On type-2 fuzzy partial differential equations and its applications

Abstract

Type-2 fuzzy sets are generally used to describe those membership values of fuzzy sets that are imprecise. This paper attempts to develop the theory of type-2 fuzzy partial differential equations (T2FPDE) using type-2 fuzzy initial condition. The theory of type-2 FPDEs could be used with type-2 fuzzy initial values, type-2 fuzzy boundary values and type-2 fuzzy parameters. Some natural phenomena can be modelled as dynamical systems whose initial conditions and/or parameters might be imprecise in nature. This imprecision of initial values and/or parameters are generally modelled by fuzzy sets. In this paper the concept of generalized H₂-differentiability is applied. This concept is based on the enlargement of the class of differentiable type-2 fuzzy mappings which are commonly known as Hukuhara derivatives. Some theorem is presented to show the solution of a fuzzy type-1 and type-2 PDE could be obtained by solving the corresponding embedded systems based on fuzzy differential inclusions. An algorithm is also developed to simulate of type-2 fuzzy partial differential equations and obtain its solution numerically. Some illustrative examples have also been provided for different type-2 FPDE models.

Keywords

Type-2 FPDE Type-2 Hukuhara partial derivatives Type-2 fuzzy differential inclusion numerical solutions Type-2 fuzzy heat equation Type-2 fuzzy wave equation Type-2 fuzzy advection equation

1 Introduction

The concept of fuzzy derivative was first introduced by Chang and Zadeh [1]. It was further developed by Dubois and Prade [2] using the extension principle. Few other concepts related to fuzzy derivative have been discussed by Puri and Ralescu [3]. The fuzzy initial value problem (FIVP) has been studied by Kaleva [4] and Seikkala [5]. Quite a few definitions of the differentiability of fuzzy functions have been presented as a consequence. Among these, the Hukuhara differentiability [H-differentiability] and the strongly generalized differentiability [3, 6], became most acceptable and have attracted more attention. Therefore, in recent times, many papers about FDEs are found using H-differentiability and the strongly generalized differentiability.

In 1975 Zadeh presented a more general concept of Type-2 Fuzzy Sets (T2FSs) to model imprecision and uncertainty [7]. T2FS is a fuzzy set in which membership grades are Type-1 Fuzzy Sets (T1FSs). Hence, a T2FS includes not only data uncertainty, but also the uncertainty on the membership function is presented. T2FSs are helpful in those cases where the exact form of the membership functions cannot be determined or where membership function grades are themself fuzzy or imprecise in nature. T1FSs and T2FSs differ in that, a T1FS is an aggregation of different individuals’ (experts’) experience and presented in the form of one single type-1 fuzzy membership function. On the other hand a T2FS is the union of all different individuals’ (experts’) experiences. T2FSs are better to deal with high levels of uncertainties present in real problems which are highly complex in terms of computational effort. Besides the parameters and state variables of different dynamical systems like environmental and biological systems are highly imprecise, and hence could be modelled more appropriately by type 2 fuzzy variables and by type 2 fuzzy differential equations.

Differential inclusions (DI) is an important generalization of differential equations. The solution to a DI is a reachable set, instead of a single trajectory. Although these concepts are well known in control theory, they are yet to be explored properly in the field of modelling and simulation. The main applications may be related to the uncertainty in dynamical systems. The theory of DI is developed by Blasi [8], followed by Raczynski [9] and Diamond [10]. Abbasbandy et al. [11], Khastan et al. [12] applied fuzzy differential inclusion to solve fuzzy differential equations.

Differentiability of type 2 fuzzy functions was first defined by Mazandarani et al. [13], based upon type-2 Hukuhara differentiability [H₂-differentiability] and also Perfect Type-2 Fuzzy Numbers [T2FNs] and perfect Quasi Type-2 Fuzzy Numbers [QT2FNs] have been defined. However the representation and different algebraic operations of T2FS was introduced by Hamrawi and Coupland [14], and Hung [15]. Using the $\tilde{a}$ -plane representation, Mazandarani et al. [13] proved that the H₂-difference reduces to H-difference.

There are a few research work focused on the solution of type-1 fuzzy ODEs and PDEs have published is last two decades. The major drawback of early methods was that straightforward use of interval calculus for the solution of fuzzy differential equation can produce incorrect results for even type-1 fuzzy systems of ODEs. This is due to the fact that the fuzzy formalism is unable to represent the interaction that the differential equation establishes between variables. In recent past some efficient methods were developed to obtain the numerical solution of type-1 fuzzy ODEs Allahviranloo et al. [16], Abbasbandy et al. [17], Hosseini et al. [18], Allahviranloo et al. [19] and PDEs Allahviranloo et al. [20], Arqub et al. [21]; but the numerical solution of type-2 FPDE is yet to achieve reliable methodologies. In this paper we’ve introduced the idea of embedded system of a type-1 and type-2 FPDE and developed a methodology to obtain numerical solution of type-1 and type-2 FPDE by solving the corresponding embedded systems.

