\bar{F}}_{α, q} (l, l^{*}; λ) & = \sum_{ς = 0}^{l} \sum_{ζ = 0}^{l^{*}} a_{ς, l - ς} (q) a_{ζ, l^{*} - ζ} (q) {\bar{U}}_{q} (l^{*} - ζ; l - ς; λ) {\bar{V}}_{q} (ζ; ς; λ), 0 \leq λ \leq 1 . \end{matrix}$ Let that f is (ii)-differentiability, using similar techniques, can be obtained the result. The proof is completed.

Theorem 3.9 Let us consider U_α,q (l, l^*) and V_α,q (l, l^*) are the two-dimensional q-differential transformations of u (x, t) and v (x, t) are positive real-valued function, respectively. If w (x, t) = u (x, t) · v (x, t) , where (x₀, t₀) = (0, 0) then under (i)-differentiability of w, we have ${\underline{W}}_{α, q} (l, l^{*}; λ) = \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\underline{V}}_{q} (l; l^{*}; λ) {\underline{U}}_{q} (l - ς; l^{*} - ι; λ), 0 \leq λ \leq 1,$ (3.50) ${\bar{W}}_{α, q} (l, l^{*}; λ) = \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\bar{V}}_{q} (l; l^{*}; λ) {\bar{U}}_{q} (l - ς; l^{*} - ι; λ), 0 \leq λ \leq 1,$ (3.51) and (ii)-differentiability of f as: $\begin{matrix} {\underline{W}}_{α, q} (l, l^{*}; λ) & = \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\underline{V}}_{q} (l; l^{*}; λ) {\underline{U}}_{q} (l - ς; l^{*} - ι; λ) \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\bar{V}}_{q} (l; l^{*}; λ) {\bar{U}}_{q} (l - ς; l^{*} - ι; λ), 0 \leq λ \leq 1, \end{matrix}$ (3.52) $both of l, l^{*} are even or odd,$ and $\begin{matrix} {\bar{W}}_{α, q} (l, l^{*}; λ) & = \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\bar{V}}_{q} (l; l^{*}; λ) {\bar{U}}_{q} (l - ς; l^{*} - ι; λ) \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\underline{V}}_{q} (l; l^{*}; λ) {\underline{U}}_{q} (l - ς; l^{*} - ι; λ), 0 \leq λ \leq 1, \end{matrix}$ (3.53) $both of l, l^{*} are even or odd .$

Proof. Using definition of two-dimensional fuzzy q-DTM and under (i)-differentiability of f we have for 0 ≤ λ ≤ 1:

$\begin{matrix} W_{q} (l, l^{*}; λ) & = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\frac{\partial_{q}^{l^{*}}}{\partial_{q} t^{l^{*}}} \sum_{ς = 0}^{l} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} (\frac{\partial_{q}^{l - ς}}{\partial_{q} x^{l - ς}} \underline{u} ({xq}^{ς}, t; λ); \frac{\partial_{q}^{l - ς}}{\partial_{q} x^{l - ς}} \bar{u} ({xq}^{ς}, t; λ)) \times (\frac{\partial_{q}^{ς}}{\partial_{q} x^{ς}} \underline{v} (x, t; λ); \frac{\partial_{q}^{ς}}{\partial_{q} x^{ς}} \bar{v} (x, t; λ))]}_{x = 0, t = 0} \\ = \frac{1}{[l]_{q}! [l^{*}]_{q}!} {[\sum_{ς = 0}^{l} {[\begin{matrix} l \\ ς \end{matrix}]}_{q} \sum_{ι = 0}^{l^{*}} {[\begin{matrix} l^{*} \\ ι \end{matrix}]}_{q} \times (\frac{\partial_{q}^{l^{*} - ι}}{\partial_{q} x^{h - ι}} \frac{\partial_{q}^{l - ς}}{\partial_{q} x^{l - ς}} \underline{u} ({xq}^{ς}, {tq}^{ι}; λ); \frac{\partial_{q}^{l^{*} - ι}}{\partial_{q} x^{l^{*} - ι}} \frac{\partial_{q}^{l - ς}}{\partial_{q} x^{l - ς}} \bar{u} ({xq}^{ς}, {tq}^{ι}; λ)) \times (\frac{\partial_{q}^{ι}}{\partial_{q} t^{ι}} \frac{\partial_{q}^{ς}}{\partial_{q} x^{ς}} \underline{v} (x, t; λ); \frac{\partial_{q}^{ι}}{\partial_{q} t^{ι}} \frac{\partial_{q}^{ς}}{\partial_{q} x^{ς}} \bar{v} (x, t; λ))]}_{x = 0, t = 0} \end{matrix}$

$\begin{matrix} = \frac{1}{[l]_{q}! [l^{*}]_{q}!} [\sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} (\frac{[l^{*}]_{q}!}{[ι]_{q}! [l^{*} - ι]_{q}!} \frac{[l]_{q}!}{[ς]_{q}! [l - ς]_{q}!} \times [\frac{\partial_{q}^{(l + l^{*}) - (ς + ι)}}{\partial_{q} x^{l - ς} \partial_{q} t^{l^{*} - ι}} \underline{u} ({xq}^{ς}, {tq}^{ι}; λ); \frac{\partial_{q}^{(l + l^{*}) - (ς + ι)}}{\partial_{q} x^{l - ς} \partial_{q} t^{l^{*} - ι}} \bar{u} ({xq}^{ς}, {tq}^{ι}; λ)]) \times (\frac{\partial_{q}^{ς + ι}}{\partial_{q} x^{ς} \partial_{q} t^{ι}} \underline{v} (x, t; λ); \frac{\partial_{q}^{ς + ι}}{\partial_{q} x^{ς} \partial_{q} t^{ι}} \bar{v} (x, t; λ))] \\ = \frac{1}{[l]_{q}! [l^{*}]_{q}!} [\sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} (\frac{1}{[l - ς]_{q}! [l^{*} - ι]_{q}!} \times [\frac{\partial_{q}^{(l + l^{*}) - (ς + ι)}}{\partial_{q} x^{l - ς} \partial_{q} t^{l^{*} - ι}} \underline{u} ({xq}^{ς}, {tq}^{ι}; λ); \frac{\partial_{q}^{(l + l^{*}) - (ς + ι)}}{\partial_{q} x^{l - ς} \partial_{q} t^{l^{*} - ι}} \bar{u} ({xq}^{ς}, {tq}^{ι}; λ)]) \times (\frac{1}{[ς]_{q}! [ι]_{q}!} \frac{\partial_{q}^{ς + ι}}{\partial_{q} x^{ς} \partial_{q} t^{ι}} \underline{v} (x, t; λ); \frac{\partial_{q}^{ς + ι}}{\partial_{q} x^{ς} \partial_{q} t^{ι}} \bar{v} (x, t; λ))] . \end{matrix}$ Using the definition of fuzzy q-DTM, we get

$\begin{matrix} {\underline{W}}_{α, q} (l, l^{*}; λ) & = \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\underline{V}}_{q} (l; l^{*}; λ) {\underline{U}}_{q} (l - ς; l^{*} - ι; λ), 0 \leq λ \leq 1, \\ {\bar{W}}_{α, q} (l, l^{*}; λ) & = \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\bar{V}}_{q} (l; l^{*}; λ) {\bar{U}}_{q} (l - ς; l^{*} - ι; λ), 0 \leq λ \leq 1 . \end{matrix}$ Assume that w is (ii)-differentiability, using similar techniques, can be obtained the result. The proof is completed.

Lemma 3.10 We consider the following, it is valid

$\frac{\partial_{q}}{\partial_{q} x} (x - x_{0})^{n} = [n]_{q} (x - x_{0})^{n - 1}, (x, x_{0} \in ℝ, n \in ℕ) .$

Theorem 3.11 Let us consider w (x, t) = (x - x₀) ⁿ (t - t₀) ^μ, then W_q (l, l^*) = δ (l - n) δ (l^* - μ) , where

$δ (l - n) = {\begin{matrix} 1, & l = n, \\ 0, & l \neq n, \end{matrix}$ , and $δ (l^{*} - μ) = {\begin{matrix} 1, & l^{*} = μ, \\ 0, & l^{*} \neq μ, \end{matrix}$ are the q-differential transformations of w.

Proof. Using the definition of two-dimensional fuzzy q-DTM, for any λ ∈ [0, 1] ,

$\begin{matrix} \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} w (x, t; λ) \\ = [\frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} \underline{w} (x, t; λ), \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} \bar{w} (x, t; λ)], \end{matrix}$

$\begin{matrix} \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} \underline{w} (x, t; λ) \\ = \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} {(x - x_{0})^{n} (t - t_{0})^{μ}} (λ), \\ \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} \bar{w} (x, t; λ) = \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} \\ {(x - x_{0})^{n} (t - t_{0})^{μ}} (λ) . \end{matrix}$ Using the previous lemma 3.10, we obtain

$\frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} {(x - x_{0})^{n} (t - t_{0})^{μ}} (λ) = \tilde{0}, l > n, l^{*} > μ .$ In order cases, we get $\begin{matrix} \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} {(x - x_{0})^{n} (t - t_{0})^{μ}} (λ) = [n]_{q} \dots \\ [n - l + 1]_{q} (x - x_{0})^{n - l} [μ]_{q} \dots [μ - l^{*}]_{q} (t - t_{0})^{μ - l^{*}} . \end{matrix}$ The assumed expansion is valid within some neighborhood of (x₀, t₀). Putting x = x₀ and t = t₀, all members of the sum vanish, except for n = l and μ = l^*. Then

$\begin{matrix} \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} \underline{w} (x_{0}, t_{0}; λ) = [[l]_{q}! [l^{*}]_{q}!] (λ), \\ \frac{\partial^{l + l^{*}}}{\partial_{q} x^{l} \partial_{q} t^{l^{*}}} \bar{w} (x_{0}, t_{0}; λ) = [[l]_{q}! [l^{*}]_{q}!] (λ) . \end{matrix}$ Using the definition of fuzzy two-dimensional q-differential transform, we obtain

$W_{q} (l, l^{*}; λ) = [δ (l - n) δ (l^{*} - μ)] (λ), 0 \leq λ \leq 1,$ the proof is complete. □

3.4 Application and result

In this section, we investigated the solutions of fuzzy partial q-differential equations using fuzzy q-differential transformation to demonstrate the usefulness of the proposed method. To this end, two examples are provided.

Example 3.1. We consider the following fuzzy q-diffusion equation $\frac{\partial_{q} w}{\partial_{q} t} = \frac{\partial_{q}^{2} w}{\partial_{q} x^{2}}, 0 < q < 1, t > 0$ (3.54) with the initial condition $w (x, 0) = [(1 + 2 λ)^{n}, (5 - 2 λ)^{n}] ⊙ {exp}_{q} (x),$ (3.55) where λ ∈ [0, 1] , (n = 1, 2, 3, . . .) , and ${exp}_{q} (x) = \sum_{i = 0}^{\infty} \frac{x^{ς}}{[ς]_{q}!}$ is the q-exponential function, and obtain the q-derivative $\frac{\partial_{q}}{\partial_{q} x} {exp}_{q} (x) = {exp}_{q} (x) .$ Applying the fuzzy two-dimensional q-differential transform of the equation (3.54), we get $[l^{*} + 1]_{q} {\underline{W}}_{q} (l, l^{*} + 1; λ) = [l + 1]_{q} [l + 2]_{q} {\underline{W}}_{q} (l + 2, l^{*}; λ),$ (3.56) $[l^{*} + 1]_{q} {\bar{W}}_{q} (l, l^{*} + 1; λ) = [l + 1]_{q} [l + 2]_{q} {\bar{W}}_{q} (l + 2, l^{*}; λ) .$ (3.57)

Utilizing the initial conditions (3.55) and properties of fuzzy two-dimensional q-differential transform, we obtain ${\underline{W}}_{q} (ς, 0; λ) = (1 + 2 λ)^{n} \frac{1}{[ς]_{q}!}, ς = 0, 1, 2, 3, . . .$ (3.58) ${\bar{W}}_{q} (ς, 0; λ) = (5 - 2 λ)^{n} \frac{1}{[ς]_{q}!}, ς = 0, 1, 2, 3, . . . .$ (3.59) Substituting (3.59) into (3.56), and using recursive method, we obtain

$\begin{matrix} {\underline{W}}_{q} (ς, ι; λ) = (1 + 2 λ)^{n} \frac{1}{[ς]_{q}! [ι]_{q}!}, \\ ς = 0, 1, 2, 3, . . ., ι = 0, 1, 2, 3, . . ., \end{matrix}$ (3.60)

$\begin{matrix} {\bar{W}}_{q} (ς, ι; λ) = (5 - 2 λ)^{n} \frac{1}{[ς]_{q}! [ι]_{q}!}, \\ ς = 0, 1, 2, 3, . . ., ι = 0, 1, 2, 3, . . . . \end{matrix}$ (3.61) Substituting W_q (ς, ι, λ), we get

$\begin{matrix} \underline{w} (x, t; λ) & = \sum_{ς = 0}^{\infty} \sum_{ι = 0}^{\infty} (\frac{x^{ς}}{[ς]_{q}!} \frac{t^{ι}}{[ι]_{q}!}) (λ), \\ \bar{w} (x, t; λ) & = \sum_{ς = 0}^{\infty} \sum_{ι = 0}^{\infty} (\frac{x^{ς}}{[ς]_{q}!} \frac{t^{ι}}{[ι]_{q}!}) (λ), \end{matrix}$ we can be represented as follows:

$\begin{matrix} w (x, t; λ) = [(1 + 2 λ)^{n}, (5 - 2 λ)^{n}] \\ ⊙ ({exp}_{q} (x) {exp}_{q} (t)), 0 \leq λ \leq 1 . \end{matrix}$

Example 3.2. We consider the following fuzzy nonlinear q-partial differential equation $\frac{\partial_{q} w}{\partial_{q} t} = w^{2} \oplus \frac{\partial_{q} w}{\partial_{q} x}, 0 < q < 1, t > 0$ (3.62) subject to the initial condition $w (x, 0) = [(λ - 1)^{n}, (1 - λ)^{n}] \oplus (1 + 2 x),$ (3.63) where λ ∈ [0, 1] , (n = 1, 2, 3, . . .) .

Applying the fuzzy two-dimensional q-differential transform of (3.62), we get

$\begin{matrix} [l^{*} + 1]_{q} {\underline{W}}_{q} (l, l^{*} + 1; λ) = [l + 1]_{q} {\underline{W}}_{q} (l + 1, l^{*}) \\ + \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\underline{W}}_{q} (ς, ι) {\underline{W}}_{q} (l - ς, l^{*} - ι; λ), \end{matrix}$ (3.64)

$\begin{matrix} [l^{*} + 1]_{q} {\bar{W}}_{q} (l, l^{*} + 1; λ) = [l + 1]_{q} {\bar{W}}_{q} (l + 1, l^{*}) \\ + \sum_{ς = 0}^{l} \sum_{ι = 0}^{l^{*}} q^{ς + ι} {\bar{W}}_{q} (ς, ι) {\bar{W}}_{q} (l - τ, l^{*} - ι; λ) . \end{matrix}$ (3.65)

Taking the initial condition (3.63), we obtain

${\begin{matrix} {\underline{W}}_{q} (0, 0; λ) & = (λ - 1)^{n} + 1 \\ {\underline{W}}_{q} (1, 0; λ) & = (λ - 1)^{n} + 2 \\ {\underline{W}}_{q} (m, 0; λ) & = (λ - 1)^{n}, m = 2, 3, . . ., \end{matrix}$ (3.66) and

${\begin{matrix} {\bar{W}}_{q} (0, 0; λ) & = (1 - λ)^{n} + 1 \\ {\bar{W}}_{q} (1, 0; λ) & = (1 - λ)^{n} + 2 \\ {\bar{W}}_{q} (m, 0; λ) & = (1 - λ)^{n}, m = 2, 3, . . ., \end{matrix}$ (3.67) From (3.67) into (3.64), we obtain

$\begin{matrix} {\underline{W}}_{q} (0, 1; λ) = (λ - 1)^{n} + 3, \\ {\underline{W}}_{q} (1, 1; λ) = (λ - 1)^{n} + (2 + 2 q), \\ {\underline{W}}_{q} (2, 1; λ) = (λ - 1)^{n} + 4 q, \\ {\underline{W}}_{q} (0, 2; λ) = (λ - 1)^{n} + 5, \\ {\underline{W}}_{q} (1, 2; λ) = (λ - 1)^{n} + (\frac{2 + 18 q + 6 q^{2} + 2 q^{3}}{1 + q}), \\ ⋮ \end{matrix}$ and

$\begin{matrix} {\bar{W}}_{q} (0, 1; λ) = (1 - λ)^{n} + 3, \\ {\bar{W}}_{q} (1, 1; λ) = (1 - λ)^{n} + (2 + 2 q), \\ {\bar{W}}_{q} (2, 1; λ) = (1 - λ)^{n} + 4 q, \\ {\bar{W}}_{q} (0, 2; λ) = (1 - λ)^{n} + 5, \\ {\bar{W}}_{q} (1, 2; λ) = (1 - λ)^{n} + (\frac{2 + 18 q + 6 q^{2} + 2 q^{3}}{1 + q}), \\ ⋮ . \end{matrix}$ Thus, we can obtained the series solutions as follows

$\begin{matrix} w (x, t; λ) = [(λ - 1)^{n}, (1 - λ)^{n}] \oplus \\ (1 + 2 x + 3 t + (2 + 2 q) xt + 4 {qx}^{2} t + [\frac{2 + 18 q + 6 q^{2} + 2 q^{3}}{1 + q}] {xt}^{2} + \dots) . \end{matrix}$

4 Fuzzy fractional q-differential transform

In this section, we present some definitions, properties of q-calculus, fractional q-integrals, and q-derivatives, which we will be using for investigating this method.

Let us consider f is a fuzzy-valued function $(f : T_{q} \to E^{1})$ ; we present the fuzzy q-fractional derivative by g H-difference [40].

Definition 4.1 Let $f : T_{q} \to E^{1}$ is a q-integrable function and $[f (x)]_{λ} = [\underline{f} (x; λ), \bar{f} (x; λ)]$ for $0 \leq λ \leq 1, t \in T_{q} - {0}$ . The fuzzy q-fractional integral of order α≠ 0, - 1, - 2, …. for f (s) is defined by $_{q}^{F} I_{a}^{α} f (x) = \frac{1}{Γ_{q} (α)} ⊙ \int_{a}^{x} (x - qs)_{q}^{α - 1} ⊙ f (s) d_{q} s$

Definition 4.2 Let $\forall m,_{q}^{F} D^{m} f$ be continuous and q integrable functions in $T_{q}$ . The α-order fuzzy Caputo q fractional derivative ( $(_{q}^{FC} [gH]$ -derivative for short) of fuzzy-valued function f is defined $\begin{matrix} _{q}^{FC} D_{*}^{α} f (x) =_{q}^{F} I_{a}^{m - α} (_{q}^{F} D^{m} f) (x) \\ = \frac{1}{Γ_{q} (m - α)} ⊙ \int_{a}^{x} (t - qs)_{q}^{m - α - 1} ⊙_{q}^{F} D^{m} f (s) d_{q} s \end{matrix}$

where $m - 1 < α \leq m, m \in ℕ, t > a$ . Consider $_{q}^{FC} [gH]$ -derivative of order 0 < α ≤ 1 for fuzzy-valued function f, so the fuzzy Caputo q-fractional gH-derivative will be expressed by $\begin{matrix} _{q}^{FC} D_{*}^{α} f (x) & =_{q}^{F} I_{a}^{1 - α} (_{q}^{F} Df) (x) \\ = \frac{1}{Γ_{q} (1 - α)} ⊙ \int_{a}^{x} (x - qs)_{q}^{- α} ⊙_{q}^{F} Df (s) d_{q} s, \end{matrix}$

for x > a

Theorem 4.3 Let $f : T_{q} \to E^{1}$ be a fuzzy-valued function on $T_{q}$

(i) The function f is $_{q}^{F} [(i) - gH]$ -differentiable at x₀∈ $T_{q} - {0}$ if and only if $_{q}^{FC} [(i) - gH]$ -differentiable at x₀,

(ii) The function f is $\underset{q}{F} [($ ii) - gH]-differentiable at x₀∈ $T_{q} - {0}$ if and only if $\underset{q}{FC} [(ii) - gH]$ -differentiable at x₀.