There are a number of different systems whose behaviour is explained and modelled by several dynamical systems, where the parameters as well as the initial values are not precisely deterministic. In practise the dynamical systems with imprecise variables and parameters could be better modelled by fuzzy dynamical systems. Moreover the consideration of membership function of a fuzzy set is an issue of debate for long time. The membership function of a fuzzy set is often couldn’t be precisely determined or might have different forms. In these cases the fuzzy set is eventually a T2FS. Thus the dynamical systems whose parameter values and/or initial condition are imprecise or if the membership grade of any of these is imprecise; then it should be modelled by T2DE and type-2 dynamical systems. Different types of membership of T2FS could be considered, for the sake of simplicity in this paper only triangular membership functions of T2FS is considered and the proposed method to solve the T2FDE could be very easily applied to other types of membership functions of T2FS.

The rest of the paper is organized as follows. Section 2, briefly explains type-2 fuzzy sets (T2FS), type-2 fuzzy derivative, type-2 fuzzy partial differential equation (T2FPDE), type-2 fuzzy boundary value problem (T2BVP). We also define the embedded PDE system based on fuzzy differential inclusion (T2DI), corresponding to a FPDE and some theorems are provided to show that the solution of the original FPDE is the union of solutions of the corresponding embedded systems. In Section 3, the development of a methodology for numerical solution of T2FDE is described. In Section 4, several examples of different physical and environmental phenomenon have been presented and solved by using numerical simulation. In Section 5, the results of the examples are discussed. Finally the conclusion and future research directions are presented in Section 6.

2 Basic concepts

Throughout this paper, the set of all real numbers is denoted by R, the set of all natural numbers by N, the set of all type-1 fuzzy numbers (T1FNs) on R by E₁ and the set of all perfect type-2 fuzzy numbers (T2FNs) on R by E₂.

2.1 Type-2 fuzzy sets (T2FSs)

Definition 2.1.1 (Type 2 fuzzy set). A type 2 fuzzy set (T2FS) $\tilde{A}$ can be characterized by its type-2 membership function, $μ_{\tilde{A}} : J_{x} \times [0, 1] \to [0, 1]$ , where J_x is the universe for the primary variable x ∈ X. The 3D membership of $\tilde{A}$ is denoted by $μ_{\tilde{A}} (x, u), x \in J_{x}$ and u ∈ [0, 1]

i.e. $\tilde{A} = {(x, u, μ_{\tilde{A}} (x, u)) : x \in J_{x}, u \in [0, 1]}$ , in which $0 \leq μ_{\tilde{A}} (x, u) \leq 1$ . $\tilde{A}$ can also be expressed as $\int_{x \in J_{x}} \int_{u \in [0, 1]} \frac{μ_{\tilde{A}} (x, u)}{(x, u)}$ where ∫∫ denotes union over all admissible x and u. For discrete universe ∫ is replaced by ∑.

Let $\tilde{A}$ be a T2FS. Its secondary domain at a fixed point x₀ ∈ X, J_{x
₀}, is a T1FS whose membership function, $μ_{\tilde{A}} (x_{0})$ , is called secondary membership function or vertical slice of $\tilde{A}$ at x₀ and expressed as: $μ_{\tilde{A}} (x_{0}) = \int_{u \in J_{x_{0}}} \frac{f_{x_{0}} (u)}{u}$ , where f_{x
₀} (u) is called secondary membership grade. The integral sign stands for the union and not standard integration. As well, the sign ‘/’ stands for association (or a marker) and does not imply division.

Based on this definition, a T2FS $\tilde{A}$ can be represented by its vertical slices as: $\tilde{A} = \int_{x \in X} \frac{[\int_{u \in J_{x_{0}}} \frac{f_{x_{0}} (u)}{u}]}{x}$ .

An α-cut set of the vertical slice of $\tilde{A}$ at x₀ ∈ X is defined as: $S_{\tilde{A}} (x_{0} | \tilde{α}) = {\begin{matrix} {u \in J_{x_{0}} : f_{x} (u) > \tilde{α}}, 0 < α \leq 1 . \\ cl ({u \in J_{x_{0}} : f_{x} (u) > 0}), α = 0 \end{matrix}$ cl () denotes the closure of the set, and it is compact. The vertical slice of $\tilde{A}$ at a fixed point x₀ ∈ X, J_{x
₀}.

The $\tilde{α}$ plane of a T2FS, $\tilde{A_{\tilde{α}}}$ , is an interval valued fuzzy set, at level $\tilde{a}$ , $\tilde{A_{\tilde{α}}}$ can be characterized in a parametric form denoted by the pair $(\underline{A_{\tilde{α}}}, \bar{A_{\tilde{α}}})$ , where $\underline{A_{\tilde{α}}},$ is called Lower Membership Function [LMF], and $\bar{A_{\tilde{α}}}$ is called Upper Membership Function [UMF], and both are T1FSs [14, 22].

Definition 2.1.2. The union of all primary memberships or the two-dimensional domain of a T2FS, $\tilde{A}$ is called Footprint of Uncertainty (FOU) of $\tilde{A}$ , and it is represented as: $FOU (\tilde{A}) = \int_{x \in X} J_{x}$ .

Definition 2.1.3. Assume that each of the vertical slices of a T2FS, $\tilde{A}$ has the secondary grade that equals one, i.e., f_x (u) = 1. A Principal Set (PS) of $\tilde{A}$ is the union of all such points at which this occurs and is defined as follows: $μ_{PS} (x) = \int_{x \in X} (\frac{u}{x}) : f_{x} (u) = 1$ where μ_PS (x) is the membership function of PS of $\tilde{A}$ .

Definition 2.1.4. Let $\tilde{A}$ be a T2FS, and $\tilde{A_{\tilde{α}}} = (\underline{A_{\tilde{α}}}, \bar{A_{\tilde{α}}})$ be its $\tilde{α}$ plane represented by its lower membership function (LMF) and upper membership function (UMF). Then the $\tilde{α}$ plane of $\tilde{A}$ is defined as: $\tilde{A_{\tilde{α}}^{α}} = (\underline{A_{\tilde{α}}^{α}}, \bar{A_{\tilde{α}}^{α}})$ .

Where $\underline{A_{\tilde{α}}^{α}}$ and $\bar{A_{\tilde{α}}^{α}}$ are the $\tilde{α}$ plane of the LMF and UMF of $\tilde{A}$ respectively. $\begin{matrix} \underline{A_{\tilde{α}}^{α}} & = & LMF (\tilde{A}) \\ = & \inf {u : u \in [0, 1], μ_{\tilde{A}} (x, u) > 0}, \forall x \in X \end{matrix}$ and $\begin{matrix} \bar{A_{\tilde{α}}^{α}} & = & UMF (\tilde{A}) \\ = & \sup {u : u \in [0, 1], μ_{\tilde{A}} (x, u) > 0}, \forall x \in X \end{matrix}$

Theorem 2.1.1 (α-cut representation of T2FS). A T2FS $\tilde{A}$ can be described by the union of all its $\tilde{α}$ plane as follows: $\tilde{A} = ⋃_{\tilde{α} \in [0, 1]} \tilde{α} ⋃_{\tilde{α} \in [0, 1]} α A_{\tilde{α}}^{α}$ . For proof see [14, 22].

Theorem 2.1.2. T2FS $\tilde{α}$ plane extension principle: Let $X = \prod_{i = 1}^{n} X_{i}$ be the Cartesian product of universes, and $\tilde{A_{1}}, \tilde{A_{2}}, \dots \dots, \tilde{A_{n}}$ be the T2FSs in each universe respectively. Also let Y be another universe, and $\tilde{B}$ be a T2FS such that $\tilde{B} = f (\tilde{A_{1}}, \tilde{A_{2}}, \dots \dots, \tilde{A_{n}})$ , where f : X → Y is a monotonic mapping. Then $\tilde{B}$ is equal to the union of applying the same function to all of its decomposed the $\tilde{a}$ plane as follows: [14, 22]. $\begin{matrix} \tilde{B} & = & f (\tilde{A_{1}}, \tilde{A_{2}}, \dots \dots, \tilde{A_{n}}) \\ = & \cup_{\tilde{α} \in [0, 1]} \tilde{α} \cup_{\tilde{α} \in [0, 1]} α f (A_{1 \tilde{α}}^{α}, A_{2 \tilde{α}}^{α}, \dots \dots, A_{n \tilde{α}}^{α}) . \end{matrix}$

Differentiability of the type-2 fuzzy number-valued functions is based on the metric space defined by Huang [15] as follows:

Definition 2.1.5. Let $\tilde{A}$ and $\tilde{B}$ be two T2FSs whose $\tilde{α}$ plane of the secondary membership functions at x₀ ∈ [a, b] are $S_{\tilde{A}} (x | \tilde{α})$ and $S_{\tilde{B}} (x | \tilde{α})$ respectively. Then the distance between $\tilde{A}$ and $\tilde{B}$ is defined as follows: $d_{2} (\tilde{A}, \tilde{B}) = \int_{a}^{b} H_{f} (μ_{\tilde{A}} (x), μ_{\tilde{B}} (x))$ , where $\begin{matrix} H_{f} (μ_{\tilde{A}} (x), μ_{\tilde{B}} (x)) \\ = \frac{\int_{0}^{1} \tilde{α} d_{H} (S_{\tilde{A}} (x | \tilde{α}), S_{\tilde{B}} (x | \tilde{α})) d \tilde{α}}{\int_{0}^{1} \tilde{α} d \tilde{α}} \\ = 2 \int_{0}^{1} \tilde{α} d_{H} (S_{\tilde{A}} (x | \tilde{α}), S_{\tilde{B}} (x | \tilde{α})) d \tilde{α} \end{matrix}$

Note $d_{2} (\tilde{A}, \tilde{B})$ is very similar to the weighted hamming distance of two fuzzy sets.