Definition 4.4 Let $f : T_{q} \to E^{1}$ be a fuzzy-valued function on $T_{q}$ . A point $x_{0} \in T_{q} - {0}$ is said to be a switching point for the $_{q}^{FC} [gH]$ -differentiably of f, if in any neighborhood V of x₀ there exist points x₁ < x₀ < x₂ such that

(type i) f is $_{q}^{FC} [(i) - gH]$ -differentiable at x₁ while f is not $_{q}^{FC} [(ii) - gH]$ -differentiable at x₁, and f is $_{q}^{FC} [(ii) - gH]$ -differentiable at x₂ while f is not $_{q}^{FC} [(i) - gH]$ -differentiable at x₂, or

(type ii) f is $_{q}^{FC} [(ii) - gH]$ -differentiable at x₁ while f is not _qF^C [(i) - gH]-differentiable at x₁, and f is _q^FC [(i) - gH]-differentiable at x₂ while f is not $_{q}^{FC} [(ii) - gH]$ -differentiable at x₂.

Theorem 4.5 Let $f : T_{q} \to E^{1}$ be a fuzzy-valued function on $T_{q}$

(i) If $x_{0} \in T_{q} - {0}$ is a switching point for the q-differentiability of f of (type i)

then x₀ is a switching point for the Caputo q-differentiability of f of (type i)

(ii) If $x_{0} \in T_{q} - {0}$ is a switching point for the q-differentiability of f of (type ii)

then x₀ is a switching point for the Caputo q-differentiability of f of (type ii)

Definition 4.6 Suppose that |q|<1, for any real-valued functions f (x) and g (x) , we have

$D_{q}^{n} {f (x) \cdot g (x)} = \sum_{l = 0}^{n} [\begin{matrix} n \\ l \end{matrix}] D_{q}^{n - l} f ({xq}^{l}) D_{q}^{l} g (x),$ where

$[n]_{q} = \frac{q^{n} - 1}{q - 1} = q^{n - 1} + \cdot \cdot \cdot + q + 1, n \in ℕ_{+}$ and

$[\begin{matrix} n \\ l \end{matrix}] = \frac{[n]_{q}!}{[l]_{q}! [n - l]_{q}!},$

$[n]_{q}! = {\begin{matrix} 1, & if n = 0, \\ [n]_{q} [n - 1]_{q} \dots [1]_{q} & if n \in ℕ_{+} . \end{matrix}$

Definition 4.7 The q-analogue of $(x - a)_{q}^{n}$ is the polynomial

$(x - a)_{q}^{n} = {\begin{matrix} 1, & if n & = 0, \\ \prod_{ς = 0}^{n - 1} (x - {aq}^{ς}) & if n & \in ℕ . \end{matrix}$ The q-gamma function is given by

$Γ_{q} (x) = \frac{(1 - q)^{x - 1}}{(1 - q)^{x - 1}}, x \in ℝ / {0} \cup ℕ_{-}, 0 < q < 1,$ where

$(a - b)^{α} = a^{α} \prod_{i = 0}^{\infty} \frac{(a - {bq}^{ς})}{(a - {bq}^{α + ς})}, α \in ℝ,$ and satisfies Γ_q (x + 1) = [x] _qΓ_q (x) .

We denote $C^{F} [a, b]$ as a space of all fuzzy-valued functions which are continuous on [a, b], and the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval $[a, b] \subset ℝ$ by $L^{F} [a, b],$ (see [19, 48]).

Definition 4.8 Let $f (x) \in C^{F} [a, b] \cap L^{F} [a, b]$ and $f : T_{q} \to E^{1}$ . The fuzzy q-Riemann-Liouville integral of fuzzy-valued function f is defined on (0, b) , b > 0 and a ∈ (0, b) is an arbitrary fixed point as follows $\begin{matrix} (I_{q, a}^{α} f) (x) = \frac{1}{Γ_{q} (α)} ⊙ \int_{a}^{x} (x - qt)^{α - 1} ⊙ f (t) d_{q} t, \\ α \geq 0, 0 < a < x < b, \end{matrix}$ (4.1) and the fuzzy Caputo q-fractional derivatives of order α is defined by $D_{a}^{α} f (x) = I_{q, a}^{m - α} D^{m} f (x),$ (4.2) where α, β > 0, m - 1 < α ≤ m, a ≥ 0 and γ > -1, we have the following properties

$\begin{matrix} (I_{q, a}^{α} I_{q, a}^{β}) f (x; λ) = (I_{q, a}^{β} I_{q, a}^{α}) f (x; λ) \\ = (I_{q, a}^{α + β}) f (x; λ), \end{matrix}$ (4.3)

$\begin{matrix} I_{q, a}^{α} {(x - a)}^{γ} = \frac{Γ_{q} (γ + 1)}{Γ_{q} (γ + α + 1)} (x - a)^{γ + α}, \end{matrix}$ (4.4)

$\begin{matrix} (I_{q, a}^{α} D_{q, a}^{α}) f (x; λ) = f (x; λ) \underset{gH}{⊖} \sum_{l = 0}^{m - 1} f^{l} (a^{+}; λ) \\ ⊙ \frac{(x - a)^{l}}{l!} . \end{matrix}$ (4.5)

Suppose that λ-level represent of fuzzy-valued function f as $f (x; λ) = [\underline{f} (x; λ), \bar{f} (x; λ)]$ , for λ ∈ [0, 1] then we can indicate the fuzzy q-Riemann-Liouville integral of fuzzy-valued function f can be expressed as follows:

$(I_{q}^{0} f) (x; λ) = [(I_{q}^{0} \underline{f}) (x; λ), (I_{q}^{0} \bar{f}) (x; λ)],$ where 0 ≤ λ ≤ 1

$\begin{matrix} (I_{q}^{0} \underline{f}) (x; λ) & = \underline{f} (x; λ), \\ (I_{q}^{0} \bar{f}) (x; λ) & = \bar{f} (x; λ) . \end{matrix}$ The fuzzy fractional q-derivative of the fuzzy Riemann-Liouvill type is given by

$\begin{matrix} (^{RL} D_{q, a}^{α} f) (x; λ) \\ = {\begin{matrix} (I_{q, a}^{- α} f) (x; λ), & α \leq 0, \\ (D_{q}^{[α]} I_{q, a}^{[α] - α} f) (x; λ) & α > 0 . \end{matrix} \end{matrix}$ And the fuzzy fractional q-derivative of the Caputo type is

$\begin{matrix} (D_{q, a}^{α} f) (x; λ) = \\ [(D_{q, a}^{α} \underline{f}) (x; λ); (D_{q, a}^{α} \bar{f}) (x; λ)] (x), α > 0, \end{matrix}$ where

$(D_{q, a}^{α} f) (x; λ) = (I_{q, a}^{[α] - α} D_{q}^{[α]}) f (x; λ), α > 0,$ for [α] denotes the smallest integer greater or equal to α .

Theorem 4.9 [10]. The fractional q-Leibniz rule, as follows

$D_{q}^{α} {f (x) \cdot g (x)} = \sum_{l = 0}^{[α]} [\begin{matrix} α \\ l \end{matrix}] D_{q}^{α - l} f ({xq}^{l}) D_{q}^{l} g (x), α > 0 .$ (4.6)

A fractional q-Taylor’s formula was introduced in [43, 52].

In which $f (x) : C^{F} [a, b] \cap L^{F} [a, b]$ is a continuous fuzzy-valued function, $D_{q, a}^{α}$ denotes the fuzzy Caputo q-fractional derivative of order α, $D_{q, a}^{α} f (x) : (a, b] \to E^{1}$ , and $f : T_{q} \to E^{1}$ is an arbitrary fuzzy-valued function, $D_{q, a}^{α} f (x) : T_{q} \to E^{1}$ .

Theorem 4.10 Let us consider $f (x) \in C^{F} [a, b] \cap L^{F} [a, b]$ , and $D_{q, a}^{α} f (x) \in C^{F} (a, b] \cap L^{F} (a, b],$ where 0 ≤ λ ≤ 1, then

$\begin{matrix} f (x) = f (a) \oplus \frac{1}{Γ_{q} (α)} ⊙ (D_{q, a}^{α}) f (ζ) ⊙ (x - a)^{α}, \\ 0 < α \leq 1, \end{matrix}$ (4.7) where a ≤ ζ ≤ x, ∀ x ∈ (a, b] .

Proof. Whereas f is q-integral and $_{q}^{F} [(i) - gH]$ -differentiable, by using properties (4.1) and (4.3), we get for λ ∈ [0, 1] $(I_{q, a}^{α} D_{q, a}^{α} f) (x; λ) = [(I_{q, a}^{α} D_{q, a}^{α} \underline{f}) (x; λ), (I_{q, a}^{α} D_{q, a}^{α} \bar{f}) (x; λ)]$ so $(I_{q, a}^{α} D_{q, a}^{α} \underline{f}) (x; λ) = \frac{1}{Γ_{q} (α)} \int_{a}^{x} (x - t)^{α - 1} D_{q, a}^{α} \underline{f} (t; λ) d_{q} t,$ (4.8) $(I_{q, a}^{α} D_{q, a}^{α} \bar{f}) (x; λ) = \frac{1}{Γ_{q} (α)} \int_{a}^{x} (x - t)^{α - 1} D_{q, a}^{α} \bar{f} (t; λ) d_{q} t .$ (4.9) Taking (4.8) and (4.9), we obtain

$\begin{matrix} (I_{q, a}^{α} D_{q, a}^{α} \underline{f}) (x; λ) & = \frac{1}{Γ_{q} (α)} (D_{q, a}^{α} \underline{f}) (ζ; λ) \int_{a}^{x} (x - t)^{α - 1} d_{q} t \\ = \frac{1}{Γ_{q} (α)} (D_{q, a}^{α} \underline{f}) (ζ; λ) (x - a)^{α}, 0 \leq ζ \leq x . \end{matrix}$ (4.10) and

$\begin{matrix} (I_{q, a}^{α} D_{q, a}^{α} \bar{f}) (x; λ) & = \frac{1}{Γ_{q} (α)} (D_{q, a}^{α} \bar{f}) (ζ; λ) \int_{a}^{x} (x - t)^{α - 1} d_{q} t \\ = \frac{1}{Γ_{q} (α)} (D_{q, a}^{α} \bar{f}) (ζ; λ) (x - a)^{α}, 0 \leq ζ \leq x . \end{matrix}$ (4.11) Furthermore, using (4.5), we obtain

$(I_{q, a}^{α} D_{q, a}^{α} f) (x; λ) = [\underline{f} (x; λ) - \underline{f} (a; λ), \bar{f} (x; λ) - \bar{f} (a; λ)] 0 \leq λ \leq 1 .$ (4.12) Tanking (4.12) into (4.10), we obtain:

$f (x; λ) = f (a; λ) \oplus \frac{1}{Γ_{q} (α)} ⊙ (D_{q, a}^{α} f) (ζ; λ) ⊙ (x - a)^{α}, 0 \leq λ \geq 1,$ and under $_{q}^{F} [(ii) - gH]$ -differentiable, we have $(I_{q, a}^{α} D_{q, a}^{α} f) (x; λ) = [(I_{q, a}^{α} D_{q, a}^{α} \bar{f}) (x; λ), (I_{q, a}^{α} D_{q, a}^{α} \underline{f}) (x; λ)]$ so $(I_{q, a}^{α} D_{q, a}^{α} \bar{f}) (x; λ) = \frac{1}{Γ_{q} (α)} \int_{a}^{x} (x - t)^{α - 1} D_{q, a}^{α} \bar{f} (t; λ) d_{q} t,$ (4.13) $(I_{q, a}^{α} D_{q, a}^{α} \underline{f}) (x; λ) = \frac{1}{Γ_{q} (α)} \int_{a}^{x} (x - t)^{α - 1} D_{q, a}^{α} \underline{f} (t; λ) d_{q} t .$ (4.14) Taking (4.13) and (4.14), we obtain

$(I_{q, a}^{α} D_{q, a}^{α} f) (x; λ) = [\bar{f} (x; λ) - \bar{f} (a; λ), \underline{f} (x; λ) - \underline{f} (a; λ)] 0 \leq λ \leq 1 .$ (4.17) Putting (4.17) into (4.15), we obtain:

$f (x; λ) = f (a; λ) \oplus \frac{1}{Γ_{q} (α)} ⊙ (D_{q, a}^{α} f) (ζ; λ) ⊙ (x - a)^{α}, 0 \leq λ \geq 1,$ the proof is completed.□

Theorem 4.11 Suppose that $f (x) \in C^{F} [a, b] \cap L^{F} [a, b]$ and ${(D_{q, a}^{α})}^{n} f (x), {(D_{q, a}^{α})}^{n + 1} f (x) \in C^{F} (a, b] \cap L^{F} (a, b],$ for 0 < α ≤ 1, then

$I_{q, a}^{n α} (D_{q, a}^{α})^{n} f (x) \underset{gH}{⊖} I_{q, a}^{(n + 1) α} (D_{q, a}^{α})^{n + 1} f (x) = \frac{(x - α)^{n α}}{Γ_{q} (n α + 1)} ⊙ (D_{q, a}^{α})^{n} f (a^{+}),$ (4.18) where $(D_{q, a}^{α})^{n} = D_{q, a}^{α} D_{q, a}^{α} \dots D_{q, a}^{α}$ (n-times).

Proof. Whereas f is q-integral and $_{q}^{F} [(i) - gH]$ -differentiable, using properties (4.3), we have $\begin{matrix} I_{q, a}^{n α} (D_{q, a}^{α})^{n} \underline{f} (x; λ) - I_{q, a}^{α (n + 1)} (D_{q, a}^{α})^{n + 1} \underline{f} (x; λ) \\ = I_{q, a}^{n α} ((D_{q, a}^{α})^{n} \underline{f} (x; λ) - I_{q, a}^{α} (D_{q, a}^{α})^{n + 1} \underline{f} (x; λ)) \\ = I_{q, a}^{n α} (((D_{q, a}^{α})^{n} \underline{f} (x; λ)) - (I_{q, a}^{α} D_{q, a}^{α}) ((D_{q, a}^{α})^{n} \underline{f} (x; λ))) . \end{matrix}$ and

$\begin{matrix} I_{q, a}^{n α} (D_{q, a}^{α})^{n} \bar{f} (x; λ) - I_{q, a}^{α (n + 1)} (D_{q, a}^{α})^{n + 1} \bar{f} (x; λ) \\ = I_{q, a}^{n α} ((D_{q, a}^{α})^{n} \bar{f} (x; λ) - I_{q, a}^{α} (D_{q, a}^{α})^{n + 1} \bar{f} (x; λ)) \\ = I_{q, a}^{n α} (((D_{q, a}^{α})^{n} \bar{f} (x; λ)) - (I_{q, a}^{α} D_{q, a}^{α}) ((D_{q, a}^{α})^{n} \bar{f} (x; λ))) . \end{matrix}$ From (4.5), we obtain

$\begin{matrix} I_{q, a}^{n α} (D_{q, a}^{α})^{n} \underline{f} (x; λ) - I_{q, a}^{α (n + 1)} (D_{q, a}^{α})^{n + 1} \underline{f} (x; λ) = (I_{q, a}^{n α} (D_{q, a}^{α})^{n} \underline{f} (a^{+}; λ)), \\ I_{q, a}^{n α} (D_{q, a}^{α})^{n} \bar{f} (x; λ) - I_{q, a}^{α (n + 1)} (D_{q, a}^{α})^{n + 1} \bar{f} (x; λ) = (I_{q, a}^{n α} (D_{q, a}^{α})^{n} \bar{f} (a^{+}; λ)), \end{matrix}$ using (4.4), we can obtained

$I_{q, a}^{n α} (D_{q, a}^{α})^{n} f (x; λ) \underset{gH}{⊖} I_{q, a}^{(n + 1) α} (D_{q, a}^{α})^{n + 1} f (x; λ) = \frac{(x - α)^{n α}}{Γ_{q} (n α + 1)} ⊙ (D_{q, a}^{α})^{n} f (a^{+}; λ), 0 \leq λ \geq 1 .$ The proof for $_{q}^{F} [(ii) - gH]$ -differentiable of f is similar. Thus, the proof is completed.□

Theorem 4.12 Let $f (x) \in C^{F} [a, b] \cap L^{F} [a, b]$ be denotes of the fuzzy continuous Caputo fractional q-derivative of order (n + 1) α in some open interval I containing a, and define $E_{n} (x)$ for any x in I by

$f (x) = \sum_{ς = 0}^{n} \frac{{(D_{q, a}^{α})}^{ς} f (a^{+})}{Γ_{q} (ς α + 1)} ⊙ (x - a)^{ς α} \oplus E_{n} (x), 0 < α < 1,$ (4.19) Then $E_{n} (x)$ is given by the integral

$E_{n} (x) = I_{q, a}^{α (1 + n)} (D_{q, a}^{α})^{n + 1} f (x)$ (4.20)Proof. By induction on n . For n = 1, and $_{q}^{FC} [(i) - gH]$ -differentiable, we get for λ ∈ [0, 1]

$\begin{matrix} E_{n} (x; λ) & = \underline{f} (x; λ) - \underline{f} (a^{+}; λ) - \frac{(D_{q, a^{+}}^{α}) \underline{f} (a^{+}; λ)}{Γ_{q} (α + 1)} (x - a)^{α} \\ = \underline{f} (x; λ) - \underline{f} (a^{+}; λ) - \underset{gH}{⊖} (I_{q, a}^{α} (D_{q, a}^{α}) \underline{f}) (a^{+}; λ) . \end{matrix}$ and

$\begin{matrix} E_{n} (x; λ) & = \bar{f} (x; λ) - \bar{f} (a^{+}; λ) - \frac{(D_{q, a^{+}}^{α}) \bar{f} (a^{+}; λ)}{Γ_{q} (α + 1)} (x - a)^{α} \\ = \bar{f} (x; λ) - \bar{f} (a^{+}; λ) - (I_{q, a}^{α} (D_{q, a}^{α}) \bar{f}) (a^{+}; λ) . \end{matrix}$ Applying of Theorems $(I_{q, a}^{α} D_{q, a^{+}}^{α} f) (a; λ) = (I_{q, a}^{α} D_{q, a}^{α} f) (x; λ) \underset{gH}{⊖} (I_{q, a}^{2 α} D_{q, a}^{2 α} f) (x; λ),$

$\begin{matrix} {\underline{E}}_{n} (x; λ) & = \underline{f} (x; λ) - \underline{f} (a; λ) - (I_{q, a}^{α} D_{q, a}^{α} \underline{f}) (x; λ) + (I_{q, a}^{2 α} D_{q, a}^{2 α} \underline{f}) (x; λ) = (I_{q, a}^{2 α} D_{q, a}^{2 α} \underline{f}) (x; λ), \\ {\bar{E}}_{n} (x; λ) & = \bar{f} (x; λ) - \bar{f} (a; λ) - (I_{q, a}^{α} D_{q, a}^{α} \bar{f}) (x; λ) + (I_{q, a}^{2 α} D_{q, a}^{2 α} \bar{f}) (x; λ) = (I_{q, a}^{2 α} D_{q, a}^{2 α} \bar{f}) (x; λ), \end{matrix}$

Let of equation (4.20) is true for n and prove it for n + 1 . Applying equation (4.19), we get

$\begin{matrix} {\underline{E}}_{n + 1} (x; λ) & = {\underline{E}}_{n} (x; λ) - \frac{{(D_{q, a}^{α})}^{n + 1} \underline{f} (a^{+}; λ)}{Γ_{q} (n + 1) α + 1} (x - a)^{(n + 1) α}, 0 \leq λ \leq 1, \\ {\bar{E}}_{n + 1} (x; λ) & = {\bar{E}}_{n} (x; λ) - \frac{{(D_{q, a}^{α})}^{n + 1} \bar{f} (a^{+}; λ)}{Γ_{q} (n + 1) α + 1} (x - a)^{(n + 1) α}, 0 \leq λ \leq 1 . \end{matrix}$ Since, we get

$\begin{matrix} \frac{{(D_{q, a}^{α})}^{n + 1} f (a^{+}; λ)}{Γ_{q} (n + 1) α + 1} ⊙ (n - a)^{(n + 1) α} \\ = [I_{q, a}^{α (n + 1)} {(D_{q, a}^{α})}^{n + 1} f (x; λ) \underset{gH}{⊖} I_{q, a}^{(n + 2) α} {(D_{q, a}^{α})}^{n + 2} f (x; λ)] . \end{matrix}$ where

$E_{n + 1} (x; λ) = I_{q, a}^{(n + 2) α} {(D_{q, a}^{α})}^{n + 2} f (x, λ), 0 \leq λ \leq 1,$ and under $_{q}^{FC} [(ii) - gH]$ -differentiable, we have

$\begin{matrix} {\bar{E}}_{n} (x; λ) & = \bar{f} (x; λ) - \bar{f} (a^{+}; λ) - \frac{(D_{q, a^{+}}^{α}) \bar{f} (a^{+}; λ)}{Γ_{q} (α + 1)} (x - a)^{α} \\ = \bar{f} (x; λ) - \bar{f} (a^{+}; λ) - \underset{gH}{⊖} (I_{q, a}^{α} (D_{q, a}^{α}) \bar{f}) (a^{+}; λ) . \end{matrix}$ and