Theorem 2.1.3. The distance measure d₂ is a metric on the space of type-2 fuzzy sets [13].

Definition 2.1.6. A T2FS, $\tilde{A}$ , is called a perfect T2FN if the following conditions are satisfied:

UMF and LMF of $FOU (\tilde{A})$ are T1FNs themselves,

UMF and LMF of PS of $\tilde{A}$ are T1FNs themselves - [14, 22].

A perfect T2FN can be completely determined using its FOU and PS if all its vertical slices are T1FNs, and piecewise functions of the same kind.

Definition 2.1.7. A perfect T2FN is called a perfect quasi type-2 fuzzy number (QT2FN) if it can be completely determined using its FOU and PS. Parametric closed form of a triangular perfect QT2FN.

Let E₂ be the set of all triangular perfect QT2FN and $\tilde{w} \in E_{2}$ with the core m. The closed form of the triangular perfect QT2FN can be characterized as follows: $\begin{matrix} \tilde{w_{\tilde{α}}^{α}} = [\underline{w_{\tilde{α}}^{α}}, \bar{w_{\tilde{α}}^{α}}], \underline{w_{\tilde{α}}^{α}} = [\underline{L_{w_{\tilde{α}}}^{α}}, \underline{R_{w_{\tilde{α}}}^{α}}], \\ \bar{w_{\tilde{α}}^{α}} = [\bar{L_{w_{\tilde{α}}}^{α}}, \bar{R_{w_{\tilde{α}}}^{α}}], \underline{L_{w_{\tilde{α}}}^{α}} \\ = L_{w_{1}}^{α} - (1 - \tilde{α}) (L_{w_{1}}^{α} - \underline{L_{w_{0}}^{α}}), \\ \underline{R_{w_{\tilde{α}}}^{α}} = R_{w_{1}}^{α} + (1 - \tilde{α}) (\underline{R_{w_{0}}^{α}} - R_{w_{1}}^{α}) \\ \bar{L_{w_{\tilde{α}}}^{α}} = L_{w_{1}}^{α} - (1 - \tilde{α}) (L_{w_{1}}^{α} - \bar{L_{w_{0}}^{α}}), \\ \bar{R_{w_{\tilde{α}}}^{α}} = R_{w_{1}}^{α} + (1 - \tilde{α}) (\bar{R_{w_{0}}^{α}} - R_{w_{1}}^{α}) \end{matrix}$

Where $\bar{L_{w_{0}}^{α}} \leq L_{w_{1}}^{α} \leq \underline{L_{w_{0}}^{α}} \leq \underline{R_{w_{0}}^{α}} \leq R_{w_{1}}^{α} \leq \bar{R_{w_{0}}^{α}}$ . The α-cut of LMF and UMF of $FOU (\tilde{w})$ are $\underline{w_{0}^{α}} = [\underline{L_{0}^{α}}, \underline{R_{0}^{α}}]$ and $\bar{w_{0}^{α}} = [\bar{L_{0}^{α}}, \bar{R_{0}^{α}}]$ respectively where $\underline{L_{w_{0}}^{α}} = m - (1 - α) (m - \underline{L_{w_{0}}})$ and $\underline{R_{w_{0}}^{α}} = m - (1 - α) (\underline{R_{w_{0}}} - m)$ are the left and right endpoints of $\underline{w_{0}^{α}}$ with the support $[\underline{L_{w_{0}}}, \underline{R_{w_{0}}}]$ , where $\bar{L_{w_{0}}^{α}} = m - (1 - α) (m - \bar{L_{w_{0}}})$ and $\bar{R_{w_{0}}^{α}} = m - (1 - α) (\bar{R_{w_{0}}} - m)$ are the left and right endpoints of $\bar{w_{0}^{α}}$ with the support $[\bar{L_{w_{0}}}, \bar{R_{w_{0}}}]$ .

The α-cut of PS is $w_{1}^{α} = [L_{1}^{α}, R_{1}^{α}]$ where $L_{w_{1}}^{α} = m - (1 - α) (m - L_{w_{1}})$ and $R_{w_{1}}^{α} = m - (1 - α) (R_{w_{1}} - m)$ are the left and right endpoints of $w_{1}^{α}$ respectively with the support [L_{w
₁}, R_{w
₁}]. As a result, the triangular perfect QT2FN, $\tilde{w}$ , can be represented by $\tilde{w} = (\bar{L_{w_{0}}}, L_{w_{1}}^{,} \underline{L_{w_{0}}}, m, \underline{R_{w_{0}}}, R_{w_{1}}^{,} \bar{R_{w_{0}}})$ where $\bar{L_{w_{0}}} \leq L_{w_{1}} \leq \underline{L_{w_{0}}} \leq m \leq \underline{R_{w_{0}}} \leq R_{w_{1}} \leq \bar{R_{w_{0}}}$ .

Let $\tilde{w}$ and $\tilde{z}$ be two perfect T2FNs. Then the arithmetic operations addition, subtraction, multiplication, division and scalar multiplication by a scalar k are defined as follows [14, 22]:

$\begin{matrix} [{\tilde{w}}^{o} \tilde{z}]_{\tilde{α}}^{α} = ([\underline{w_{\tilde{α}}^{α}} \circ \underline{z_{\tilde{α}}^{α}}], [\bar{w_{\tilde{α}}^{α}} \circ \bar{z_{\tilde{α}}^{α}}]), \\ \underline{w_{\tilde{α}}^{α}} \circ \underline{z_{\tilde{α}}^{α}} \\ = [\begin{matrix} min {\underline{L_{w_{α}}^{α}} \circ \underline{L_{z_{α}}^{α}}, \underline{L_{w_{α}}^{α}} \circ \underline{R_{z_{α}}^{α}}, \underline{R_{w_{α}}^{α}} \circ \underline{L_{z_{α}}^{α}}, \underline{R_{w_{α}}^{α}} \circ \underline{R_{z_{α}}^{α}}}, \\ max {\underline{L_{w_{α}}^{α}} \circ \underline{L_{z_{α}}^{α}}, \underline{L_{w_{α}}^{α}} \circ \underline{R_{z_{α}}^{α}}, \underline{R_{w_{α}}^{α}} \circ \underline{L_{z_{α}}^{α}}, \underline{R_{w_{α}}^{α}} \circ \underline{R_{z_{α}}^{α}}} \end{matrix}] \\ \bar{w_{\tilde{α}}^{α}} \circ \bar{z_{\tilde{α}}^{α}} \\ = [\begin{matrix} min {\bar{L_{w_{α}}^{α}} \circ \bar{L_{z_{α}}^{α}}, \bar{L_{w_{α}}^{α}} \circ \bar{R_{z_{α}}^{α}}, \bar{R_{w_{α}}^{α}} \circ \bar{L_{z_{α}}^{α}}, \bar{R_{w_{α}}^{α}} \circ \bar{R_{z_{α}}^{α}}}, \\ max {\bar{L_{w_{α}}^{α}} \circ \bar{L_{z_{α}}^{α}}, \bar{L_{w_{α}}^{α}} \circ \bar{R_{z_{α}}^{α}}, \bar{R_{w_{α}}^{α}} \circ \bar{L_{z_{α}}^{α}}, \bar{R_{w_{α}}^{α}} \circ \bar{R_{z_{α}}^{α}}} \end{matrix}] \\ [k \tilde{w}]_{\tilde{α}}^{α} = {\begin{matrix} ([k \underline{L_{w_{\tilde{a}}}^{α}}, k \underline{R_{w_{\tilde{a}}}^{α}}], [k \bar{L_{w_{\tilde{a}}}^{α}}, k \bar{R_{w_{\tilde{a}}}^{α}}]) \\ ([k \underline{R_{w_{\tilde{a}}}^{α}}, k \underline{L_{w_{\tilde{a}}}^{α}}], [k \bar{R_{w_{\tilde{a}}}^{α}}, k \bar{L_{w_{\tilde{a}}}^{α}}]) \end{matrix} \end{matrix}$

Where ∘ stands for +, - , × or ÷.

2.2 Differentials of type-2 fuzzy functions

In this section, we define the differentiability of the type-2 fuzzy number-valued functions whose definition is similar to that of the strongly generalized differentiability [6].

Definition 2.2.1. Let $\tilde{x}, \tilde{y} \in E_{2}$ , if there exists $\tilde{z} \in E_{2}$ such that $\tilde{x} = \tilde{y} + \tilde{z}$ , then $\tilde{z}$ is called type-2 Hukuhara difference [H₂-difference] of $\tilde{x}$ and $\tilde{y}$ ; denoted by $\tilde{x} -^{H_{2}} \tilde{y}$ .

Theorem 2.2.1. If $\tilde{x}, \tilde{y} \in E_{2}$ , then the $\tilde{α}$ -plane of the H₂-difference of $\tilde{x}$ and $\tilde{y}$ is H-difference of LMF and UMF of $\tilde{x}$ and $\tilde{y}$ [14, 22].

Definition 2.2.2. Let $T = [a, b] \subseteq R$ . Then $\tilde{f}$ is called a type-2 fuzzy number-valued function on T if $\tilde{f} : T \to E_{2}$ and is called an n-dimensional vector of type-2 fuzzy number-valued functions on T if $\tilde{f} : T \to E_{2}^{n}$ .

Definition 2.2.3. Let $\tilde{w} = \tilde{f} (t) = (\bar{L_{{\tilde{f}}_{0}}} (t), L_{{\tilde{f}}_{1}} (t), \underline{L_{{\tilde{f}}_{0}}} (t), m_{\tilde{f}} (t), \underline{R_{{\tilde{f}}_{0}}} (t), R_{{\tilde{f}}_{1}} (t), \bar{R_{{\tilde{f}}_{0}}} (t))$ . The T2FN is $\tilde{w}$ called a triangular perfect QT2FN-valued function if it is a mapping from [a, b] ⊆ R into the set of all the triangular perfect QT2FNs.