$\begin{matrix} {\underline{E}}_{n} (x; λ) & = \underline{f} (x; λ) - \underline{f} (a^{+}; λ) - \frac{(D_{q, a^{+}}^{α}) \underline{f} (a^{+}; λ)}{Γ_{q} (α + 1)} (x - a)^{α} \\ = \underline{f} (x; λ) - \underline{f} (a^{+}; λ) - (I_{q, a}^{α} (D_{q, a}^{α}) \underline{f}) (a^{+}; λ) . \end{matrix}$ Applying of Theorems $(I_{q, a}^{α} D_{q, a^{+}}^{α} f) (a; λ) = (I_{q, a}^{α} D_{q, a}^{α} f) (x; λ) \underset{gH}{⊖} (I_{q, a}^{2 α} D_{q, a}^{2 α} f) (x; λ),$

$\begin{matrix} {\bar{E}}_{n} (x; λ) & = \bar{f} (x; λ) - \bar{f} (a; λ) - (I_{q, a}^{α} D_{q, a}^{α} \bar{f}) (x; λ) + (I_{q, a}^{2 α} D_{q, a}^{2 α} \bar{f}) (x; λ) = (I_{q, a}^{2 α} D_{q, a}^{2 α} \bar{f}) (x; λ), \\ {\underline{E}}_{n} (x; λ) & = \underline{f} (x; λ) - \underline{f} (a; λ) - (I_{q, a}^{α} D_{q, a}^{α} \underline{f}) (x; λ) + (I_{q, a}^{2 α} D_{q, a}^{2 α} \underline{f}) (x; λ) = (I_{q, a}^{2 α} D_{q, a}^{2 α} \underline{f}) (x; λ), \end{matrix}$ Let of equation (4.20) is true for n and prove it for n + 1 . Applying equation (4.19), we have

$\begin{matrix} {\bar{E}}_{n + 1} (x; λ) & = {\bar{E}}_{n} (x; λ) - \frac{{(D_{q, a}^{α})}^{n + 1} \bar{f} (a^{+}; λ)}{Γ_{q} (n + 1) α + 1} (x - a)^{(n + 1) α}, 0 \leq λ \leq 1, \\ {\underline{E}}_{n + 1} (x; λ) & = {\underline{E}}_{n} (x; λ) - \frac{{(D_{q, a}^{α})}^{n + 1} \underline{f} (a^{+}; λ)}{Γ_{q} (n + 1) α + 1} (x - a)^{(n + 1) α}, 0 \leq λ \leq 1 . \end{matrix}$ Since, we obtain

$\begin{matrix} \frac{{(D_{q, a}^{α})}^{n + 1} f (a^{+}; λ)}{Γ_{q} (n + 1) α + 1} ⊙ (n - a)^{(n + 1) α} \\ = [(- 1) ⊙ (I_{q, a}^{α (n + 1)} {(D_{q, a}^{α})}^{n + 1} f (x; λ) \underset{gH}{⊖} I_{q, a}^{(n + 2) α} {(D_{q, a}^{α})}^{n + 2} f (x; λ))], 0 \leq λ \leq 1, \end{matrix}$ the proof is completed.□

4.1 Fuzzy one-dimensional fractional q-differential transform

We consider the one-dimensional fuzzy fractional q-differential transformation as follows:

Definition 4.13 Let us consider x (t) be a differentiable of order αl over time domain T, then $\begin{matrix} {\underline{X}}_{ς} (l; λ) & = {[\frac{d_{q}^{α l} \underline{x} (t; λ)}{d_{q} t^{α l}}]}_{t = t_{ς}}, \forall α l \in ℤ_{+}, \\ {\bar{X}}_{ς} (l; λ) & = {[\frac{d_{q}^{α l} \bar{x} (t; λ)}{d_{q} t^{α l}}]}_{t = t_{ς}}, \forall α l \in ℤ_{+}, \end{matrix}}$ (4.21) when x (t) is (i)-differentiable and $\begin{matrix} {\underline{X}}_{ς} (l; λ) & = \frac{d_{q}^{α l}}{d_{q} t^{α l}} \bar{x} (t; λ) |_{t = t_{ς}}, α l is odd, \\ {\bar{X}}_{ς} (l; λ) & = \frac{d_{q}^{α l}}{d_{q} t^{α l}} \underline{x} (t; λ) |_{t = t_{ς}}, α l is odd, \end{matrix}}$ (4.22) and $\begin{matrix} {\underline{X}}_{ς} (l; λ) & = \frac{d_{q}^{α l}}{d_{q} t^{α l}} \underline{x} (t; λ) |_{t = t_{ς}}, α l is even, \\ {\bar{X}}_{ς} (l; λ) & = \frac{d_{q}^{α l}}{d_{q} t^{α l}} \bar{x} (t; λ) |_{t = t_{ς}}, α l is even, \end{matrix}}$ (4.23) when x (t) is (ii)-differentiable.

Notice that ${\underline{X}}_{ς} (l; λ)$ and ${\bar{X}}_{ς} (l; λ)$ are called the lower and upper spectrum of x (t) at t = t_ς, respectively.

Thus, if x (t) be (i)-differentiable, then x (t) can be represented as: $\underline{x} (t; λ) = \sum_{l = 0}^{\infty} \frac{(t - t_{ς})^{α l}}{Γ_{q} (α l + 1)} \underline{X} (l; λ), α l \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.24) $\bar{x} (t; λ) = \sum_{l = 0}^{\infty} \frac{(t - t_{ς})^{α l}}{Γ_{q} (α l + 1)} \bar{X} (l; λ), α l \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.25) and if x (t) be (ii)-differentiable, then x (t) can be represented as: $\begin{matrix} \underline{x} (t; λ) = & \sum_{l = 1, odd}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \bar{X} (l, λ) + \sum_{l = 0, even}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \underline{X} (l; λ), 0 \leq λ \leq 1, \end{matrix}$ (4.26) and $\begin{matrix} \bar{x} (t; λ) = & \sum_{l = 1, odd}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \underline{X} (l, λ) + \sum_{l = 0, even}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \bar{X} (l; λ), 0 \leq λ \leq 1, \end{matrix}$ (4.27) The above equations are known as the inverse transformation of X (l ; λ). If X (l ; λ) is defined as follows: $\begin{matrix} \underline{X} (l; λ) & = P (l) {[\frac{d_{q}^{α l} \underline{x (t; λ)}}{d_{q} t^{α l}}]}_{t = t_{0}}, \forall α l \in ℤ_{+}, \\ \bar{X} (l; λ) & = P (l) {[\frac{d_{q}^{α l} \bar{x (t; λ)}}{d_{q} t^{α l}}]}_{t = t_{0}}, \forall α l \in ℤ_{+}, \end{matrix}}$ (4.28) when x (t) (i)-differentiable and $\begin{matrix} \underline{X} (l; λ) & = P (l) {[\frac{d_{q}^{α l}}{d_{q} t^{α l^{*}}} (\bar{x (t; λ)})]}_{t = t_{0}}, α l is odd \\ \bar{X} (l; λ) & = P (l) {[\frac{d_{q}^{α l}}{d_{q} t^{α l^{*}}} (\underline{x (t; λ)})]}_{t = t_{0}}, α l is odd \end{matrix}}$ (4.29) and $\begin{matrix} \underline{X} (l; λ) & = P (l) {[\frac{d_{q}^{α l}}{d_{q} t^{α l^{*}}} (\underline{x (t; λ)})]}_{t = t_{0}}, α l is even \\ \bar{X} (l; λ) & = P (l) {[\frac{d_{q}^{α l}}{d_{q} t^{α l^{*}}} (\bar{x (t; λ)})]}_{t = t_{0}}, α l is even \end{matrix}}$ (4.30) when x (t) is (ii)-differentiable.

Then, the function x (t) can be proposed as: $\underline{x} (t; λ) = \sum_{l = 0}^{\infty} \frac{(t - t_{ς})^{α l}}{Γ_{q} (α l + 1)} \frac{\underline{X} (l; λ)}{P (l)}, α l \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.31) $\bar{x} (t; λ) = \sum_{l = 0}^{\infty} \frac{(t - t_{ς})^{α l}}{Γ_{q} (α l + 1)} \frac{\bar{X} (l; λ)}{P (l)}, α l \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.32) when x (t) are (i)-differentiable and if x (t) can be (ii)-differentiable we obtain $\begin{matrix} \underline{x} (t; λ) = & [\sum_{l = 1, odd}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \frac{\bar{X} (l; λ)}{P (l)} + \sum_{l = 0, even}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \frac{\underline{X} (l; λ)}{P (l)}], 0 \leq λ \leq 1, \end{matrix}$ (4.33) and $\begin{matrix} \bar{x} (t; λ) = & [\sum_{l = 1, odd}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \frac{\underline{X} (l; λ)}{P (l)} + \sum_{l = 0, even}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \frac{\bar{X} (l; λ)}{P (l)}], 0 \leq λ \leq 1, \end{matrix}$ (4.34) where P (l) >0, P (l) is called the weighting factor.

In this work $P (l) = \frac{H^{α l}}{Γ_{q} (α l + 1)}$ is applied where H is the time horizon on interest. Consequently, if x (t) be (i)-differentiable, then $\underline{X} (l; λ) = \frac{H^{α l}}{Γ_{q} (α l + 1)} \frac{d_{q}^{α l} \underline{x} (t; λ)}{d_{q} t^{α l}}, α l \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.35) $\bar{X} (l; λ) = \frac{H^{α l}}{Γ_{q} (α l + 1)} \frac{d_{q}^{α l} \bar{x} (t; λ)}{d_{q} t^{α l}}, α l \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.36) and if x (t) be (ii)-differentiable, then

$\begin{matrix} \underline{X} (l; λ) & = \frac{H^{α l}}{Γ_{q} (α l + 1)} \frac{d_{q}^{α l} \bar{x} (t; λ)}{d_{q} t^{α l^{*}}}, α l is odd \\ \bar{X} (l; λ) & = \frac{H^{α l}}{Γ_{q} (α l + 1)} \frac{d_{q}^{α l} \underline{x} (t; λ)}{d_{q} t^{α l^{*}}}, α l is odd \end{matrix}}$ (4.37) and

$\begin{matrix} \underline{X} (l; λ) & = \frac{H^{α l}}{Γ_{q} (α l + 1)} \frac{d_{q}^{α l} \underline{x} (t; λ)}{d_{q} t^{α l^{*}}}, α l is even \\ \bar{X} (l; λ) & = \frac{H^{α l}}{Γ_{q} (α l + 1)} \frac{d_{q}^{α l} \bar{x} (t; λ)}{d_{q} t^{α l^{*}}}, α l is even \end{matrix}}$ (4.38) Unitizing the fuzzy fractional q-differential transform, a fuzzy fractional partial q-differential equation in the domain of interest can be transformed to an algebraic equation in the domain $ℤ_{+}$ and x (t) can be obtained as the finite-term q-Taylor series plus a reminder as:

$\begin{matrix} \underline{x} (t; λ) & = \sum_{l = 0}^{n} \frac{(t - t_{0})^{α l}}{Γ_{q} (α l + 1)} \frac{\underline{X} (l; λ)}{P (l)} + R_{(n + 1)} (t) \\ = \sum_{l = 0}^{n} {(\frac{t - t_{0}}{H})}^{α L} \frac{\underline{X} (l; λ)}{P (l)} + R_{(n + 1)} (t), α l \in ℤ_{+}, 0 \leq λ \leq 1, \end{matrix}$ (4.39) and

$\begin{matrix} \bar{x} (t; λ) & = \sum_{l = 0}^{n} \frac{(t - t_{0})^{α l}}{Γ_{q} (α l + 1)} \frac{\bar{X} (l; λ)}{P (l)} + R_{(n + 1)} (t) \\ = \sum_{l = 0}^{n} {(\frac{t - t_{0}}{H})}^{α l} \frac{\bar{X} (l; λ)}{P (l)} + R_{(n + 1)} (t), α l \in ℤ_{+}, 0 \leq λ \leq 1, \end{matrix}$ (4.40) when x (t) is (i)-differentiable and $\begin{matrix} \underline{x} (t; λ) = & [\sum_{l = 1, odd}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \frac{\bar{X} (l; λ)}{P (l)} + \sum_{l = 0, even}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \frac{\underline{X} (l; λ)}{P (l)}] + R_{(n + 1)} (t), 0 \leq λ \leq 1, \end{matrix}$ (4.41) and $\begin{matrix} \bar{x} (t; λ) = & [\sum_{l = 1, odd}^{\infty} \frac{{(t - t_{ς})}^{α l}}{Γ_{q} (α l + 1)} \frac{\underline{X} (l; λ)}{P (l)} + \sum_{l = 0, even}^{\infty} \frac{{(t - t_{ι})}^{α l}}{Γ_{q} (α l + 1)} \frac{\bar{X} (l; λ)}{P (l)}] + R_{(n + 1)} (t), 0 \leq λ \leq 1, \end{matrix}$ (4.42) when x (t) is (ii)-differentiable.

The objective of this section is to find the solution of fuzzy fractional partial q-differential equations at the equally spaced grid points [t₀, t₁, . . . , t_n] where t_ς = a + il^* for each (ς = 0, 1, 2, . . . , n), and $l^{*} = \frac{b - a}{n}$ . That is, the domain of interest is divided to is n subdomain, and the fuzzy approximation functions in each subdomain are x_ς (t ; α) for ς = 0, 1, 2, . . , n - 1, respectively.

Taking the initial conditions, we obtain

$\underline{X} (0; λ) = \underline{x} (0; λ), \bar{X} (0; λ) = \bar{x} (0; λ), 0 \leq λ \leq 1 .$

In the first subdomain, $\underline{x} (t; λ)$ and $\bar{x} (t; λ)$ can be depict by $\underline{x} (0; λ) = {\underline{x}}_{0} (λ)$ and $\bar{x} (0; λ) = {\bar{x}}_{0} (λ),$ respectively. They can be represented in terms of their n-th order q-Taylor series with respect to t₀, that is

$\underline{x} (t_{0}; λ) = {\underline{X}}_{0} (0; λ) + {\underline{X}}_{0} (1; λ) (t - t_{0}) + {\underline{X}}_{0} (2; λ) (t - t_{0})^{2} + . . . + {\underline{X}}_{0} (n; λ) (t - t_{0})^{n},$ and

$\bar{x} (t_{0}; λ) = {\bar{X}}_{0} (0; λ) + {\bar{X}}_{0} (1; λ) (t - t_{0}) + {\bar{X}}_{0} (2; λ) (t - t_{0})^{2} + . . . + {\bar{X}}_{0} (n; λ) (t - t_{0})^{n},$ Using q-Taylor series for x (t_ς, λ), we get

$\begin{matrix} \underline{x} (t_{0}; λ) & = {\underline{X}}_{0} (0; λ) + {\underline{X}}_{0} (1; λ) (t - t_{0}) + {\underline{X}}_{0} (2; λ) (t - t_{0})^{2} + . . . + {\underline{X}}_{0} (n; λ) (t - t_{0})^{n} \\ = \sum_{ι = 0}^{n} X_{0} (ι; λ) l^{* ι}, \end{matrix}$ and

$\begin{matrix} \bar{x} (t_{0}; λ) & = {\bar{X}}_{0} (0; λ) + {\bar{X}}_{0} (1; λ) (t - t_{0}) + {\bar{X}}_{0} (2; λ) (t - t_{0})^{2} + . . . + {\bar{X}}_{0} (n; λ) (t - t_{0})^{n} \\ = \sum_{ι = 0}^{n} X_{0} (ι; λ) l^{* ι} . \end{matrix}$ The last value x₀ (t₁) of the first subdomian is the initial value of the second subdomian, i.e., x₁ (t₁ ; λ) = X₁ (0) = x₀ (t₁ ; λ) . In the similar manner x (t₂ ; λ) can be obtained as follows:

$\begin{matrix} \underline{x} (t_{2}; λ) \approx {\underline{x}}_{1} (t_{2}; λ) & = {\underline{X}}_{1} (0; λ) + {\underline{X}}_{1} (1; λ) (t_{2} - t_{1}) + {\underline{X}}_{1} (2; λ) (t_{2} - t_{1})^{2} \\ + . . . + {\underline{X}}_{1} (n; α) (t_{2} - t_{1})^{n} \\ = \sum_{ι = 0}^{n} X_{1} (ι; λ) l^{* ι}, \end{matrix}$ and

$\begin{matrix} \bar{x} (t_{2}; λ) \approx {\bar{x}}_{1} (t_{2}; λ) & = {\bar{X}}_{1} (0; λ) + {\bar{X}}_{1} (1; λ) (t_{2} - t_{1}) + {\bar{X}}_{1} (2; λ) (t_{2} - t_{1})^{2} \\ + . . . + {\bar{X}}_{1} (n; α) (t_{2} - t_{1})^{n} \\ = \sum_{ι = 0}^{n} X_{1} (ι; λ) l^{* ι} . \end{matrix}$ Thus, the solution on the grid points t_ς+1 can be represented as

$\begin{matrix} \underline{x} (t_{ς + 1}; λ) \approx {\underline{x}}_{ς} (t_{ς + 1}; λ) & = {\underline{X}}_{ς} (0; λ) + {\underline{X}}_{ς} (1; λ) (t_{ς + 1} - t_{ς}) + {\underline{X}}_{ς} (2; λ) (t_{ς + 2} - t_{ς})^{2} \\ + . . . + {\underline{X}}_{ς} (n; λ) (t_{ς + 1} - t_{ς})^{n} \\ = \sum_{ι = 0}^{n} {\underline{X}}_{ς} (ι; λ) l^{* ι}, \end{matrix}$ and

$\begin{matrix} \bar{x} (t_{ς + 1}; λ) \approx {\bar{x}}_{ς} (t_{ς + 1}; λ) & = {\bar{X}}_{ς} (0; λ) + {\bar{X}}_{ς} (1; λ) (t_{ς + 1} - t_{ς}) + {\bar{X}}_{ς} (2; λ) (t_{ς + 2} - t_{ς})^{2} \\ + . . . + {\bar{X}}_{ς} (n; λ) (t_{ς + 1} - t_{ς})^{n} \\ = \sum_{ι = 0}^{n} {\bar{X}}_{ς} (ι; λ) l^{* ι} . \end{matrix}$

4.1.1 The properties of fuzzy fractional q-differential transform

We present the properties of fuzzy one-dimensional fractional q-differential transform as follows.

Theorem 4.14 Let us consider u (t) and v (t) are fuzzy-valued functions and their fuzzy q-DTM denoted by U_α,q (l) and V_α,q (l), respectively. Then

If f (x) = u (t) ⊕ v (x) , then $F_{α, q} (l) = U_{α, q} (l) \oplus V_{α, q} (l), l \in ℤ_{+}$

If $f (t) = u (x) \underset{gH}{⊖} v (x),$ then $F_{α, q} (l) = U_{α, q} (l) \underset{gH}{⊖} V_{α, q} (l), l \in ℤ_{+}$

If f (x) = λ ⊙ u (t) , then $F_{α, q} (l) = λ ⊙ U_{α, q} (l), l \in ℤ_{+},$

provided the generalized Hukuhara difference (gH-difference) exists.

Proof. According to Definition 4.13, the proof is obvious.

Theorem 4.15 Let us consider the fuzzy-valued function $u \in E^{1}$ and if $f (t) = \frac{d_{q}^{μ} v (x)}{d_{q} x^{μ}}$ , then we obtain $F_{α, q} (l) = \frac{Γ_{q} (α l + μ + 1)}{Γ_{q} (α l + 1)} V_{α, q} (l + \frac{μ}{α})$ , where F_α,q (l) and V_α,q (l) are the fuzzy fractional one-dimensional q-differential transformations of f and v, respectively.