Definition 2.2.4 (Type 2 fuzzy derivatives). Let ${\tilde{f}}_{\tilde{α}} : T \to E_{2}$ and t₀ ∈ [a, b] ⊆ R . Then ${\tilde{f}}_{\tilde{α}}$ is differentiable at t₀ if there exists an element ${\tilde{f}}_{\tilde{α}} (t_{0}) \in E_{2}$ , such that for all h > 0, sufficiently small,

The function ${\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})$ , ${\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} - h)$ , and the following limits converges $lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{h} = lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} - h)}{h} = {\overset{´}{\tilde{f}}}_{\tilde{α}} (t_{0})$ (1st form) Or

The function ${\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} + h)$ , ${\tilde{f}}_{\tilde{α}} (t_{0} - h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})$ , and the following limits converges: $lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} + h)}{- h} = lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} - h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{- h} = {\overset{´}{\tilde{f}}}_{\tilde{α}} (t_{0})$ (1st form) Or

The function ${\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})$ , ${\tilde{f}}_{\tilde{α}} (t_{0} - h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})$ , and the following limits converges: $lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{h} = lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} - h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{- h} = {\overset{´}{\tilde{f}}}_{\tilde{α}} (t_{0})$ (2nd form) Or

The function ${\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} + h)$ , ${\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} - h)$ , and the following limits converges: $lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} + h)}{- h} = lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} - h)}{h} = {\overset{´}{\tilde{f}}}_{\tilde{α}} (t_{0})$ (2nd form) [6, 13].

Theorem 2.2.2. Let $\tilde{f} : T \to E_{2}$ be a type-2 fuzzy number-valued function and ${\tilde{f}}_{\tilde{α}} (t) = [\underline{f_{\tilde{α}}} (t), \bar{f_{\tilde{α}}} (t)]$ ; where $\underline{f_{\tilde{α}}} (t)$ and $\bar{f_{\tilde{α}}} (t)$ are the LMF and UMF of $\tilde{α}$ -plane of $\tilde{f} (t)$ , then

If $\tilde{f} (t)$ is H₂-differentiable in the first form, then $\underline{f_{\tilde{α}}} (t)$ and $\bar{f_{\tilde{α}}} (t)$ are differentiable functions in the first form and $\overset{´}{{\tilde{f}}_{\tilde{α}}} (t) = [\overset{´}{\underline{f_{\tilde{α}}}} (t), \overset{´}{\bar{f_{\tilde{α}}}} (t)]$ .

If $\tilde{f} (t)$ is H₂-differentiable in the second form, then $\underline{f_{\tilde{α}}} (t)$ and $\bar{f_{\tilde{α}}} (t)$ are differentiable functions in the second form and $\overset{´}{{\tilde{f}}_{\tilde{α}}} (t) = [\overset{´}{\bar{f_{\tilde{α}}}} (t), \overset{´}{\underline{f_{\tilde{α}}}} (t),]$ [13].

Theorem 2.2.3. Let $\tilde{f} (t) = (\bar{L_{{\tilde{f}}_{0}}} (t), L_{{\tilde{f}}_{1}} (t), \underline{L_{{\tilde{f}}_{0}}} (t), m_{\tilde{f}} (t), \underline{R_{{\tilde{f}}_{0}}} (t), R_{{\tilde{f}}_{1}} (t), \bar{R_{{\tilde{f}}_{0}}} (t))$ be the triangular perfect QT2FN-valued function; then it satisfies the following: [13].

if $\tilde{f} (t)$ is H₂-differentiable in the first form then $\overset{'}{\tilde{f} (t)} = (\overset{'}{\bar{L_{{\tilde{f}}_{0}}} (t)}, \overset{'}{L_{{\tilde{f}}_{1}} (t)}, \overset{'}{\underline{L_{{\tilde{f}}_{0}}} (t)}, \overset{'}{m_{\tilde{f}} (t)}, \overset{'}{\underline{R_{{\tilde{f}}_{0}}} (t)}, \overset{'}{R_{{\tilde{f}}_{1}} (t)}, \overset{'}{\bar{R_{{\tilde{f}}_{0}}} (t)})$

if $\tilde{f} (t)$ is H₂-differentiable in the second form then $\tilde{f} (t) = (\overset{'}{\bar{R_{{\tilde{f}}_{0}}} (t)}, \overset{'}{R_{{\tilde{f}}_{1}} (t)}, \overset{'}{\underline{R_{{\tilde{f}}_{0}}} (t)}, \overset{'}{m_{\tilde{f}} (t)}, \overset{'}{\underline{L_{{\tilde{f}}_{0}}} (t)}, \overset{'}{L_{{\tilde{f}}_{1}} (t)}, \overset{'}{\bar{L_{{\tilde{f}}_{0}}} (t)})$

Definition 2.2.5 (Type 2 fuzzy partial derivatives). Let T_i = (a_i, b_i) ⊆ R , i = 1, 2, …, n; then $\tilde{f}$ is called a type-2 fuzzy number-valued function on $Φ = \prod_{i = 1}^{n} T_{i} \subseteq R^{n}$ . $\tilde{f} : Φ \to E_{2}^{n}$ and is called an n-dimensional vector of type-2 fuzzy number-valued functions on T.

Let $\tilde{f} (X) = (\bar{L_{{\tilde{f}}_{0}}} (X), L_{{\tilde{f}}_{1}} (X), \underline{L_{{\tilde{f}}_{0}}} (X), m_{\tilde{f}} (X), \underline{R_{{\tilde{f}}_{0}}} (X), R_{{\tilde{f}}_{1}} (X), \bar{R_{{\tilde{f}}_{0}}} (X))$ , then $\tilde{f} (X)$ is called a triangular perfect QT2FN-valued function if it is a mapping from Φ ⊆ R ⁿ into the set of all the triangular perfect QT2FNs.

Let $\tilde{f} : Φ \to E_{2}$ and $x_{0}^{i} \in (a_{i}, b_{i}) \in R$ . Then $\tilde{f}$ is differentiable at $X_{0} = (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{n}) \in Φ$ if there exists an element $\frac{\partial \tilde{f} (X_{0})}{\partial x^{i}} \in E_{2}, i = 1, 2, \dots, n$ , such that for all h > 0, sufficiently small, then

The functions $\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} + h, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n})$ and $\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} - h, \dots, x_{0}^{n})$ and the following limits converges:

$\begin{matrix} \lim_{h \to 0} \frac{\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} + h, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n})}{h} \\ = \lim_{h \to 0} \frac{\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} - h, \dots, x_{0}^{n})}{h} \\ = \frac{\partial}{\partial x^{i}} \tilde{f} (X_{0}) \end{matrix}$

The functions $\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} + h, \dots, x_{0}^{n})$ and $\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} - h, \dots, x_{0}^{n}) -^{H_{2}}$ $\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n})$ and the following limits converges: $\begin{matrix} \lim_{h \to 0} \frac{\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} + h, \dots, x_{0}^{n})}{h} \\ = \lim_{h \to 0} \frac{\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} - h, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n})}{h} \\ = \frac{\partial}{\partial x^{i}} \tilde{f} (X_{0}) \end{matrix}$

The functions $\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} + h, \dots, x_{0}^{n})$ and $\tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} - h, \dots, x_{0}^{n}) -^{H_{2}} \tilde{f} (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n})$ and the following limits converges: $\begin{matrix} \lim_{h \to 0} \frac{f (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n}) -^{H_{2}} f (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} + h, \dots, x_{0}^{n})}{- h} \\ = \lim_{h \to 0} \frac{f (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i} - h, \dots, x_{0}^{n}) -^{H_{2}} f (x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{i}, \dots, x_{0}^{n})}{- h} \\ = \frac{\partial}{\partial x^{i}} \tilde{f} (X_{0}) \end{matrix}$

Here the limits are taken in the metric space (E₂, d₂).

Theorem 2.2.4. Let $\tilde{f} : Φ \to E_{2}$ be a type-2 fuzzy number-valued function and ${\tilde{f}}_{\tilde{α}} (X) = [\underline{f_{\tilde{α}}} (X), \bar{f_{\tilde{α}}} (X)]$ ; where $\underline{f_{\tilde{α}}} (X)$ and $\bar{f_{\tilde{α}}} (X)$ are the LMF and UMF of $\tilde{α}$ -plane of $\tilde{f} (X)$ respectively, where X = (x¹, x², …, xⁿ) ∈ Φ. Then the following holds. If $\tilde{f} (X)$ is H₂-differentiable in the first form, then $\underline{f_{\tilde{α}}} (X)$ and $\bar{f_{\tilde{α}}} (X)$ are differentiable functions in the first form and $\frac{\partial}{\partial x^{i}} \tilde{f_{\tilde{α}}} (X) = (\frac{\partial}{\partial x^{i}} \underline{f_{\tilde{α}}} (X), \frac{\partial}{\partial x^{i}} \bar{f_{\tilde{α}}} (X))$ . If $\tilde{f} (X)$ is H₂-differentiable in the second form, then $\underline{f_{\tilde{α}}} (X)$ and $\bar{f_{\tilde{α}}} (X)$ are differentiable functions in the second form and $\frac{\partial}{\partial x^{i}} \tilde{f_{\tilde{α}}} (X) = (\frac{\partial}{\partial x^{i}} \bar{f_{\tilde{α}}} (X), \frac{\partial}{\partial x^{i}} \underline{f_{\tilde{α}}} (X))$ .