Proof. Using definition 4.13, we obtain for 0 ≤ λ ≤ 1 $f (t; λ) = \frac{d_{q}^{μ} v (x)}{d_{q} x^{μ}} = {[\frac{d_{q}^{μ} \underline{v} (x; λ)}{d_{q} x^{μ}}, \frac{d_{q}^{μ} \bar{v} (x; λ)}{d_{q} x^{μ}}]}_{x = x_{0}},$ then

$\begin{matrix} F_{α, q} (l; λ) & = \frac{1}{Γ_{q} (α l + 1)} {[\frac{d_{q}^{α l}}{d_{q} x^{α l}} [\frac{d_{q}^{μ} \underline{v} (x; λ)}{d_{q} x^{μ}}, \frac{d_{q}^{μ} \bar{v} (x; λ)}{d_{q} x^{μ}}]]}_{x = x_{0}} \\ = \frac{1}{Γ_{q} (α l + 1)} {[\frac{d_{q}^{α l + μ} \underline{v} (x; λ)}{d_{q} x^{α l + μ}}, \frac{d_{q}^{α l + μ} \bar{v} (x; λ)}{d_{q} x^{α l + μ}}]}_{x = x_{0}} \\ = \frac{[α l + 1]_{q} [α l + 2]_{q} [α l + 3]_{q} \dots [α l + μ]_{q}}{[α l + 1]_{q} [α l + 2]_{q} [α l + 3]_{q} \dots [α l + μ]_{q} Γ_{q} (α l + 1)} {[\frac{d_{q}^{α l + μ} \underline{v} (x; λ)}{d_{q} x^{α l + μ}}, \frac{d_{q}^{α l + μ} \bar{v} (x; λ)}{d_{q} x^{α l + μ}}]}_{x = x_{0}} \\ = \frac{[α l + 1]_{q} [α l + 2]_{q} [α l + 3]_{q} \dots [α l + μ]_{q}}{Γ_{q} (α (l + \frac{μ}{α}) + 1)} {[\frac{d_{q}^{α (l + \frac{μ}{α})} \underline{v} (x; λ)}{d_{q} x^{α (l + \frac{μ}{α})}}, \frac{d_{q}^{α (l + \frac{μ}{α})} \bar{v} (x; λ)}{d_{q} x^{α (l + \frac{μ}{α})}}]}_{x = x_{0}} . \end{matrix}$ Hence

$F_{α, q} (l; λ) = \frac{Γ_{q} (α l + μ + 1)}{Γ_{q} (α l + 1)} V_{α, q} (l + \frac{μ}{α}; λ), 0 \leq λ \leq 1,$ the proof is completed.□

Theorem 4.16 Let us consider U_α,q (l) and V_α,q (l) are the one-dimensional fractional q-differential transformations of u (x) and v (x) are positive real-valued function, respectively. If f (x) = u (x) · v (x) , x₀ = 0, then under (i)-differentiability of f we have ${\underline{F}}_{α, q} (l; λ) = \sum_{j = 0}^{l} a_{α ς, α (l - ς)} (q) {\underline{U}}_{α, q} (ς; λ) {\underline{V}}_{α, q} (l - ς; λ), 0 \leq λ \leq 1,$ (4.43) ${\bar{F}}_{α, q} (l; λ) = \sum_{j = 0}^{l} a_{α ς, α (l - ς)} (q) {\bar{U}}_{α, q} (ς; λ) {\bar{V}}_{α, q} (l - ς; λ), 0 \leq λ \leq 1,$ (4.44) and (ii)-differentiability of f as:

$\begin{matrix} {\underline{F}}_{α, q} (l; λ) & = \sum_{j = 0, even}^{l} a_{α ς, α (l - ς)} (q) {\underline{U}}_{α, q} (ς; λ) {\underline{V}}_{α, q} (l - ς; λ) + \sum_{j = 1, odd}^{l} a_{α ς, α (l - ς)} (q) {\bar{U}}_{α, q} (ς; λ) {\bar{V}}_{α, q} (l - ς; λ), \\ {\bar{F}}_{α, q} (l; λ) & = \sum_{j = 0, even}^{l} a_{α ς, α (l - ς)} (q) {\bar{U}}_{α, q} (ς; λ) {\bar{V}}_{α, q} (l - ς; λ) + \sum_{j = 1, odd}^{l} a_{α ς, α (l - ς)} (q) {\underline{U}}_{α, q} (ς; λ) {\underline{V}}_{α, q} (l - ς; λ) . \end{matrix}$

Proof. Using definition of one-dimensional fuzzy fractional q-DTM and under (i)-differentiability of f we have for 0 ≤ λ ≤ 1:

$F_{α, q} (l, λ) = \frac{1}{Γ_{q} (α l + 1)} [\frac{d_{q}^{α l}}{d_{q} x^{α l}} (\underline{u} (x) \cdot \underline{v} (x)), \frac{d_{q}^{α l}}{d_{q} x^{α l}} (\bar{u} (x) \cdot \bar{v} (x))] .$ (4.45) Taking the fractional q-Leibniz rule (4.6), one obtain

$\begin{matrix} F_{α, q} (l; λ) & = \frac{1}{Γ_{q} (α l + 1)} {[\sum_{ι = 0}^{[α l]} {[\begin{matrix} [α l] \\ ς \end{matrix}]}_{q} [D_{q}^{ς} \underline{v} (x; λ), D_{q}^{ς} \bar{v} (x; λ)] [D_{q}^{α l - 1} \underline{u} (q^{ς} x; λ); D_{q}^{α l - 1} \bar{u} (q^{ς} x; λ)]]}_{x = 0} \\ = \frac{1}{Γ_{q} (α l + 1)} \\ \times {[\sum_{ι = 0}^{[α l]} \frac{Γ_{q} (α l + 1)}{Γ_{q} (ς + 1) Γ_{q} (α l - ς + 1)} [D_{q}^{ς} \underline{v} (x; λ), D_{q}^{i} \bar{v} (x; λ)] [D_{q}^{α l - 1} \underline{u} (q^{ς} x; λ); D_{q}^{α l - 1} \bar{u} (q^{ς} x; λ)]]}_{x = 0} \\ = \sum_{ι = 0}^{[α l]} \frac{1}{Γ_{q} (α (\frac{ς}{α}) + 1)} {[D_{q}^{α (\frac{ς}{α})} \underline{v} (x; λ); D_{q}^{α (\frac{ς}{α})} \bar{v} (x; λ)]}_{x = 0} \\ \times \frac{1}{Γ_{q} (α (l - \frac{ς}{α}) + 1)} {[D_{q}^{α (l - \frac{ς}{α})} \underline{u} (q^{ς} x; λ); D_{q}^{α (l - \frac{ς}{α})} \bar{u} (q^{ς} x; λ)]}_{x = 0} . \end{matrix}$ Taking the relation (3.27), we obtain

$\begin{matrix} F_{α, q} (l; λ) & = \sum_{ι = 0}^{[α l]} \frac{1}{Γ_{q} (α (\frac{ς}{α}) + 1)} {[D_{q}^{α (\frac{ς}{α})} \underline{v} (t; λ); D_{q}^{α (\frac{ς}{α})} \bar{v} (x; λ)]}_{x = 0} \\ \times \frac{a_{α l - ς, ς} (q)}{Γ_{q} (α (l - \frac{ς}{α}) + 1)} {[D_{q}^{α (l - \frac{ς}{α})} \underline{u} (x; λ); D_{q}^{α (l - \frac{ς}{α})} \bar{u} (q^{ς} t; λ)]}_{x = 0} \end{matrix}$ we get

$\begin{matrix} {\underline{F}}_{α, q} (l; λ) & = \sum_{ι = 0}^{[α l]} a_{α l - ς, ς} (q) {\underline{V}}_{α, q} (\frac{ς}{α}) {\underline{U}}_{α, q} (l - \frac{ς}{α}), l - \frac{ς}{α} \geq 0, \\ {\bar{F}}_{α, q} (l; λ) & = \sum_{ι = 0}^{[α l]} a_{α l - ς, ς} (q) {\bar{V}}_{α, q} (\frac{ι}{α}) {\bar{U}}_{α, q} (l - \frac{ς}{α}), l - \frac{ς}{α} \geq 0, \end{matrix}$ Let $ɛ = l - \frac{ς}{α}$

$\begin{matrix} {\underline{F}}_{α, q} (l; λ) & = \sum_{ɛ = 0}^{l} a_{α ɛ, α (l - ɛ)} (q) {\underline{V}}_{α, q} (l - ɛ; λ) {\underline{U}}_{α, q} (ɛ; λ), \\ {\bar{F}}_{α, q} (l; λ) & = \sum_{ɛ = 0}^{l} a_{α ɛ, α (l - ɛ)} (q) {\bar{V}}_{α, q} (l - ɛ; λ) {\bar{U}}_{α, q} (ɛ; λ) . \end{matrix}$ Thus, we obtain

$\begin{matrix} {\underline{F}}_{α, q} (l; λ) & = \sum_{j = 0}^{l} a_{α ς, α (l - ς)} (q) {\underline{U}}_{α, q} (ς; λ) {\underline{V}}_{α, q} (l - ς; λ), 0 \leq λ \leq 1, \\ {\bar{F}}_{α, q} (l; λ) & = \sum_{j = 0}^{l} a_{α ς, α (l - ς)} (q) {\bar{U}}_{α, q} (ς; λ) {\bar{V}}_{α, q} (l - ς; λ), 0 \leq λ \leq 1 . \end{matrix}$ Assume that f is (ii)-differentiability, using similar techniques, can be obtained the result. The proof completed.

Theorem 4.17 Let us consider $f \in E^{1}$ and $f (x) = (x - x_{0})_{q}^{μ}, μ \in ℤ_{+}$ , then $F_{α, q} (l) = δ (l - \frac{μ}{α})$ where F_α,q (l) is the fuzzy fractional q-differential transformations of f, for

$δ (l - \frac{μ}{α}) = {\begin{matrix} 1, & l = \frac{μ}{α}, \\ 0, & l \neq \frac{μ}{α} . \end{matrix}$ (4.46)

Proof. Using the definition of fuzzy one-dimensional fractional q-differential transform, for λ ∈ [0, 1] ,

$F_{α, q} (l; λ) = \frac{1}{Γ_{q} (α l + 1)} {[\frac{d_{q}^{α l}}{d_{q} x^{α l}} (x - x_{0})_{q}^{μ} (λ)]}_{x = x_{0}} .$ According to the previous lemma, we obtain

$F_{α, q} (l; λ) = \frac{1}{Γ_{q} (α l + 1)} \frac{Γ_{q} (μ + 1)}{Γ_{q} (μ - α l + 1)} {[(x - x_{0})_{q}^{μ - α l} (λ)]}_{x = x_{0}} .$

Using (4.46), we obtained the proof as

$F_{α, q} (l; λ) = δ (l - \frac{μ}{α}) (λ), μ \in Z_{+}, 0 \leq λ \leq 1,$ the proof is completed.□

Theorem 4.18 Assume that $u \in E^{1}$ and if $w (x) = \int_{x_{0}}^{x} u (t) d_{q} t$ , then $W_{α, q} (l) = \frac{1}{[α l]_{q}} U_{α, q} (l - \frac{1}{α})$ , where $l \geq \frac{1}{α}$ for W_α,q (l) and U_α,q (l) are fuzzy one-dimensional fractional q-DTM of fuzzy-valued functions w and u, respectively.

Proof. Using Definition 4.13, we obtain

$\begin{matrix} w (x; λ) = \int_{x_{0}}^{x} u (t; λ) d_{q} t & = [\int_{x_{0}}^{x} \underline{u} (t; λ) d_{q} t, \int_{x_{0}}^{x} \bar{u} (t; λ) d_{q} t] \\ = [\int_{x_{0}}^{x} \sum_{l = 0}^{\infty} {\underline{U}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l} d_{q} t, \int_{x_{0}}^{x} \sum_{l = 0}^{\infty} {\bar{U}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l} d_{q} t] \\ = [\sum_{l = 0}^{\infty} \frac{{\underline{U}}_{α, q} (l; λ)}{[α l + 1]_{q}} (t - x_{0})_{q}^{α l + 1} |_{x_{0}}^{x}, \sum_{l = 0}^{\infty} \frac{{\bar{U}}_{α, q} (l; λ)}{[α l + 1]_{q}} (t - x_{0})_{q}^{α l + 1} |_{x_{0}}^{x}] \\ = [\sum_{l = 0}^{\infty} \frac{{\underline{U}}_{α, q} (l; λ)}{[α l + 1]_{q}} (x - x_{0})_{q}^{α l + 1}, \sum_{l = 0}^{\infty} \frac{{\bar{U}}_{α, q} (l; λ)}{[α l + 1]_{q}} (x - x_{0})_{q}^{α l + 1}] . \end{matrix}$

Then, we assume that $l = l - \frac{1}{α},$ we get

$w (x; λ) = [\sum_{l = \frac{1}{α}}^{\infty} \frac{{\underline{U}}_{α, q} (l - \frac{1}{α}; λ)}{[α l]_{q}} (x - x_{0})_{q}^{α l}, \sum_{l = \frac{1}{α}}^{\infty} \frac{{\bar{U}}_{α, q} (l - \frac{1}{α}; λ)}{[α l]_{q}} (x - x_{0})_{q}^{α l}] .$ Thus, via definition 4.13, we get

$W_{α, q} (l; λ) = \frac{1}{[α l]_{q}} U_{α, q} (l - \frac{1}{α}; λ), l \geq \frac{1}{α}, 0 \leq λ \leq 1,$ the proof completed.□

Theorem 4.19 Suppose that $u \in E^{1}$ and if $w (x) = \int_{x_{0}}^{x} \int_{x_{0}}^{x_{n - 1}} \dots \int_{x_{0}}^{x_{2}} \int_{x_{0}}^{x_{1}} u (t) d_{q} {td}_{q} x_{1} d_{q} x_{2} \dots d_{q} x_{n - 1}$ , then $W_{α, q} (l) = \frac{Γ_{q} (α l - n + 1)}{Γ_{q} (α l + 1)} U_{α, q} (l - \frac{n}{α})$ , where $l \geq \frac{n}{α}$ for W_α,q (l) and U_α,q (l) are fuzzy fractional q-DTM of fuzzy-valued functions w and u, respectively.

Proof. Taking definition 4.13, we obtain

$\begin{matrix} \underline{w} (x; λ) & = \int_{x_{0}}^{x} \int_{x_{0}}^{x_{n - 1}} \dots \int_{x_{0}}^{x_{2}} \int_{x_{0}}^{x_{1}} \sum_{l = 0}^{\infty} {\underline{U}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l} d_{q} {td}_{q} x_{1} d_{q} x_{2} \dots d_{q} x_{n - 1} \\ = \sum_{l = 0}^{\infty} \int_{x_{0}}^{x} \int_{x_{0}}^{x_{n - 1}} \dots \int_{x_{0}}^{x_{2}} \int_{x_{0}}^{x_{1}} {\underline{U}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l} d_{q} {td}_{q} x_{1} d_{q} x_{2} \dots d_{q} x_{n - 1} \\ = \sum_{l = 0}^{\infty} \int_{x_{0}}^{x} \int_{x_{0}}^{x_{n - 1}} \dots \int_{x_{0}}^{x_{2}} \frac{{\underline{U}}_{α, q} (l; λ)}{[α l + 1]_{q}} (x_{1} - x_{0})_{q}^{α l + 1} d_{q} x_{1} d_{q} x_{2} \dots d_{q} x_{n - 1} \\ ⋮ \end{matrix}$ and

$\begin{matrix} \bar{w} (x; λ) & = \int_{x_{0}}^{x} \int_{x_{0}}^{x_{n - 1}} \dots \int_{x_{0}}^{x_{2}} \int_{x_{0}}^{x_{1}} \sum_{l = 0}^{\infty} {\bar{U}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l} d_{q} {td}_{q} x_{1} d_{q} x_{2} \dots d_{q} x_{n - 1} \\ = \sum_{l = 0}^{\infty} \int_{x_{0}}^{x} \int_{x_{0}}^{x_{n - 1}} \dots \int_{x_{0}}^{x_{2}} \int_{x_{0}}^{x_{1}} {\bar{U}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l} d_{q} {td}_{q} x_{1} d_{q} x_{2} \dots d_{q} x_{n - 1} \\ = \sum_{l = 0}^{\infty} \int_{x_{0}}^{x} \int_{x_{0}}^{x_{n - 1}} \dots \int_{x_{0}}^{x_{2}} \frac{{\bar{U}}_{α, q} (l; λ)}{[α l + 1]_{q}} (x_{1} - x_{0})_{q}^{α l + 1} d_{q} x_{1} d_{q} x_{2} \dots d_{q} x_{n - 1} \\ ⋮ . \end{matrix}$ Consequently, we obtain

$\begin{matrix} \underline{w} (x; λ) & = \sum_{l = 0}^{\infty} \frac{Γ_{q} (α l + 1)}{Γ_{q} (α l + n + 1)} {\underline{U}}_{α, q} (l; λ) (x - x_{0})_{q}^{α l + n}, 0 \leq λ \leq 1, \\ \bar{w} (x; λ) & = \sum_{l = 0}^{\infty} \frac{Γ_{q} (α l + 1)}{Γ_{q} (α l + n + 1)} {\bar{U}}_{α, q} (l; λ) (x - x_{0})_{q}^{α l + n}, 0 \leq λ \leq 1 . \end{matrix}$ Then, we put $l = l - \frac{n}{α},$ we get

$\begin{matrix} \underline{w} (l; λ) & = \sum_{l = \frac{n}{α}}^{\infty} \frac{Γ_{q} (α l - n + 1)}{Γ_{q} (α l + 1)} {\underline{U}}_{α, q} (l - \frac{n}{α}; λ) (x - x_{0})_{q}^{α l} \\ \bar{w} (l; λ) & = \sum_{l = \frac{n}{α}}^{\infty} \frac{Γ_{q} (α l - n + 1)}{Γ_{q} (α l + 1)} {\bar{U}}_{α, q} (l - \frac{n}{α}; λ) (x - x_{0})_{q}^{α l} \end{matrix}$ Thus, by the definition of fuzzy fractional q-DTM, we obtain

$\begin{matrix} {\underline{W}}_{α, q} (l; λ) & = \frac{Γ_{q} (α l - n + 1)}{Γ_{q} (α l + 1)} {\underline{U}}_{α, q} (l - \frac{n}{α}; λ), l \geq \frac{n}{α}, 0 \leq λ \leq 1, \\ {\bar{W}}_{α, q} (l; λ) & = \frac{Γ_{q} (α l - n + 1)}{Γ_{q} (α l + 1)} {\bar{U}}_{α, q} (l - \frac{n}{α}; λ), l \geq \frac{n}{α}, 0 \leq λ \leq 1, \end{matrix}$ the proof is completed.

Theorem 4.20 Let us consider U_α,q (l) and V_α,q (l) are the one-dimensional fractional q-differential transformations of u (t) and v (t) are positive real-valued function, respectively. If $w (x) = \int_{x_{0}}^{x} u (t) v (t) d_{q} (t),$ then under (i)-differentiability of w we have ${\underline{W}}_{α, q} (l; λ) = \frac{1}{[α l]_{q}} \sum_{ς = 0}^{[l - \frac{1}{α}]} a_{α ς, α (l - ς - \frac{1}{α})} (q) U_{α, q} (ς) \cdot {\underline{V}}_{α, q} (l - ς - \frac{1}{α}; λ), 0 \leq λ \leq 1$ (4.47) ${\bar{W}}_{α, q} (l; λ) = \frac{1}{[α l]_{q}} \sum_{ς = 0}^{[l - \frac{1}{α}]} a_{α ς, α (l - ς - \frac{1}{α})} (q) U_{α, q} (ς) \cdot {\bar{V}}_{α, q} (l - ς - \frac{1}{α}; λ), 0 \leq λ \leq 1$ (4.48) and (ii)-differentiability of f as:

$\begin{matrix} {\underline{W}}_{α, q} (l; λ) & = \frac{1}{[α l]_{q}} \sum_{ς = 0, even}^{[l - \frac{1}{α}]} a_{α ς, α (l - ς - \frac{1}{α})} (q) U_{α, q} (ς) \cdot {\underline{V}}_{α, q} (l - ς - \frac{1}{α}; λ) \\ + \frac{1}{[α l]_{q}} \sum_{ς = 0, odd}^{[l - \frac{1}{α}]} a_{α ς, α (l - ς - \frac{1}{α})} (q) U_{α, q} (ς) \cdot {\bar{V}}_{α, q} (l - ς - \frac{1}{α}; λ), 0 \leq λ \leq 1 \end{matrix}$ (4.49) and

$\begin{matrix} {\bar{W}}_{α, q} (l; λ) & = \frac{1}{[α l]_{q}} \sum_{ς = 0, even}^{[l - \frac{1}{α}]} a_{α ς, α (l - ς - \frac{1}{α})} (q) U_{α, q} (ς) \cdot {\bar{V}}_{α, q} (l - ς - \frac{1}{α}; λ) \\ + \frac{1}{[α l]_{q}} \sum_{ς = 0, odd}^{[l - \frac{1}{α}]} a_{α ς, α (l - ς - \frac{1}{α})} (q) U_{α, q} (ς) \cdot {\underline{V}}_{α, q} (l - ς - \frac{1}{α}; λ), 0 \leq λ \leq 1 . \end{matrix}$ (4.50)

Proof. Taking definition one-dimensional fuzzy fractional q-DTM, Theorem (4.18), and under (i)-differentiability for 0 ≤ λ ≤ 1, we obtain

$\begin{matrix} \underline{w} (x; λ) & = \int_{x_{0}}^{x} \sum_{l = 0}^{\infty} U_{α, q} (l) \cdot {\underline{V}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l} d_{q} t \\ = \sum_{l = 0}^{\infty} U_{α, q} (l) \cdot \frac{{\underline{V}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l + 1}}{[α l + 1]_{q}} ∣_{x_{0}}^{x} \\ = \sum_{l = 0}^{\infty} U_{α, q} (l) \cdot \frac{{\underline{V}}_{α, q} (l; λ) (x - x_{0})_{q}^{α l + 1}}{[α l + 1]_{q}}, \end{matrix}$ and

$\begin{matrix} \bar{w} (x; λ) & = \int_{x_{0}}^{x} \sum_{l = 0}^{\infty} U_{α, q} (l) \cdot {\bar{V}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l} d_{q} t \\ = \sum_{l = 0}^{\infty} U_{α, q} (l) \cdot \frac{{\bar{V}}_{α, q} (l; λ) (t - x_{0})_{q}^{α l + 1}}{[α l + 1]_{q}} ∣_{x_{0}}^{x} \\ = \sum_{l = 0}^{\infty} U_{α, q} (l) \cdot \frac{{\bar{V}}_{α, q} (l; λ) (x - x_{0})_{q}^{α l + 1}}{[α l + 1]_{q}} . \end{matrix}$ put $l = l - \frac{1}{α}$ , we obtain

$\begin{matrix} \underline{w} (x; λ) = \sum_{l = \frac{1}{α}}^{\infty} U_{α, q} (l - \frac{1}{α}) \cdot \frac{{\underline{V}}_{α, q} (l - \frac{1}{α}; λ)}{[α l]_{q}} (x - x_{0})_{q}^{α l}, \\ \bar{w} (x; λ) = \sum_{l = \frac{1}{α}}^{\infty} U_{α, q} (l - \frac{1}{α}) \cdot \frac{{\bar{V}}_{α, q} (l - \frac{1}{α}; λ)}{[α l]_{q}} (x - x_{0})_{q}^{α l} . \end{matrix}$ Thus,

$\begin{matrix} {\underline{W}}_{α, q} (l; λ) = \frac{1}{[α l]_{q}} U_{α, q} (l - \frac{1}{α}) \cdot {\underline{V}}_{α, q} (l - \frac{1}{α}; λ), l \geq \frac{1}{α}, 0 \leq λ \leq 1, \\ {\bar{W}}_{α, q} (l; λ) = \frac{1}{[α l]_{q}} U_{α, q} (l - \frac{1}{α}) \cdot {\bar{V}}_{α, q} (l - \frac{1}{α}; λ), l \geq \frac{1}{α}, 0 \leq λ \leq 1 . \end{matrix}$ Applying Theorem (4.16), we obtain

$\begin{matrix} {\underline{W}}_{α, q} (l; λ) & = \frac{1}{[α l]_{q}} \sum_{ς = 0}^{[l - \frac{1}{α}]} a_{α ς, α (l - ς - \frac{1}{α})} (q) U_{α, q} (ς) \cdot {\underline{V}}_{α, q} (l - ς - \frac{1}{α}; λ), 0 \leq λ \leq 1 \\ {\bar{W}}_{α, q} (l; λ) & = \frac{1}{[α l]_{q}} \sum_{ς = 0}^{[l - \frac{1}{α}]} a_{α ς, α (l - ς - \frac{1}{α})} (q) U_{α, q} (ς) \cdot {\bar{V}}_{α, q} (l - ς - \frac{1}{α}; λ), 0 \leq λ \leq 1 . \end{matrix}$ At the same time, suppose that considering w is (ii)-differentiability, then by the similar way, one easily gets a result. The proof is completed.

Theorem 4.21 Let us consider U_α,q (l) and V_α,q (l) are the one-dimensional fractional q-differential transformations of u (t) and v (t) are positive real-valued function, respectively. If $w (x) = u (x) \int_{x_{0}}^{x} {vd}_{q} (t)$ then under (i)-differentiability of w we have ${\underline{W}}_{α, q} (l; λ) = \sum_{ς \geq \frac{1}{α}}^{l} \frac{1}{[α i]_{q}} a_{α (ς - \frac{1}{α}), α (l - ς)} (q) V_{α, q} (ς - \frac{1}{α}) {\underline{U}}_{α, q} (l - ς; λ), 0 \leq λ \leq 1,$ (4.51) ${\bar{W}}_{α, q} (l; λ) = \sum_{ς \geq \frac{1}{α}}^{l} \frac{1}{[α i]_{q}} a_{α (ς - \frac{1}{α}), α (l - ς)} (q) V_{α, q} (ς - \frac{1}{α}) {\bar{U}}_{α, q} (l - ς; λ), 0 \leq λ \leq 1,$ (4.52) and (ii)-differentiability of f as:

$\begin{matrix} {\underline{W}}_{α, q} (l; λ) & = \sum_{ς \geq \frac{1}{α}, even}^{l} \frac{1}{[α i]_{q}} a_{α (ς - \frac{1}{α}), α (l - ς)} (q) V_{α, q} (ς - \frac{1}{α}) {\underline{U}}_{α, q} (l - ς; λ) \\ + \sum_{ς = 1, odd}^{l} \frac{1}{[α i]_{q}} a_{α (ς - \frac{1}{α}), α (l - ς)} (q) V_{α, q} (ς - \frac{1}{α}) {\bar{U}}_{α, q} (l - ς; λ), 0 \leq λ \leq 1, \end{matrix}$ (4.53) and

$\begin{matrix} {\bar{W}}_{α, q} (l; λ) & = \sum_{ς \geq \frac{1}{α}, even}^{l} \frac{1}{[α i]_{q}} a_{α (ς - \frac{1}{α}), α (l - ς)} (q) V_{α, q} (ς - \frac{1}{α}) {\bar{U}}_{α, q} (l - ς; λ) \\ + \sum_{ς = 1, odd}^{l} \frac{1}{[α i]_{q}} a_{α (ς - \frac{1}{α}), α (l - ς)} (q) V_{α, q} (ς - \frac{1}{α}) {\underline{U}}_{α, q} (l - ς; λ), 0 \leq λ \leq 1 . \end{matrix}$ (4.54)

Proof. The proof was removed because it is the same as the previous above.

Remark 4.22 [42]. Let x, t be fuzzy numbers, and x ≥ 0 (it means $\underline{x} (λ) \geq 0$ and $\bar{x} (λ) \geq 0)$ . Then $(x ⊙ t) (λ) = [\underline{x} (λ) \underline{t} (λ), \bar{x} (λ) \bar{t} (λ)] .$

Assumption 4.23 We only discuss the case of u ≥ 0 satisfying $\frac{d_{q} u}{d_{q} x} \geq 0$ . In order to simplify the presentation of our results, we only consider as follows a case ${[u ⊙ \frac{d_{q} u}{d_{q} x}]}_{λ} = {[\underline{u} (x) \frac{d_{q} \underline{u}}{d_{q} x}, \bar{u} (x) \frac{d_{q} \bar{u}}{d_{q} x}]}_{λ} .$ (4.55)

Next, we propose some examples illustrating the method is very effective, convenient for solving some kinds of one-dimensional fuzzy fractional q-differential equations.

Example 4.1. We consider the following fuzzy nonlinear fractional q-differential equation

$u ⊙ \frac{d_{q} u}{d_{q} x} \underset{gH}{⊖} \frac{d_{q}^{2} u}{d_{q} x^{2}} \oplus \frac{d_{q}^{0.5} u}{d_{q} x^{0.5}} = 0, 0 < q < 1,$ (4.56) subject to

$u (0) = \tilde{0}, \frac{d_{q} u (x)}{d_{q} x} |_{x = 0} = [(0.2 + 0.6 λ)^{n}, (1.5 - 0.7 λ)^{n}] ⊙ 0.5,$ (4.57) where u is a fuzzy-valued function satisfying u (x) ≥0 (see [42]), for n = 1, 2, 3, ⋯ .

The parametric form of (4.56) is $\underline{u} \frac{d_{q} \underline{u}}{d_{q} x} - \frac{d_{q}^{2} \underline{u}}{d_{q} x^{2}} + \frac{d_{q}^{0.5} \underline{u}}{d_{q} x^{0.5}} = \tilde{0}, 0 < q < 1,$ (4.58) $\bar{u} \frac{d_{q} \bar{u}}{d_{q} x} - \frac{d_{q}^{2} \bar{u}}{d_{q} x^{2}} + \frac{d_{q}^{0.5} \bar{u}}{d_{q} x^{0.5}} = \tilde{0}, 0 < q < 1,$ (4.59) for λ ∈ [0, 1] , where $\underline{u}$ stands for $\underline{u} (x; λ),$ and similarly for $\bar{u} .$

Applying the one-dimensional fuzzy fractional q-differential transform of (4.58)and (4.59), we get $\begin{matrix} {\underline{U}}_{0.5, q} (l + 4; λ) & = \frac{1}{{[\frac{l}{2} + 1]}_{q} {[\frac{l}{2} + 2]}_{q}} [\frac{Γ_{q} (\frac{l}{2} + \frac{3}{2})}{Γ_{q} (\frac{l}{2} + 1)} {\underline{U}}_{0.5, q} (l + 1; λ) + \sum_{ς = 0}^{l} {[\frac{l - ς}{2} + 1]}_{q} U_{0.5, q} (ς) {\underline{U}}_{0.5, q} (l - ς + 2; λ)], \end{matrix}$ (4.60) $\begin{matrix} {\bar{U}}_{0.5, q} (l + 4; λ) & = \frac{1}{{[\frac{l}{2} + 1]}_{q} {[\frac{l}{2} + 2]}_{q}} [\frac{Γ_{q} (\frac{l}{2} + \frac{3}{2})}{Γ_{q} (\frac{l}{2} + 1)} {\bar{U}}_{0.5, q} (l + 1; λ) + \sum_{ς = 0}^{l} {[\frac{l - ς}{2} + 1]}_{q} U_{0.5, q} (ς) {\bar{U}}_{0.5, q} (l - ς + 2; λ)] . \end{matrix}$ (4.61) Using the initial conditions (4.57), we obtain ${\underline{U}}_{0.5, q} (0; λ) = {\underline{U}}_{0.5, q} (1; λ) = \tilde{0}, {\underline{U}}_{0.5, q} (2; λ) = (0.2 + 0.6 λ)^{n} 0.5, {\underline{U}}_{0.5, q} (3; λ) = \tilde{0},$ (4.62) ${\bar{U}}_{0.5, q} (0; λ) = {\bar{U}}_{0.5, q} (1; λ) = \tilde{0}, {\bar{U}}_{0.5, q} (2; λ) = (1.5 - 0.7 λ)^{n} 0.5, {\bar{U}}_{0.5, q} (3; λ) = \tilde{0},$ (4.63) Taking (4.63) into (4.60) and by recursive method, we obtain ${\underline{U}}_{0.5, q} (4; λ) = \tilde{0}, {\underline{U}}_{0.5, q} (5; λ) = \frac{(0.2 + 0.6 λ)^{n}}{2 {[\frac{5}{2}]}_{q} Γ_{q} (\frac{5}{2})}, {\underline{U}}_{0.5, q} (6; λ) = \frac{(0.2 + 0.6 λ)^{n}}{4 {[3]}_{q}!}, \dots,$ (4.64) ${\bar{U}}_{0.5, q} (4; λ) = \tilde{0}, {\bar{U}}_{0.5, q} (5; λ) = \frac{(1.5 - 0.7 λ)^{n}}{2 {[\frac{5}{2}]}_{q} Γ_{q} (\frac{5}{2})}, {\bar{U}}_{0.5, q} (6; λ) = \frac{(1.5 - 0.7 λ)^{n}}{4 {[3]}_{q}!}, \dots .$ (4.65) From all U_0.5,q (l ; λ), we can be obtained the solution as follows:

$u (x; λ) = [(0.2 + 0.6 λ)^{n}, (1.5 - 0.7 λ)^{n}] ⊙ [\frac{x}{2} + \frac{x^{2.5}}{2 [2.5]_{q} Γ_{q} (2.5)} + \frac{x^{3}}{4 [3]_{q}!} + \dots], 0 \leq λ \leq 1 .$

Example 4.2. We consider the following fuzzy fractional q-differential equation

$\frac{d_{q}^{α} u}{d_{q} t^{α}} = ρ ⊙ u, 0 \leq α \leq 1, 0 < q < 1, ρ \in R, t > 0,$ (4.66) subject to

$u (0) = [(0.4 + 0.2 λ)^{n}, (0.8 - 0.2 λ)^{n}] ⊙ u_{0},$ (4.67) where ρ ≥ 0, (n = 1, 2, 3, ⋯) .

Applying the one-dimensional fuzzy fractional q-DTM of (4.66), and assume α = βφ, and $_{q}^{FC} [(i) - gH]$ -differentiable, we get for λ ∈ [0, 1] ${\underline{U}}_{β, q} (l + φ; λ) = \frac{ρ Γ_{q} (β l + 1)}{Γ_{q} (β l + α + 1)} {\underline{U}}_{β, q} (l; λ),$ (4.68) ${\bar{U}}_{β, q} (l + φ; λ) = \frac{ρ Γ_{q} (β l + 1)}{Γ_{q} (β l + α + 1)} {\bar{U}}_{β, q} (l; λ) .$ (4.69) Using the initial condition (4.67), we obtain

${\begin{matrix} {\underline{U}}_{β, q} (0; λ) & = (0.4 + 0.2 λ)^{n} {\underline{u}}_{0} (t; λ), \\ {\underline{U}}_{β, q} (l; λ) & = \tilde{0}, l = 1, 2, . . ., φ - 1, \end{matrix}$ (4.70) and

${\begin{matrix} {\bar{U}}_{β, q} (0; λ) & = (0.8 - 0.2 λ)^{n} {\bar{u}}_{0} (t; λ), \\ {\bar{U}}_{β, q} (l; λ) & = \tilde{0}, l = 1, 2, . . ., φ - 1 . \end{matrix}$ (4.71) Taking (4.71) into (4.68), and using the recursive method, we obtain

$\begin{matrix} {\underline{U}}_{β, q} (φ + 1; λ) = {\underline{U}}_{β, q} (φ + 2; λ) = \dots = \tilde{0} \\ {\underline{U}}_{β, q} (μ φ; λ) = (0.4 + 0.2 λ)^{n} [\frac{ρ^{μ} {\underline{u}}_{0} (t; λ)}{Γ_{q} (μ α + 1)}], μ = 1, 2, . . . \end{matrix}$ and

$\begin{matrix} {\bar{U}}_{β, q} (φ + 1; λ) = {\bar{U}}_{β, q} (φ + 2; λ) = \dots = \tilde{0} \\ {\bar{U}}_{β, q} (μ φ; λ) = (0.8 - 0.2 λ)^{n} [\frac{ρ^{μ} {\bar{u}}_{0} (t; λ)}{Γ_{q} (μ α + 1)}], μ = 1, 2, . . . \end{matrix}$ Thus, we can be obtained the series solutions as follows:

$\begin{matrix} u (t; λ) & = [(0.4 + 0.2 λ)^{n}, (0.8 - 0.2 λ)^{n}] ⊙ [u_{0} [1 + \frac{ρ}{Γ_{q} (α + 1)} t^{α} + \frac{ρ^{2}}{Γ_{q} (2 α + 1)} t^{2 α} + \dots + \frac{ρ^{μ}}{Γ_{q} (μ α + 1)} t^{μ α} + \dots]], 0 \leq λ \leq 1 . \end{matrix}$ and under $_{q}^{F} [(ii) - gH]$ -differentiable, we have ${\bar{U}}_{β, q} (l + φ; λ) = \frac{ρ Γ_{q} (β l + 1)}{Γ_{q} (β l + α + 1)} {\bar{U}}_{β, q} (l; λ),$ (4.72) ${\underline{U}}_{β, q} (l + φ; λ) = \frac{ρ Γ_{q} (β l + 1)}{Γ_{q} (β l + α + 1)} {\underline{U}}_{β, q} (l; λ) .$ (4.73) Using the initial condition (4.67), we obtain

${\begin{matrix} {\bar{U}}_{β, q} (0; λ) & = (0.8 - 0.2 λ)^{n} {\bar{u}}_{0} (t; λ), \\ {\bar{U}}_{β, q} (l; λ) & = \tilde{0}, l = 1, 2, . . ., φ - 1, \end{matrix}$ (4.74) and

${\begin{matrix} {\underline{U}}_{β, q} (0; λ) & = (0.4 + 0.2 λ)^{n} {\underline{u}}_{0} (t; λ), \\ {\underline{U}}_{β, q} (l; λ) & = \tilde{0}, l = 1, 2, . . ., φ - 1 . \end{matrix}$ (4.75) Taking (4.75) into (4.72), and using the recursive method, we obtain

$\begin{matrix} u (t; λ) & = (- 1) ⊙ [(0.8 - 0.2 λ)^{n}, (0.4 + 0.2 λ)^{n}] ⊙ [u_{0} [1 + \frac{ρ}{Γ_{q} (α + 1)} t^{α} + \frac{ρ^{2}}{Γ_{q} (2 α + 1)} t^{2 α} + \dots + \frac{ρ^{μ}}{Γ_{q} (μ α + 1)} t^{μ α} + \dots]], 0 \leq λ \leq 1 . \end{matrix}$

Example 4.3. We consider the following fuzzy Bagley-Torvic equation

$ϒ_{1} ⊙ \frac{d_{q}^{2} u}{d_{q} x^{2}} \oplus ϒ_{2} ⊙ \frac{d_{q}^{1.5} u}{d_{q} x^{1.5}} \oplus ϒ_{3} ⊙ u = g (x), 0 < q < 1, ϒ_{1}, ϒ_{2}, ϒ_{3} \geq 0$ (4.76) subject to the initial conditions

$u (0) = [(1.5 λ - 0.5)^{n}, (1.5 - 0.5 λ)^{n}] \underset{gH}{⊖} 1, u^{'} (0) = [(1.5 λ - 0.5)^{n}, (1.5 - 0.5 λ)^{n}] \underset{gH}{⊖} 1,$ (4.77) where (n = 1, 2, 3, ⋯) .

Applying the one-dimensional fuzzy fractional q-DTM of equation (4.76), we get $\frac{Γ_{q} (\frac{l}{2} + 3)}{Γ_{q} (\frac{l}{2} + 1)} {\underline{U}}_{0.5, q} (l + 4; λ) + \frac{ϒ_{2}}{ϒ_{1}} \frac{Γ_{q} (\frac{l}{2} + \frac{5}{2})}{Γ_{q} (\frac{l}{2} + 1)} {\underline{U}}_{0.5, q} (l + 3; λ) + \frac{ϒ_{3}}{ϒ_{1}} {\underline{U}}_{0.5, q} (l; λ) = \frac{{\underline{G}}_{0.5, q} (l; λ)}{ϒ_{1}},$ (4.78) $\frac{Γ_{q} (\frac{l}{2} + 3)}{Γ_{q} (\frac{l}{2} + 1)} {\bar{U}}_{0.5, q} (l + 4; λ) + \frac{ϒ_{2}}{ϒ_{1}} \frac{Γ_{q} (\frac{l}{2} + \frac{5}{2})}{Γ_{q} (\frac{l}{2} + 1)} {\bar{U}}_{0.5, q} (l + 3; λ) + \frac{ϒ_{3}}{ϒ_{1}} {\bar{U}}_{0.5, q} (l; λ) = \frac{{\bar{G}}_{0.5, q} (l; λ)}{ϒ_{1}} .$ (4.79) Taking the initial condition (4.77), we obtain

${\begin{matrix} {\underline{U}}_{0.5, q} (0; λ) & = (1.5 λ - 0.5)^{n} - 1, \\ {\underline{U}}_{0.5, q} (1; λ) & = (1.5 λ - 0.5)^{n}, \\ {\underline{U}}_{0.5, q} (2; λ) & = (1.5 λ - 0.5)^{n} - 1, \\ {\underline{U}}_{0.5, q} (3; λ) & = (1.5 λ - 0.5)^{n}, \end{matrix}$ (4.80) and

${\begin{matrix} {\bar{U}}_{0.5, q} (0; λ) & = (1.5 - 0.5 λ)^{n} - 1, \\ {\bar{U}}_{0.5, q} (1; λ) & = (1.5 - 0.5 λ)^{n}, \\ {\bar{U}}_{0.5, q} (2; λ) & = (1.5 - 0.5 λ)^{n} - 1, \\ {\bar{U}}_{0.5, q} (3; λ) & = (1.5 - 0.5 λ)^{n} . \end{matrix}$ (4.81) From (4.81) into (4.78), and taking the recursive method, we obtain

$\begin{matrix} {\underline{U}}_{0.5, q} (4; λ) & = (1.5 λ - 0.5)^{n} - [\frac{1}{ϒ_{1} Γ_{q} (3)} [{\underline{G}}_{0.5, q} (0; λ) - ϒ_{3}]], \\ {\underline{U}}_{0.5, q} (5; λ) & = (1.5 λ - 0.5)^{n} - [\frac{1}{ϒ_{1} Γ_{q} (\frac{7}{2})} [Γ_{q} (\frac{3}{2}) {\underline{G}}_{0.5, q} (1; λ) - \frac{ϒ_{2}}{ϒ_{1}} ({\underline{U}}_{0.5, q} (0; λ) - ϒ_{3})]], \\ ⋮ \end{matrix}$ and

$\begin{matrix} {\bar{U}}_{0.5, q} (4; λ) & = (1.5 - 0.5 λ)^{n} - [\frac{1}{ϒ_{1} Γ_{q} (3)} [{\bar{G}}_{0.5, q} (0; λ) - ϒ_{3}]], \\ {\bar{U}}_{0.5, q} (5; λ) & = (1.5 - 0.5 λ)^{n} - [\frac{1}{ϒ_{1} Γ_{q} (\frac{7}{2})} [Γ_{q} (\frac{3}{2}) {\bar{G}}_{0.5, q} (1; λ) - \frac{ϒ_{2}}{ϒ_{1}} ({\bar{U}}_{0.5, q} (0; λ) - ϒ_{3})]], \\ ⋮ . \end{matrix}$ Thus, we can be obtained the series solution as follows:

$\begin{matrix} u (x; λ) & = [(1.5 λ - 0.5)^{n}, (1.5 - 0.5 λ)^{n}] \underset{gH}{⊖} [1 + x + \frac{x^{2}}{ϒ_{1} Γ_{q} (3)} [G_{0.5, q} (0; λ) - ϒ_{3}] + \frac{x^{\frac{5}{2}}}{ϒ_{1} Γ_{q} (\frac{7}{2})} [Γ_{q} (\frac{3}{2}) G_{0.5, q} (1; λ) - \frac{ϒ_{2}}{ϒ_{1}} (U_{0.5, q} (0; λ) - ϒ_{3})] + \dots], 0 \leq λ \leq 1 . \end{matrix}$

Example 4.4. We consider the following fuzzy fractional q-integro-differential equation

$\frac{d_{q}^{\frac{3}{4}} u (x)}{d_{q} x^{\frac{3}{4}}} = {(0.3, 0.6, 0, 9)}^{n} \underset{gH}{⊖} \frac{1}{5} x^{2} e_{q}^{x} ⊙ u (x) \oplus \frac{6 x^{\frac{9}{4}}}{Γ_{q} (3.25)} \oplus e_{q}^{x} \int_{0}^{x} t ⊙ u (t) d_{q} t, 0 < q < 1, t > 0,$ (4.82) with the initial condition

$u (0) = {(0.3, 0.6, 0, 9)}^{n},$ (4.83) where n = 1, 2, 3, ⋯.

Applying the one-dimensional fuzzy fractional q-differential transform of (4.82), we get

$\begin{matrix} {\underline{U}}_{\frac{1}{4}, q} (l + 3; λ) & = \frac{Γ_{q} (\frac{l}{4} + 1)}{Γ_{q} (\frac{l}{4} + \frac{7}{4})} [(0.3 + 0.3 λ)^{n} - \frac{1}{5} \sum_{l_{2} = 0}^{l} \sum_{l_{1} = 0}^{l_{2}} a_{\frac{l_{1}}{4}, (\frac{l_{2} - l_{1}}{4})} (q), a_{\frac{l_{2}}{4}, (\frac{l - l_{2}}{4})} (q) \frac{δ (l_{1} - 8) {\underline{U}}_{\frac{1}{4}, q} (l - l_{2}; λ)}{Γ_{q} (\frac{l_{2} - l_{1}}{4} + 1)} + \frac{6 δ (l - 9)}{Γ_{q} (\frac{13}{4})} + \sum_{l_{1} = 0}^{l} \frac{1}{{[\frac{l_{1}}{4}]}_{q}!} \sum_{l_{2} = 0}^{l_{1} - 4} a_{\frac{l_{2}}{4}, (\frac{l_{1} - l_{2} - 4}{4})} (q) \frac{δ (l_{2} - 4) {\underline{U}}_{\frac{1}{4}, q} (l_{1} - l_{2} - 4; λ)}{Γ_{q} (\frac{l - l_{1}}{4} + 1)}], \end{matrix}$ (4.84)

and $\begin{matrix} {\bar{U}}_{\frac{1}{4}, q} (l + 3; λ) & = \frac{Γ_{q} (\frac{l}{4} + 1)}{Γ_{q} (\frac{l}{4} + \frac{7}{4})} [(0.9 - 0.3 λ)^{n} - \frac{1}{5} \sum_{l_{2} = 0}^{l} \sum_{l_{1} = 0}^{l_{2}} a_{\frac{l_{1}}{4}, (\frac{l_{2} - l_{1}}{4})} (q), a_{\frac{l_{2}}{4}, (\frac{l - l_{2}}{4})} (q) \frac{δ (l_{1} - 8) {\bar{U}}_{\frac{1}{4}, q} (l - l_{2}; λ)}{Γ_{q} (\frac{l_{2} - l_{1}}{4} + 1)} + \frac{6 δ (l - 9)}{Γ_{q} (\frac{13}{4})} + \sum_{l_{1} = 0}^{l} \frac{1}{{[\frac{l_{1}}{4}]}_{q}!} \sum_{l_{2} = 0}^{l_{1} - 4} a_{\frac{l_{2}}{4}, (\frac{l_{1} - l_{2} - 4}{4})} (q) \frac{δ (l_{2} - 4) {\bar{U}}_{\frac{1}{4}, q} (l_{1} - l_{2} - 4; λ)}{Γ_{q} (\frac{l - l_{1}}{4} + 1)}] . \end{matrix}$ (4.85)

Taking the initial conditions (4.83), we obtain ${\underline{U}}_{\frac{1}{4}, q} (0; λ) = {\underline{U}}_{\frac{1}{4}, q} (1; λ) = {\underline{U}}_{\frac{1}{4}, q} (2; λ) = (0.3 + 0.3 λ)^{n},$ (4.86) ${\bar{U}}_{\frac{1}{4}, q} (0; λ) = {\bar{U}}_{\frac{1}{4}, q} (1; λ) = {\bar{U}}_{\frac{1}{4}, q} (2; λ) = (0.9 - 0.3 λ)^{n} .$ (4.87)

According to (4.87) into (4.84), with using recursive method, we get ${\underline{U}}_{\frac{1}{4}, q} (l; λ) = {\begin{matrix} (0.3 + 0.3 λ)^{n} + \frac{6}{[3]_{q}!}, & l = 12, \\ 0 & l \neq 12, \end{matrix}$ (4.88) and ${\bar{U}}_{\frac{1}{4}, q} (l; λ) = {\begin{matrix} (0.9 - 0.3 λ)^{n} + \frac{6}{[3]_{q}!}, & l = 12, \\ 0 & l \neq 12 . \end{matrix}$ (4.89)

Thus, we can be obtained the solution as:

$u (x; λ) = [(0.3 + 0.3 λ)^{n}, (0.9 - 0.3 λ)^{n}] \oplus [\frac{6}{[3]_{q}!} x^{3}], 0 \leq λ \leq 1 .$

4.2 Fuzzy two-dimensional fractional q-differential transform

In this part, we consider the following two-dimensional fuzzy fractional q-differential transform for solving fuzzy fractional partial q-differential equations.

Definition 4.24 Let us consider x (ϑ, t) be a differentiable of order α (l + l^*) over time domain T, then $\begin{matrix} {\underline{X}}_{ς ι} (l, l^{*}; λ) & = {[\frac{\partial_{q}^{α (l + l^{*})} \underline{x} (ϑ, t; λ)}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}]}_{ϑ = ϑ_{ς}, t = t_{ι}}, α l, α l^{*} \in ℤ_{+}, 0, α l, α l^{*} \notin ℤ_{+}, \\ {\bar{X}}_{ς ι} (l, l^{*}; λ) & = {[\frac{\partial_{q}^{α (l + l^{*})} \bar{x} (ϑ, t; λ)}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}]}_{ϑ = ϑ_{ς}, t = t_{ι}}, α l, α l^{*} \in ℤ_{+}, 0, α l, α l^{*} \notin ℤ_{+}, \end{matrix}}$ (4.90) when x (ϑ, t) is (i)-differentiable and $\begin{matrix} {\underline{X}}_{ς ι} (l, l^{*}; λ) & = \frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}} \bar{x} (ϑ, t; λ), (α l is odd, α l^{*} is even) or inverse, \\ {\bar{X}}_{ς ι} (l, l^{*}; λ) & = \frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}} \underline{x} (ϑ, t; λ), (α l is odd, α l^{*} is even) or inverse, \end{matrix}}$ (4.91) and $\begin{matrix} {\underline{X}}_{ς ι} (l, l^{*}; λ) & = \frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}} \underline{x} (ϑ, t; λ), both of l, l^{*} are even or odd, \\ {\bar{X}}_{ς ι} (l, l^{*}; λ) & = \frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}} \bar{x} (ϑ, t; λ), both of l, l^{*} are even or odd, \end{matrix}}$ (4.92) when x (ϑ, t) is (ii)-differentiable.

Notice that ${\underline{X}}_{ς ι} (l, l^{*}; λ)$ and ${\bar{X}}_{ς ι} (l, l^{*}; λ)$ are called the lower and upper spectrum of x (ϑ, t) at ϑ = ϑ_ς and t = t_ι, respectively.

Thus, if x (ϑ, t) be (i)-differentiable, then x (ϑ, t) can be represented as: $\underline{x} (ϑ, t; λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{(ϑ - ϑ_{ς})^{α l} (t - t_{ι})^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \underline{X} (l, l^{*}; λ), α l, α l^{*} \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.93) $\bar{x} (ϑ, t; λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{(ϑ - ϑ_{ς})^{α l} (t - t_{ι})^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \bar{X} (l, l^{*}; λ), α l, α l^{*} \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.94) and if x (ϑ, t) be (ii)-differentiable, then x (ϑ, t) can be represented as: $\begin{matrix} \underline{x} (ϑ, t, λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{α l} {(t - t_{ι})}^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \bar{X} (l, l^{*}; λ) \\ (α l is odd, α l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{α l} {(t - t_{ι})}^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \underline{X} (l, l^{*}; λ), 0 \leq λ \leq 1, \\ both of α l, α l^{*} are even or odd, \end{matrix}$ (4.95) and $\begin{matrix} \bar{x} (ϑ, t, λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{α l} {(t - t_{ι})}^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \underline{X} (l, l^{*}; λ) \\ (α l is odd, α l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{α l} {(t - t_{ι})}^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \bar{X} (l, l^{*}; λ), 0 \leq λ \leq 1, \\ both of α l, α l^{*} are even or odd . \end{matrix}$ (4.96) The above equations are known as the inverse transformation of X (l, l^* ; λ). If X (l, l^* ; λ) is defined as follows: $\begin{matrix} \underline{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{α (l + l^{*})} \underline{x (ϑ, t; λ)}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}]}_{ϑ = ϑ_{0}, t = t_{0}}, \forall α l, α l^{*} \in ℤ_{+}, \\ \bar{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{α (l + l^{*})} \bar{x (ϑ, t; λ)}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}]}_{ϑ = ϑ_{0}, t = t_{0}}, \forall α l, α l^{*} \in ℤ_{+}, \end{matrix}}$ (4.97) when x (ϑ, t) (i)-differentiable and $\begin{matrix} \underline{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}} (\bar{x (ϑ, t; λ)})]}_{ϑ = ϑ_{0}, t = t_{0}}, (α l is odd, α l^{*} is even) or inverse \\ \bar{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}} (\underline{x (ϑ, t; λ)})]}_{ϑ = ϑ_{0}, t = t_{0}}, (α l is odd, α l^{*} is even) or inverse \end{matrix}}$ (4.98) and $\begin{matrix} \underline{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}} (\underline{x (ϑ, t; λ)})]}_{ϑ = ϑ_{0}, t = t_{0}}, both of α l, α l^{*} are even or odd \\ \bar{X} (l, l^{*}; λ) & = P (l, l^{*}) {[\frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} α^{α l} \partial_{q} t^{α l^{*}}} (\bar{x (ϑ, t; λ)})]}_{ϑ = ϑ_{0}, t = t_{0}}, both of α l, α l^{*} are even or odd \end{matrix}}$ (4.99) when x (ϑ, t) is (ii)-differentiable, then, the function x (ϑ, t) can be introduced as: $\underline{x} (ϑ, t; λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{(ϑ - ϑ_{ς})^{α l} (t - t_{ι})^{l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\underline{X} (l, l^{*}; λ)}{P (l, l^{*})}, α l, α l^{*} \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.100) $\bar{x} (ϑ, t; λ) = \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{(ϑ - ϑ_{ς})^{α l} (t - t_{ι})^{l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\bar{X} (l, l^{*}; λ)}{P (l, l^{*})}, α l, α l^{*} \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.101) when x (ϑ, t) are (i)-differentiable.

Then, the function x (ϑ, t) can be (ii)-differentiable we obtain: $\begin{matrix} \underline{x} (ϑ, t; λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{α l} {(t - t_{ι})}^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\bar{X} (l, l^{*}; λ)}{P (l, l^{*})} \\ (α l is odd, α l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{α l} {(t - t_{ι})}^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\underline{X} (l, l^{*}; λ)}{P (l, l^{*})}, 0 \leq λ \leq 1, \\ both of α l, α l^{*} are even or odd \end{matrix}$ (4.102) and $\begin{matrix} \bar{x} (ϑ, t; λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{α l} {(t - t_{ι})}^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\underline{X} (l, l^{*}; λ)}{P (l, l^{*})} \\ (α l is odd, α l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} \frac{{(ϑ - ϑ_{ς})}^{α l} {(t - t_{ι})}^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\bar{X} (l, l^{*}; λ)}{P (l, l^{*})}, 0 \leq λ \leq 1, \\ both of α l, α l^{*} are even or odd \end{matrix}$ (4.103) where P (l, l^*) >0, P (l, l^*) is called the weighting factor. In this work $P (l, l^{*}) = \frac{G^{α l} H^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)}$ is applied where G and H are the time horizon on interest. Consequently, if x (ϑ, t) be (i)-differentiable, then $\underline{X} (l, l^{*}; λ) = \frac{G^{α l} H^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\partial_{q}^{α (l + l^{*})} \underline{x} (ϑ, t; λ)}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}, α l, α l^{*} \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.104) $\bar{X} (l, l^{*}; λ) = \frac{G^{α l} H^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\partial_{q}^{α (l + l^{*})} \bar{x} (ϑ, t; λ)}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}, α l, α l^{*} \in ℤ_{+}, 0 \leq λ \leq 1,$ (4.105) and if x (ϑ, t) be (ii)-differentiable, then

$\begin{matrix} \underline{X} (l, l^{*}; λ) & = \frac{G^{α l} H^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\partial_{q}^{α (l + l^{*})} \bar{x} (ϑ, t; λ)}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}, (α l is odd, α l^{*} is even) or inverse \\ \bar{X} (l, l^{*}; λ) & = \frac{G^{α l} H^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\partial_{q}^{α (l + l^{*})} \underline{x} (ϑ, t; λ)}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}, (α l is odd, α l^{*} is even) or inverse \end{matrix}}$ (4.106) and

$\begin{matrix} \underline{X} (l, l^{*}; λ) & = \frac{G^{α l} H^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\partial_{q}^{α (l + l^{*})} \underline{x} (ϑ, t; λ)}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}, both of α l, α l^{*} are even or odd \\ \bar{X} (l, l^{*}; λ) & = \frac{G^{α l} H^{α l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\partial_{q}^{α (l + l^{*})} \bar{x} (ϑ, t; λ)}{\partial_{q} ϑ^{α l} \partial_{q} t^{α l^{*}}}, both of α l, α l^{*} are even or odd, \end{matrix}}$ (4.107) Applying the fuzzy fractional q-DTM, a fuzzy fractional partial q-differential equation in the domain of interest can be transformed to an algebraic equation in the domain $ℤ_{+}$ and x (ϑ, t) can be obtained as the finite-term q-Taylor series plus a reminder as:

$\begin{matrix} \underline{x} (ϑ, t; λ) & = \sum_{l = 0}^{n} \sum_{l^{*} = 0}^{m} \frac{(ϑ - ϑ_{0})^{l} (t - t_{0})^{l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\underline{X} (l, l^{*}; λ)}{P (l, l^{*})} + R_{(n + 1, m + 1)} (ϑ, t) \\ = \sum_{l = 0}^{n} \sum_{l^{*} = 0}^{m} {(\frac{ϑ - ϑ_{0}}{G})}^{l} {(\frac{t - t_{0}}{H})}^{l^{*}} \underline{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), α l, α l^{*} \in ℤ_{+}, 0 \leq λ \leq 1, \end{matrix}$ and

$\begin{matrix} \bar{x} (ϑ, t; λ) & = \sum_{l = 0}^{n} \sum_{l^{*} = 0}^{m} \frac{(ϑ - ϑ_{0})^{l} (t - t_{0})^{l^{*}}}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{\bar{X} (l, l^{*}; λ)}{P (l, l^{*})} + R_{(n + 1, m + 1)} (ϑ, t) \\ = \sum_{l = 0}^{n} \sum_{l^{*} = 0}^{m} {(\frac{ϑ - ϑ_{0}}{G})}^{l} {(\frac{t - t_{0}}{H})}^{l^{*}} \bar{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), α l, α l^{*} \in ℤ_{+}, 0 \leq λ \leq 1, \end{matrix}$ when x (ϑ, t) is (i)-differentiable and

$\begin{matrix} \underline{x} (ϑ, t, λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} {(\frac{ϑ - ϑ_{0}}{G})}^{α l} {(\frac{t - t_{0}}{H})}^{α l^{*}} \bar{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), \\ (α l is odd, α l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} {(\frac{ϑ - ϑ_{0}}{G})}^{α l} {(\frac{t - t_{0}}{H})}^{α l^{*}} \underline{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), \\ both of α l, α l^{*} are even or odd \end{matrix}}$ (4.108) and

$\begin{matrix} \bar{x} (ϑ, t, λ) = & \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} {(\frac{ϑ - ϑ_{0}}{G})}^{α l} {(\frac{t - t_{0}}{H})}^{α l^{*}} \underline{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), \\ (α l is odd, α l^{*} is even) or inverse \\ + \sum_{l = 0}^{\infty} \sum_{l^{*} = 0}^{\infty} {(\frac{ϑ - ϑ_{0}}{G})}^{α l} {(\frac{t - t_{0}}{H})}^{α l^{*}} \bar{X} (l, l^{*}; λ) + R_{(n + 1, m + 1)} (ϑ, t), \\ both of α l, α l^{*} are even or odd \end{matrix}}$ (4.109) when x (ϑ, t) is (ii)-differentiable.

The objective of this section is to find the solution of fuzzy fractional partial q-differential equations at the equally spaced grid points [t₀, t₁, . . . , t_n] and $[ϑ_{0}, ϑ_{1}, . . ., ϑ_{n}^{'}]$ where ϑ_ς = a + ςl^* and t_ι = a + ιl^*′ for each (ς = 0, 1, 2, . . . n) and (ι = 0, 1, 2, . . . n′), and $l^{*} = \frac{b - a}{n}$ and $l^{*'} = \frac{b^{'} - a^{'}}{n^{'}}$ . That is, the domain of interest are divided to [a, b] × [a′, b′] is divided to n × n′ sub-domain, and the fuzzy approximation functions in each sub-domain are x_ς (ϑ, t ; λ) for ς = 0, 1, 2, . . , n - 1, and x_ι (ϑ, t ; λ) for ι = 0, 1, 2, . . , n′ - 1, respectively.

According to the initial conditions the following can be obtained:

$\underline{X} (0, 0; λ) = \underline{x} (0, 0; λ), \bar{X} (0, 0; λ) = \bar{x} (0, 0; λ), 0 \leq λ \leq 1 .$

In the first sub-domain, $\underline{x} (ϑ, t; λ)$ and $\bar{x} (ϑ, t; λ)$ can be described by $\underline{x} (0, 0; λ) = {\underline{x}}_{0, 0} (λ)$ and $\bar{x} (0, 0; λ) = {\bar{x}}_{0, 0} (λ)$ , respectively. They can be represented in terms of their (n, n′) - t (l^*, l^*′) order bivariate q-Taylor series with respect to ϑ₀, t₀. That is

$\begin{matrix} \underline{x} (ϑ_{0}, t_{0}; λ) & = {\underline{X}}_{0} (0, 0; λ) + {\underline{X}}_{0} (0, 1; λ) (t - t_{0}) + {\underline{X}}_{0} (0, 2; λ) (t - t_{0})^{2} + . . . + {\underline{X}}_{0} (0, n^{'}; λ) (t - t_{0})^{n^{'}}, \\ + {\underline{X}}_{0} (1, 0; λ) (ϑ - ϑ_{0}) + {\underline{X}}_{0} (1, 1; ϑ) (ϑ - ϑ_{0}) (t - t_{0}) + {\underline{X}}_{0} (1, 2; λ) (ϑ - ϑ_{0}) (t - t_{0})^{2} \\ + . . . + {\underline{X}}_{0} (1, n^{'} - 1; λ) (ϑ - ϑ_{0}) (t - t_{0})^{n^{'}} + {\underline{X}}_{0} (n, n^{'} - 1; λ) (ϑ - ϑ_{0})^{n} (t - t_{0})^{n^{'} - 1} \\ + {\underline{X}}_{0} (n, n^{'}; λ) (ϑ - ϑ_{0})^{n} (t - t_{0})^{n^{'}}, \end{matrix}$ and

$\begin{matrix} \bar{x} (ϑ_{0}, t_{0}; λ) & = {\bar{X}}_{0} (0, 0; λ) + {\bar{X}}_{0} (0, 1; λ) (t - t_{0}) + {\bar{X}}_{0} (0, 2; λ) (t - t_{0})^{2} + . . . + {\bar{X}}_{0} (0, n^{'}; α) (t - t_{0})^{n^{'}}, \\ + {\bar{X}}_{0} (1, 0; λ) (ϑ - ϑ_{0}) + {\bar{X}}_{0} (1, 1; ϑ) (ϑ - ϑ_{0}) (t - t_{0}) + {\bar{X}}_{0} (1, 2; λ) (ϑ - ϑ_{0}) (t - t_{0})^{2} \\ + . . . + {\bar{X}}_{0} (1, n^{'} - 1; λ) (ϑ - ϑ_{0}) (t - t_{0})^{n^{'}} + {\bar{X}}_{0} (n, n^{'} - 1; λ) (ϑ - ϑ_{0})^{n} (t - t_{0})^{n^{'} - 1} \\ + {\bar{X}}_{0} (n, n^{'}; λ) (ϑ - ϑ_{0})^{n} (t - t_{0})^{n^{'}} . \end{matrix}$ Using q-Taylor series for x (ϑ_ς, t_ι ; λ), the solution on the grid points ϑ_ς+1 and t_ι+1 can be obtained as:

$\begin{matrix} \underline{x} (ϑ_{ς + 1}, t_{ι + 1}; λ) & = {\underline{X}}_{ς ι} (ϑ_{ς + 1}, t_{ι + 1}; λ) = {\underline{X}}_{ς ι} (0, 0; λ), \\ + {\underline{X}}_{ς ι} (0, 1; λ) (t_{ς + 1} - t_{ς}) + {\underline{X}}_{ς ι} (0, 2; λ) (t_{ς + 1} - t_{ς})^{2} + . . . + {\underline{X}}_{ς ι} (0, n^{'}; λ) (t_{ς + 1} - t_{ς})^{n^{'}}, \\ + {\underline{X}}_{ς ι} (1, 0; λ) (ϑ_{i + 1} - ϑ_{ς}) + {\underline{X}}_{ς ι} (1, 1; λ) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς}), \\ + {\underline{X}}_{ς ι} (1, 2; λ) (ϑ_{ς + 1} - ϑ_{i}) (t_{ς + 1} - t_{ς})^{2} + . . . + {\underline{X}}_{ς ι} (1, n^{'} - 1; α) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς})^{n^{'}} \\ + {\underline{X}}_{ς ι} (n, n^{'} - 1; λ) (ϑ_{ς + 1} - ϑ_{ς})^{n} (t_{ς + 1} - t_{ς})^{n^{'} - 1} \\ + {\underline{X}}_{ς ι} (n, n^{'}; λ) (ϑ_{ς + 1} - ϑ_{ς})^{n} (t_{ς + 1} - t_{ς})^{n^{'}}, \\ = \sum_{μ = 0}^{n} \sum_{i = 0}^{n^{'}} {\underline{X}}_{ς ι} (μ . n . λ) l^{* μ} l^{*' i}, \end{matrix}$ and

$\begin{matrix} \bar{x} (ϑ_{ς + 1}, t_{ι + 1}; λ) & = {\bar{X}}_{ς ι} (ϑ_{ς + 1}, t_{ι + 1}; λ) = {\bar{X}}_{ς ι} (0, 0; λ), \\ + {\bar{X}}_{ς ι} (0, 1; λ) (t_{ς + 1} - t_{ς}) + {\bar{X}}_{ς ι} (0, 2; λ) (t_{ς + 1} - t_{ς})^{2} + . . . + {\bar{X}}_{ς ι} (0, N^{'}; λ) (t_{ς + 1} - t_{ς})^{n^{'}}, \\ + {\bar{X}}_{ς ι} (1, 0; λ) (ϑ_{i + 1} - ϑ_{ς}) + {\bar{X}}_{ς ι} (1, 1; λ) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς}), \\ + {\bar{X}}_{ς ι} (1, 2; λ) (ϑ_{ς + 1} - ϑ_{i}) (t_{ς + 1} - t_{ς})^{2} + . . . + {\bar{X}}_{ς ι} (1, N^{'} - 1; α) (ϑ_{ς + 1} - ϑ_{ς}) (t_{ς + 1} - t_{ς})^{n^{'}} \\ + {\bar{X}}_{ς ι} (n, n^{'} - 1; λ) (ϑ_{ς + 1} - ϑ_{ς})^{N} (t_{ς + 1} - t_{ς})^{n^{'} - 1} \\ + {\bar{X}}_{ς ι} (n, n^{'}; λ) (ϑ_{ς + 1} - ϑ_{ς})^{n} (t_{ς + 1} - t_{ς})^{n^{'}}, \\ = \sum_{μ = 0}^{n} \sum_{i = 0}^{n^{'}} {\bar{X}}_{ς ι} (μ . n . λ) l^{* μ} l^{*' i} . \end{matrix}$

4.2.1 The properties of fuzzy fractional q-differential transform

In this part, we will give the very important properties of the two-dimensional fuzzy fractional q-differential transform as follows.

Theorem 4.25 Suppose that u (x, t) and v (x, t) are fuzzy-valued functions and their fuzzy fractional q-differential transformations denoted by U_α,q (l, l^*) and V_α,q (l, l^*), respectively. Then

If f (x, t) = u (x, t) ⊕ v (x, t) , then $F_{α, q} (l, l^{*}) = U_{α, q} (l, l^{*}) \oplus V_{α, q} (l, l^{*}), l, l^{*} \in ℤ_{+}$

If $f (x, t) = u (x, t) \underset{gH}{⊖} v (x, t),$ then $F_{α, q} (l, l^{*}) = U_{α, q} (l, l^{*}) \underset{gH}{⊖} V_{α, q} (l, l^{*}), l, l^{*} \in ℤ_{+}$

If f (x, t) = λ ⊙ u (x, t) , then $F_{α, q} (l, l^{*}) = λ ⊙ U_{α, q} (l, l^{*}), l, l^{*} \in ℤ_{+},$

provided the generalized Hukuhara difference (gH-difference) exists.

Proof. According to Definition 4.24, the proof is obvious.

Theorem 4.26 Assume that $u \in E^{1}$ and, if $f (x, t) = \frac{\partial_{q}^{μ} u (x, t)}{\partial_{q} x^{μ}}$ , then we obtain $F_{α, q} (l, l^{*}) = \frac{Γ_{q} (α l + μ + 1)}{Γ_{q} (α l + 1)} U_{α, q} (l + \frac{μ}{α}, l^{*})$ , where F_α,q (l, l^*) and U_α,q (l, l^*) are the fuzzy fractional q-differential transformations of f and u, respectively.

Proof. Using definition 4.24, we obtain for 0 ≤ λ ≤ 1 $f (x, t; λ) = \frac{\partial_{q}^{μ} u (x, t; λ)}{\partial_{q} x^{μ}} = {[\frac{\partial_{q}^{μ} \underline{u} (x, t; λ)}{\partial_{q} x^{μ}}, \frac{\partial_{q}^{μ} \bar{u} (x, t; λ)}{\partial_{q} x^{μ}}]}_{x = x_{0}, t = t_{0}},$ then

$\begin{matrix} F_{α, q} (l, l^{*}; λ) & = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} {[\frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} x^{α l} \partial_{q} t^{α l^{*}}} [\frac{\partial_{q}^{μ} \underline{u} (x, t; λ)}{\partial_{q} x^{μ}}, \frac{\partial_{q}^{μ} \bar{u} (x, t; λ)}{\partial_{q} x^{μ}}]]}_{x = x_{0}, t = t_{0}}, \\ = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} {[\frac{\partial_{q}^{α l + α l^{*} + μ}}{\partial_{q} x^{α l + μ} \partial_{q} t^{α l^{*}}} \underline{u} (x, t; λ), \frac{\partial_{q}^{α l + α l^{*} + μ}}{\partial_{q} x^{α l + μ} \partial_{q} t^{α l^{*}}} \bar{u} (x, t; λ)]}_{x = x_{0}, t = t_{0}} \\ = \frac{[α l + 1]_{q} [α l + 2]_{q} [α l + 3]_{q} \dots [α l + μ]_{q}}{[α l + 1]_{q} [α l + 2]_{q} [α l + 3]_{q} \dots [α l + μ]_{q} Γ_{q} (α l + 1)_{q} Γ_{q} (α l^{*} + 1)} \\ \times {[\frac{\partial_{q}^{α (l + \frac{μ}{α}) + α l^{*}} \underline{u} (x; λ)}{\partial_{q} x^{α (l + \frac{μ}{α})} \partial_{q} t^{α l^{*}}}, \frac{\partial_{q}^{α (l + \frac{μ}{α}) + α l^{*}} \bar{u} (x; λ)}{\partial_{q} x^{α (l + \frac{μ}{α})} \partial_{q} t^{α l^{*}}}]}_{x = x_{0}, t = t_{0}} \\ = \frac{[α l + 1]_{q} [α l + 2]_{q} [α l + 3]_{q} \dots [α l + μ]_{q}}{Γ_{q} (α (l + \frac{μ}{α}) + 1) Γ_{q} (α l^{*} + 1)} \\ \times {[\frac{\partial_{q}^{α (l + \frac{μ}{α}) + α l^{*}} \underline{u} (x; λ)}{\partial_{q} x^{α (l + \frac{μ}{α})} \partial_{q} t^{α l^{*}}}, \frac{\partial_{q}^{α (l + \frac{μ}{α}) + α l^{*}} \bar{u} (x; λ)}{\partial_{q} x^{α (l + \frac{μ}{α})} \partial_{q} t^{α l^{*}}}]}_{x = x_{0}, t = t_{0}} . \end{matrix}$ Thus, we obtain

$F_{α, q} (l, l^{*}; λ) = \frac{Γ_{q} (α l + μ + 1)}{Γ_{q} (α l + 1)} U_{α, q} (l + \frac{μ}{α}, l^{*}; λ), 0 \leq λ \leq 1 .$

Theorem 4.27 Let us consider $u \in E^{1}$ and, if $f (x, t) = \frac{\partial_{q}^{n} u (x, t)}{\partial_{q} t^{n}}$ , then we obtain $F_{α, q} (l, l^{*}) = \frac{Γ_{q} (α l^{*} + n + 1)}{Γ_{q} (α l^{*} + 1)} U_{α, q} (l, l^{*} + \frac{n}{α})$ , where F_α,q (l, l^*) and U_α,q (l, l^*) are the fuzzy fractional q-differential transformations of fuzzy-valued functions f and u, respectively.

Proof. Using definition (4.24), we obtain for 0 ≤ λ ≤ 1 $f (x, t; λ) = \frac{\partial_{q}^{n} u (x, t; λ)}{\partial_{q} t^{n}} = {[\frac{\partial_{q}^{n} \underline{u} (x, t; λ)}{\partial_{q} t^{n}}, \frac{\partial_{q}^{n} \bar{u} (x, t; λ)}{\partial_{q} t^{n}}]}_{x = x_{0}, t = t_{0}},$ then

$\begin{matrix} = \frac{[α l^{*} + 1]_{q} [α l^{*} + 2]_{q} [α l^{*} + 3]_{q} \dots [α l^{*} + n]_{q}}{Γ_{q} (α l + 1) Γ_{q} (α (l^{*} + \frac{n}{α}) + 1)} \\ \times {[\frac{\partial_{q}^{α l + α (l^{*} + \frac{n}{α})} \underline{u} (x; λ)}{\partial_{q} x^{α l} \partial_{q} t^{α (l^{*} + \frac{n}{α})}}, \frac{\partial_{q}^{α l + α (l^{*} + \frac{n}{α})} \bar{u} (x; λ)}{\partial_{q} x^{α l} \partial_{q} t^{α (l^{*} + \frac{n}{α})}}]}_{x = x_{0}, t = t_{0}} . \end{matrix}$ Thus, we obtain

$F_{α, q} (l, l^{*}; λ) = \frac{Γ_{q} (α l^{*} + n + 1)}{Γ_{q} (α l^{*} + 1)} U_{α, q} (l, l^{*} + \frac{n}{α}; λ), 0 \leq λ \leq 1 .$

Theorem 4.28 Let us consider the fuzzy-valued function $u \in E^{1}$ and, if $f (x, t) = \frac{\partial^{ζ + η} u (x, t)}{\partial_{q} x^{ζ} \partial_{q} t^{η}}$ , then we obtain $F_{α, q} (l, l^{*}) = \frac{Γ_{q} (α l + ζ + 1)}{Γ_{q} (α l + 1)} \frac{Γ_{q} (α l^{*} + η + 1)}{Γ_{q} (α l^{*} + 1)} U_{α, q} (l + \frac{ζ}{α}, l^{*} + \frac{η}{α})$ where F_α,q (l, l^*) and U_α,q (l, l^*) are the fuzzy fractional q-differential transformations of f and u, respectively.

Proof. Using definition (4.24), we obtain for 0 ≤ λ ≤ 1: $f (x, t; λ) = \frac{\partial_{q}^{ζ + η} \underline{u} (x, t; λ)}{\partial_{q} x^{ζ} \partial_{q} t^{η}} = [\frac{\partial_{q}^{ζ + η} \underline{u} (x, t; λ)}{\partial_{q} x^{ζ} \partial_{q} t^{η}}, \frac{\partial_{q}^{ζ + η} \bar{u} (x, t; λ)}{\partial_{q} x^{ζ} \partial_{q} t^{η}}], 0 \leq λ \leq 1$ then

$\begin{matrix} F_{α, q} (l, l^{*}; λ) & = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} {[\frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} x^{α l} \partial_{q} t^{α l^{*}}} [\frac{\partial_{q}^{ζ + η} \underline{u} (x, t; λ)}{\partial_{q} x^{ζ} \partial_{q} t^{η}}, \frac{\partial_{q}^{ζ + η} \bar{u} (x, t; λ)}{\partial_{q} x^{ζ} \partial_{q} t^{η}}]]}_{x = x_{0}, t = t_{0}}, \\ = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} {[\frac{\partial_{q}^{α l + α l^{*} + ζ + η} \underline{u} (x, t; λ)}{\partial_{q} x^{α l + ζ} \partial_{q} t^{α l^{*} + η}}, \frac{\partial_{q}^{α l + α l^{*} + ζ + η} \bar{u} (x, t; λ)}{\partial_{q} x^{α l + ζ} \partial_{q} t^{α l^{*} + η}}]}_{x = x_{0}, t = t_{0}} \\ = \frac{[α l + 1]_{q} [α l + 2]_{q} \dots [α l + ζ]_{q} \cdot [α l^{*} + 1]_{q} [α l^{*} + 2]_{q} \dots [α l^{*} + η]_{q}}{[α l + 1]_{q} [α l + 2]_{q} \dots [α l + ζ]_{q} \cdot [α l^{*} + 1]_{q} [α l^{*} + 2]_{q} \dots [α l^{*} + η]_{q} \cdot Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \\ \times {[\frac{\partial_{q}^{α l + α l^{*} + ζ + η} \underline{u} (x, t; λ)}{\partial_{q} x^{α l + ζ} \partial_{q} t^{α l^{*} + η}}, \frac{\partial_{q}^{α l + α l^{*} + ζ + η} \bar{u} (x, t; λ)}{\partial_{q} x^{α l + ζ} \partial_{q} t^{α l^{*} + η}}]}_{x = x_{0}, t = t_{0}} \\ = \frac{[α l + 1]_{q} [α l + 2]_{q} \dots [α l + ζ]_{q} \cdot [α l^{*} + 1]_{q} [α l^{*} + 2]_{q} \dots [α l^{*} + η]_{q}}{Γ_{q} (α (l + \frac{ζ}{α}) + 1) Γ_{q} (α (l^{*} + \frac{η}{α}) + 1)} \\ \times {[\frac{\partial_{q}^{α (l + \frac{ζ}{α}) + α (l^{*} + \frac{η}{α})} \underline{u} (x; λ)}{\partial_{q} x^{α (l + \frac{ζ}{α})} \partial_{q} t^{α (l^{*} + \frac{υ}{α})}}, \frac{\partial_{q}^{α (l + \frac{ζ}{α}) + α (l^{*} + \frac{η}{α})} \bar{u} (x; λ)}{\partial_{q} x^{α (l + \frac{ζ}{α})} \partial_{q} t^{α (l^{*} + \frac{η}{α})}}]}_{x = x_{0}, t = t_{0}} \end{matrix}$ Thus, we obtain

$F_{α, q} (l, l^{*}; λ) = \frac{Γ_{q} (α l + ζ + 1)}{Γ_{q} (α l + 1)} \frac{Γ_{q} (α l^{*} + η + 1)}{Γ_{q} (α l^{*} + 1)} U_{α, q} (l + \frac{ζ}{α}, l^{*} + \frac{η}{α}; λ), 0 \leq λ \leq 1,$ which complete the proof.□

Theorem 4.29 Let us consider U_α,q (l, l^*) and V_α,q (l, l^*) are the two-dimensional fractional q-differential transformations of u (x, t) and v (x, t) are positive real-valued function, respectively. If f (x, t) = u (x, t) v (x, t) where (x₀, t₀) = (0, 0) , then under (i)-differentiability of f we have ${\underline{F}}_{α, q} (l, l^{*}; λ) = \sum_{ζ = 0}^{l} \sum_{η = 0}^{l^{*}} a_{α (l - ζ), α ζ} (q) a_{α (l^{*} - η), α η} (q) {\underline{U}}_{α, q} (ζ, η; λ) {\underline{V}}_{α, q} (l - ζ, l^{*} - η; λ), 0 \leq λ \leq 1,$ (4.110) ${\bar{F}}_{α, q} (l, l^{*}; λ) = \sum_{ζ = 0}^{l} \sum_{η = 0}^{l^{*}} a_{α (l - ζ), α ζ} (q) a_{α (l^{*} - η), α η} (q) {\bar{U}}_{α, q} (ζ, η; λ) {\bar{V}}_{α, q} (l - ζ, l^{*} - η; λ), 0 \leq λ \leq 1,$ (4.111) and (ii)-differentiability of f as:

$\begin{matrix} {\underline{F}}_{α, q} (l, l^{*}; λ) & = \sum_{ζ = 0}^{l} \sum_{η = 0}^{l^{*}} a_{α (l - ζ), α ζ} (q) a_{α (l^{*} - η), α η} (q) {\underline{U}}_{α, q} (ζ, η; λ) {\underline{V}}_{α, q} (l - ζ, l^{*} - η; λ), 0 \leq λ \leq 1, \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{ζ = 0}^{l} \sum_{η = 0}^{l^{*}} a_{α (l - ζ), α ζ} (q) a_{α (l^{*} - η), α η} (q) {\bar{U}}_{α, q} (ζ, η; λ) {\bar{V}}_{α, q} (l - ζ, l^{*} - η; λ), 0 \leq λ \leq 1, \\ both of l, l^{*} are even or odd, 0 \leq λ \leq 1, \end{matrix}$ (4.112) and

$\begin{matrix} {\bar{F}}_{α, q} (l, l^{*}; λ) & = \sum_{ζ = 0}^{l} \sum_{η = 0}^{l^{*}} a_{α (l - ζ), α ζ} (q) a_{α (l^{*} - η), α η} (q) {\bar{U}}_{α, q} (ζ, η; λ) {\bar{V}}_{α, q} (l - ζ, l^{*} - η; λ), 0 \leq λ \leq 1, \\ (l is odd, l^{*} is even) or inverse \\ + \sum_{ζ = 0}^{l} \sum_{η = 0}^{l^{*}} a_{α (l - ζ), α ζ} (q) a_{α (l^{*} - η), α η} (q) {\underline{U}}_{α, q} (ζ, η; λ) {\underline{V}}_{α, q} (l - ζ, l^{*} - η; λ), 0 \leq λ \leq 1, \\ both of l, l^{*} are even or odd, 0 \leq λ \leq 1, \end{matrix}$ (4.113)

Proof. Using definition of two-dimensional fuzzy fractional q-DTM and under (i)-differentiability of f we have for 0 ≤ λ ≤ 1:

$\begin{matrix} F_{α, q} (l, l^{*}; λ) & = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} [[\frac{\partial_{q}^{α (l + l^{*})} \underline{u} (x, t; λ)}{\partial_{q} x^{α l} \partial_{q} t^{α l^{*}}}, \frac{\partial_{q}^{α (l + l^{*})} \bar{u} (x, t; λ)}{\partial_{q} x^{α l} \partial_{q} t^{α l^{*}}}] \\ {\times [\frac{\partial_{q}^{α (l + l^{*})} \underline{v} (x, t; λ)}{\partial_{q} x^{α l} \partial_{q} t^{α l^{*}}}, \frac{\partial_{q}^{α (l + l^{*})} \bar{v} (x, t; λ)}{\partial_{q} x^{α l} \partial_{q} t^{α l^{*}}}]]}_{x = x_{0}, t = t_{0}} . \end{matrix}$ (4.114) Taking the fractional q-Leibniz rule (4.6), we obtain

$\begin{matrix} F_{α, q} (l, l^{*}; λ) & = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \\ \times {[\frac{\partial_{q}^{α l^{*}}}{\partial_{q} t^{α l^{*}}} \sum_{c - 0}^{[α l]} {[\begin{matrix} [α l] \\ c \end{matrix}]}_{q} \frac{\partial_{q}^{α l - c}}{\partial_{q} x^{α l - c}} [\underline{u} (x q^{c}, t; λ); \bar{u} (x q^{c}, t; λ)] \frac{\partial_{q}^{c}}{\partial_{q} t^{c}} [\underline{v} (x, t; λ), \bar{v} (x, t; λ)]]}_{t = 0} \\ F_{α, q} (l, l^{*}, λ) & = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \\ \times [\frac{\partial_{q}^{α l^{*}}}{\partial_{q} t^{α l^{*}}} \sum_{c - 0}^{[α l]} {[\begin{matrix} [α l] \\ c \end{matrix}]}_{q} [\frac{\partial_{q}^{α l - c}}{\partial_{q} x^{α l - c}} \underline{u} (x q^{c}, t; λ), \frac{\partial_{q}^{α l - c}}{\partial_{q} x^{α l - c}} \bar{u} (x q^{c}, t; λ)] \\ {\times [\frac{\partial_{q}^{c}}{\partial_{q} t^{c}} \underline{v} (x, t; λ), \frac{\partial_{q}^{c}}{\partial_{q} t^{c}} \bar{v} (x, t; λ)]]}_{x = x_{0}, t = t_{0}} . \end{matrix}$ Consequently,

$\begin{matrix} F_{α, q} (l, l^{*}; λ) & = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \\ \times [\sum_{c - 0}^{[α l]} {[\begin{matrix} [α l] \\ c \end{matrix}]}_{q} \sum_{ς - 0}^{[α l^{*}]} {[\begin{matrix} [α l^{*}] \\ ς \end{matrix}]}_{q} [\frac{\partial_{q}^{α (l^{*} + l) - (c + ς)}}{\partial_{q} x^{α l - c} \partial_{q} t^{α l^{*} - ς}} \underline{u} ({xq}^{c}, {tq}^{ς}), \frac{\partial_{q}^{α (l^{*} + l) - (c + ς)}}{\partial_{q} x^{α l - c} \partial_{q} t^{α l^{*} - ς}} \bar{u} ({xq}^{c}, {tq}^{ς})] \\ {\times [\frac{\partial_{q}^{ς + c}}{\partial_{q} x^{c} \partial_{q} t^{ς}} \underline{v} (x, t; λ), \frac{\partial_{q}^{ς + c}}{\partial_{q} x^{c} \partial_{q} t^{ς}} \bar{v} (x, t; λ)]]}_{x = x_{0}, t = t_{0}} . \\ = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} [\sum_{c - 0}^{[α l]} \sum_{ς - 0}^{[α l^{*}]} \frac{[α l]_{q}!}{[c]_{q}! [α l - c]_{q}!} \frac{[α l^{*}]_{q}!}{[ς]_{q}! [α l^{*} - ς]_{q}!} \\ \times [\frac{\partial_{q}^{α (l^{*} + l) - (c + ς)}}{\partial_{q} x^{α l - c} \partial_{q} t^{α l^{*} - ς}} \underline{u} ({xq}^{c}, {tq}^{ς}; λ), \frac{\partial_{q}^{α (l^{*} + l) - (c + ς)}}{\partial_{q} x^{α l - c} \partial_{q} t^{α l^{*} - ς}} \bar{u} ({xq}^{c}, {tq}^{ς}; λ)] \\ {\times [\frac{\partial_{q}^{ς + c}}{\partial_{q} x^{c} \partial_{q} t^{ς}} \underline{v} (x, t; λ), \frac{\partial_{q}^{ς + c}}{\partial_{q} x^{c} \partial_{q} t^{ς}} \bar{v} (x, t; λ)]]}_{x = x_{0}, t = t_{0}}, \end{matrix}$ we get

$\begin{matrix} F_{α, q} (l, l^{*}; λ) & = [\sum_{c - 0}^{[α l]} \sum_{ς - 0}^{[α l^{*}]} \frac{[\frac{\partial_{q}^{α (l^{*} + l) - (c + ς)}}{\partial_{q} x^{α l - c} \partial_{q} t^{α l^{*} - ς}} \underline{u} ({xq}^{c}, {tq}^{ς}; λ), \frac{\partial_{q}^{α (l^{*} + l) - (c + ς)}}{\partial_{q} x^{α l - c} \partial_{q} t^{α l^{*} - ς}} \bar{u} ({xq}^{c}, {tq}^{ς}; λ)]}{Γ_{q} (α l - c + 1) Γ_{q} (α l^{*} - ς + 1)} \\ {\times \frac{[\frac{\partial_{q}^{ς + c}}{\partial_{q} x^{c} \partial_{q} t^{ς}} \underline{v} (x, t; λ), \frac{\partial_{q}^{ς + c}}{\partial_{q} x^{c} \partial_{q} t^{ς}} \bar{v} (x, t; λ)]}{Γ_{q} (c + 1) Γ_{q} (ς + 1)}]}_{x = x_{0}, t = t_{0}}, 0 \leq λ \leq 1 . \end{matrix}$ According to equation (3.27), we obtain

$\begin{matrix} F_{α, q} (l, l^{*}; λ) & = \sum_{c - 0}^{[α l]} \sum_{ς - 0}^{[α l^{*}]} \frac{a_{α (\frac{c}{α}), α (l - \frac{c}{α})} (q) a_{α (\frac{ς}{α}), α (l^{*} - \frac{ς}{α})} (q)}{Γ_{q} (α (l - \frac{c}{α}) + 1) Γ_{q} (α (l^{*} - \frac{ς}{α}) + 1)} \\ \times {[\frac{\partial_{q}^{α (l - \frac{c}{α}) + α (l^{*} - \frac{ς}{α})}}{\partial_{q} x^{α (l - \frac{c}{α})} \partial_{q} t^{α (l^{*} - \frac{ς}{α})}} \underline{u} (x, t; λ), \frac{\partial_{q}^{α (l - \frac{c}{α}) + α (l^{*} - \frac{ς}{α})}}{\partial_{q} x^{α (l - \frac{c}{α})} \partial_{q} t^{α (l^{*} - \frac{ς}{α})}} \bar{u} (x, t; λ)]}_{x = x_{0}, t = t_{0}} \\ \times \frac{1}{Γ_{q} (α (\frac{c}{α}) + 1) Γ_{q} (α (\frac{ς}{α}) + 1)} \\ \times {[\frac{\partial_{q}^{α (\frac{c}{α}) + α (\frac{ς}{α})}}{\partial_{q} x^{α (\frac{c}{α})} \partial_{q} t^{α (\frac{ς}{α})}} \underline{v} (x, t; λ), \frac{\partial_{q}^{α (\frac{c}{α}) + α (\frac{ς}{α})}}{\partial_{q} x^{α (\frac{c}{α})} \partial_{q} t^{α (\frac{ς}{α})}} \bar{v} (x, t; λ)]}_{x = x_{0}, t = t_{0}} . \end{matrix}$ so

$\begin{matrix} {\underline{F}}_{α, q} (l, l^{*}; λ) & = \sum_{c - 0}^{[α l]} \sum_{ς - 0}^{[α l^{*}]} a_{α (\frac{c}{α}), α (l - \frac{c}{α})} (q) a_{α (\frac{ς}{α}), α (l^{*} - \frac{ς}{α})} (q) {\underline{U}}_{α, q} (l - \frac{c}{α}, l^{*} - \frac{ς}{α}; λ) {\underline{V}}_{α, q} (\frac{c}{α}, \frac{ς}{α}; λ), \\ {\bar{F}}_{α, q} (l, l^{*}; λ) & = \sum_{c - 0}^{[α l]} \sum_{ς - 0}^{[α l^{*}]} a_{α (\frac{c}{α}), α (l - \frac{c}{α})} (q) a_{α (\frac{ς}{α}), α (l^{*} - \frac{ς}{α})} (q) {\bar{U}}_{α, q} (l - \frac{c}{α}, l^{*} - \frac{ς}{α}; λ) {\bar{V}}_{α, q} (\frac{c}{α}, \frac{ς}{α}; λ) . \end{matrix}$ Thus, assume that $ζ = l - \frac{c}{α}, η = l^{*} - \frac{ς}{α},$ we obtain

$\begin{matrix} {\underline{F}}_{α, q} (l, l^{*}; λ) & = \sum_{ζ = 0}^{l} \sum_{η = 0}^{l^{*}} a_{α (l - ζ), α ζ} (q) a_{α (l^{*} - η), α η} (q) {\underline{U}}_{α, q} (ζ, η; λ) {\underline{V}}_{α, q} (l - ζ, l^{*} - η; λ), 0 \leq λ \leq 1, \\ {\bar{F}}_{α, q} (l, l^{*}; λ) & = \sum_{ζ = 0}^{l} \sum_{η = 0}^{l^{*}} a_{α (l - ζ), α ζ} (q) a_{α (l^{*} - η), α η} (q) {\bar{U}}_{α, q} (ζ, η; λ) {\bar{V}}_{α, q} (l - ζ, l^{*} - η; λ), 0 \leq λ \leq 1 . \end{matrix}$

Assume that f is (ii)-differentiability, using similar techniques, can be obtained the result. The proof completed.

Theorem 4.30 Let us consider $f \in E^{1}$ and $f (x, t) = (x - x_{0})_{q}^{μ} (t - t_{0})_{q}^{n}$ , then $F_{α, q} (l, l^{*}) = δ (l - \frac{μ}{α}) δ (l^{*} - \frac{n}{α}), μ, n \in ℤ_{+}$ where F_α,q (l, l^*) is the q-differential transformations of f, respectably. where

$δ (l - n) = {\begin{matrix} 1, & l = \frac{μ}{α}, l^{*} = \frac{n}{α} \\ 0, & l \neq \frac{μ}{α}, l^{*} \neq \frac{n}{α} \end{matrix}$ (4.115)

Proof. Using the definition of fuzzy two-dimensional fractional q-differential transform, for λ ∈ [0, 1] ,

$F_{α, q} (l, l^{*}; λ) = \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} {[\frac{\partial_{q}^{α (l + l^{*})}}{\partial_{q} x^{α l} \partial_{q} t^{α l^{*}}} ((x - x_{0})_{q}^{μ} (t - t_{0})_{q}^{n}) (λ)]}_{x = x_{0}, t = t_{0}} .$ According to the previous lemma, we obtain

$\begin{matrix} F_{α, q} (l, l^{*}; λ) = \\ \frac{1}{Γ_{q} (α l + 1) Γ_{q} (α l^{*} + 1)} \frac{Γ_{q} (μ + 1) Γ_{q} (n + 1)}{Γ_{q} (μ - α l + 1) Γ_{q} (n - α l^{*} + 1)} {[((x - x_{0})_{q}^{μ - α l} (t - t_{0})_{q}^{n - α l^{*}}) (λ)]}_{x = x_{0}, t = t_{0}}, \end{matrix}$ Using (4.115), we obtain

$F_{α, q} (l, l^{*}; λ) = δ (l - \frac{μ}{α}) δ (l^{*} - \frac{n}{α}) (λ), μ, n \in ℤ_{+}, 0 \leq λ \leq 1,$ the proof is completed □

Next, we provide some examples illustrating the technique is powerful for solving some types of fuzzy fractional q-differential equations.

Example 4.5. We consider the following fuzzy fractional partial q-differential equation

$\frac{\partial_{q}^{0.5} u}{\partial_{q} t^{0.5}} = \frac{\partial_{q} u}{\partial_{q} x}, 0 < q < 1, t > 0,$ (4.116) subject to the initial conditions

$\begin{matrix} u (x, 0) & = [(0.7 + 0.3 λ)^{n}, (1.3 - 0.3 λ)^{n}] ⊙ x, \\ u (0, t) & = [(0.7 + 0.3 λ)^{n}, (1.3 - 0.3 λ)^{n}] ⊙ [\frac{2 \sqrt{t}}{\sqrt{π}}], \end{matrix}$ (4.117) where n = 0, 1, 2, 3, ⋯ .

Applying the two-dimensional fuzzy fractional q-DTM of (4.116), we get $\frac{Γ_{q} (\frac{l^{*}}{2} + \frac{3}{2})}{Γ_{q} (\frac{l^{*}}{2} + 1)} {\underline{U}}_{0.5, q} (l, l^{*} + 1; λ) = {[\frac{l}{2} + 1]}_{q} {\underline{U}}_{0.5, q} (l + 2, l^{*}; λ),$ (4.118) $\frac{Γ_{q} (\frac{l^{*}}{2} + \frac{3}{2})}{Γ_{q} (\frac{l^{*}}{2} + 1)} {\bar{U}}_{0.5, q} (l, l^{*} + 1; λ) = {[\frac{l}{2} + 1]}_{q} {\bar{U}}_{0.5, q} (l + 2, l^{*}; λ),$ (4.119) Using the initial condition (4.117), we obtain ${\underline{U}}_{0.5, q} (l, 0; λ) = {\begin{matrix} (0.7 + 0.3 λ)^{n}, & l = 2, \\ 0, & l \neq 2, \end{matrix}$ (4.120) ${\underline{U}}_{0.5, q} (0, l^{*}; λ) = {\begin{matrix} (0.7 + 0.3 λ)^{n} [\frac{2}{\sqrt{π}}], & l^{*} = 1, \\ 0, & l^{*} \neq 1, \end{matrix}$ (4.121) and ${\bar{U}}_{0.5, q} (l, 0; λ) = {\begin{matrix} (1.3 - 0.3 λ)^{n}, & l = 2, \\ 0, & l \neq 2, \end{matrix}$ (4.122) ${\bar{U}}_{0.5, q} (0, l^{*}; λ) = {\begin{matrix} (1.3 - 0.3 λ)^{n} [\frac{2}{\sqrt{π}}], & l^{*} = 1, \\ 0, & l^{*} \neq 1 . \end{matrix}$ (4.123) Applying (4.123) into (4.118), and using recursive method, we obtain

${\underline{U}}_{0.5, q} (l, l^{*}; λ) = {\begin{matrix} (0.7 + 0.3 λ)^{n}, & l = 2, l^{*} = 0, \\ (0.7 + 0.3 λ)^{n} [\frac{2}{\sqrt{π}}], & l^{*} = 0, l^{*} = 1, \\ 0, & otherwise, \end{matrix}$ and

${\bar{U}}_{0.5, q} (l, l^{*}; λ) = {\begin{matrix} (1.3 - 0.3 λ)^{n}, & l = 2, l^{*} = 0, \\ (1.3 - 0.3 λ)^{n} [\frac{2}{\sqrt{π}}], & l^{*} = 0, l^{*} = 1, \\ 0, & otherwise . \end{matrix}$ Thus, we can be obtained the solution as follows:

$u (x, t; λ) = [(0.7 + 0.3 λ)^{n}, (1.3 - 0.3 λ)^{n}] ⊙ [x + \frac{2 \sqrt{t}}{\sqrt{π}}], 0 \leq λ \leq 1 .$

Example 4.6. We consider the following fuzzy fractional partial q-differential equation

$\frac{\partial_{q}^{1.5} u}{\partial_{q} t^{1.5}} = \frac{\partial_{q}^{2} u}{\partial_{q} x^{2}}, 0 < q < 1, t \geq 0,$ (4.124) under the initial conditions

$\begin{matrix} u (x, 0) & = [(0.5 + 0.5 λ)^{n}, (1.5 - 0.5 λ)^{n}] ⊙ x^{3}, \\ \frac{\partial_{q} u (x, 0)}{\partial_{q} t} & = [(0.5 + 0.5 λ)^{n}, (1.5 - 0.5 λ)^{n}] ⊙ x, \end{matrix}$ (4.125) and

$u (0, t) = \frac{\partial_{q} u (x, 0)}{\partial_{q} x} = 0,$ (4.126) where n = 0, 1, 2, 3, ⋯ .

Using the two-dimensional fuzzy fractional q-DTM of (4.117), we get

$\begin{matrix} \frac{Γ_{q} (\frac{l^{*}}{2} + \frac{5}{2})}{Γ_{q} (\frac{l^{*}}{2} + 1)} {\underline{U}}_{0.5, q} (l, l^{*} + 3; λ) = {[\frac{l}{2} + 2]}_{q} \\ {[\frac{l}{2} + 1]}_{q} {\underline{U}}_{0.5, q} (l + 4, l^{*}; λ), \end{matrix}$ (4.127)

$\begin{matrix} \frac{Γ_{q} (\frac{l^{*}}{2} + \frac{5}{2})}{Γ_{q} (\frac{l^{*}}{2} + 1)} {\bar{U}}_{0.5, q} (l, l^{*} + 3; λ) = {[\frac{l}{2} + 2]}_{q} \\ {[\frac{l}{2} + 1]}_{q} {\bar{U}}_{0.5, q} (l + 4, l^{*}; λ) . \end{matrix}$ (4.128) Putting the initial conditions (4.125) and (4.126), we obtain

${\underline{U}}_{0.5, q} (l, 0; λ) = {\begin{matrix} (0.5 + 0.5 λ)^{n}, & l = 6, \\ 0, & l \neq 6, \end{matrix}$ (4.129)

${\underline{U}}_{0.5, q} (l, 2; λ) = {\begin{matrix} (0.5 + 0.5 λ)^{n}, & l^{*} = 2, \\ 0, & l^{*} \neq 2, \end{matrix}$ (4.130) and

${\bar{U}}_{0.5, q} (l, 0; λ) = {\begin{matrix} (1.5 - 0.5 λ)^{n}, & l = 6, \\ 0, & l \neq 6, \end{matrix}$ (4.131)

${\bar{U}}_{0.5, q} (l, 2; λ) = {\begin{matrix} (1.5 - 0.5 λ)^{n}, & l^{*} = 2, \\ 0, & l^{*} \neq 2 . \end{matrix}$ (4.132) Substituting (4.132) into (4.127), and using recursive method, we obtain

${\begin{matrix} {\underline{U}}_{0.5, q} (0, 3; λ) & = \tilde{0} \\ {\underline{U}}_{0.5, q} (1, 3; λ) & = \tilde{0} \\ {\underline{U}}_{0.5, q} (2, 3; λ) & = (0.5 + 0.5 λ)^{n} [\frac{[3]_{q}!}{Γ_{q} (\frac{5}{2})}] \\ ⋮ \end{matrix}$ and

${\begin{matrix} {\bar{U}}_{0.5, q} (0, 3; λ) & = \tilde{0} \\ {\bar{U}}_{0.5, q} (1, 3; λ) & = \tilde{0} \\ {\bar{U}}_{0.5, q} (2, 3; λ) & = (1.5 - 0.5 λ)^{n} [\frac{[3]_{q}!}{Γ_{q} (\frac{5}{2})}] \\ ⋮ . \end{matrix}$ Consequently, we can be obtained the series solution as follows:

$\begin{matrix} u (x, t; λ) = [(0.5 + 0.5 λ)^{n}, (1.5 - 0.5 λ)^{n}] \\ ⊙ [xt + \frac{[3]_{q}!}{Γ_{q} (\frac{5}{2})} t^{1.5} x + x^{3} + \dots], 0 \leq λ \leq 1 . \end{matrix}$

5 Conclusions

In this paper, we presented a new generalization of the fuzzy differential transform method that extends its applicability to the fuzzy q-differential equations and fuzzy fractional q-differential equations in this study. In addition, the problem does not need to be transformed into a form suitable for the application of linear theory or perturbation, and also that no symbolic computing is required, which might be challenging in nonlinear instances. The method investigated is based on gH-differentiability, fuzzy q-derivative, and fuzzy q-fractional derivative. Finally, the results are quite reliable, and many concrete problems were tested by applying the new technique and the results have shown remarkable performance.

Conflict of interest

The authors declare that they have no conflict of interest.

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