Proof. Let us consider $\tilde{f} (X)$ to be H₂-differentiable function in the first form, and X ∈ Φ, then we have $\begin{matrix} [\tilde{f} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \\ {\tilde{f} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})]}_{\tilde{α}} \\ = [\underline{f_{\tilde{α}}} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \\ \underline{f_{\tilde{α}}} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n}), \\ {\bar{f}}_{\tilde{α}} (x^{1}, x^{2}, \dots x^{i} + h, \dots, x^{n}) -^{H_{2}} \\ {\bar{f}}_{\tilde{α}} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})] \end{matrix}$ multiplying both side by $\frac{1}{h} > 0$ and taking lim h → 0 we obtain $\begin{matrix} \lim_{h \to 0} {[\frac{\tilde{f} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \tilde{f} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})}{h}]}_{\tilde{α}} \\ = \lim_{h \to 0} [\begin{matrix} \frac{\underline{f_{\tilde{α}}} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \underline{f_{\tilde{α}}} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})}{h}, \\ \frac{\bar{f_{\tilde{α}}} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \bar{f_{\tilde{α}}} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})}{h} \end{matrix}] \end{matrix}$

From the Definition 2.2.5 of type 2 partial differentiability in first form it follows $\frac{\partial}{\partial x^{i}} \tilde{f_{\tilde{α}}} (X) = (\frac{\partial}{\partial x^{i}} \underline{f_{\tilde{α}}} (X), \frac{\partial}{\partial x^{i}} \bar{f_{\tilde{α}}} (X)) .$

$\tilde{f} (X)$ to be H₂-differentiable function in the second form, and X ∈ φ, then we have $\begin{matrix} [\tilde{f} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \\ {\tilde{f} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})]}_{\tilde{α}} \\ = [\underline{f_{\tilde{α}}} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \\ \underline{f_{\tilde{α}}} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n}), \\ \bar{f_{\tilde{α}}} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \\ \bar{f_{\tilde{α}}} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})] \end{matrix}$ multiplying both side by $\frac{1}{- h} > 0$ and taking lim h → 0 we obtain $\begin{matrix} \lim_{h \to 0} {[\frac{\tilde{f} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \tilde{f} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})}{- h}]}_{\tilde{α}} \\ = \lim_{h \to 0} [\begin{matrix} \frac{\bar{f_{\tilde{α}}} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \bar{f_{\tilde{α}}} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})}{- h}, \\ \frac{\underline{f_{\tilde{α}}} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \underline{f_{\tilde{α}}} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})}{- h} \end{matrix}] \end{matrix}$

Therefore from Definition 2.2.5 we obtain $\frac{\partial}{\partial x^{i}} \tilde{f_{\tilde{α}}} (X) = (\frac{\partial}{\partial x^{i}} \bar{f_{\tilde{α}}} (X), \frac{\partial}{\partial x^{i}} \underline{f_{\tilde{α}}} (X))$ . This completes the proof.

Theorem 2.2.5. Let $\tilde{f} (X) = (\bar{L_{{\tilde{f}}_{0}}} (X), L_{{\tilde{f}}_{1}} (X), \underline{L_{{\tilde{f}}_{0}}} (X), m_{\tilde{f}} (X), \underline{R_{{\tilde{f}}_{0}}} (X), R_{{\tilde{f}}_{1}} (X), \bar{R_{{\tilde{f}}_{0}}} (X))$ be the triangular perfect QT2FN-valued function.

If $\tilde{f} (X)$ is H₂-differentiable in the first form then $\frac{\partial}{\partial x^{i}} \tilde{f} (X) = (\begin{matrix} \frac{\partial}{\partial x^{i}} \bar{L_{{\tilde{f}}_{0}}^{}} (X), \frac{\partial}{\partial x^{i}} L_{{\tilde{f}}_{1}}^{} (X), \frac{\partial}{\partial x^{i}} \underline{L_{{\tilde{f}}_{0}}^{}} (X), \frac{\partial}{\partial x^{i}} m_{\tilde{f}} (X), \\ \frac{\partial}{\partial x^{i}} \underline{R_{{\tilde{f}}_{0}}^{}} (X), \frac{\partial}{\partial x^{i}} R_{{\tilde{f}}_{1}}^{} (X), \frac{\partial}{\partial x^{i}} \bar{R_{{\tilde{f}}_{0}}^{}} (X) \end{matrix})$

If $\tilde{f} (X)$ is H₂-differentiable in the second form then $\begin{matrix} \frac{\partial}{\partial x^{i}} \tilde{f} (X) = (\begin{matrix} \frac{\partial}{\partial x^{i}} \bar{R_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} R_{{\tilde{f}}_{1}} (X), \frac{\partial}{\partial x^{i}} \underline{R_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} m_{\tilde{f}} (X), \frac{\partial}{\partial x^{i}} \underline{R_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} \underline{L_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} \bar{L_{{\tilde{f}}_{0}}} (X) \end{matrix}) \end{matrix}$

Proof. Let us suppose $\tilde{f} (X)$ is H₂-differentiable in the first form, then

$\begin{matrix} \lim_{h \to 0} [\frac{\tilde{f} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \tilde{f} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})}{h}] \\ = \lim_{h \to 0} [\begin{matrix} \frac{\bar{L_{{\tilde{f}}_{o}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \bar{L_{{\tilde{f}}_{o}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{h} \\ \frac{\bar{L_{{\tilde{f}}_{1}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \bar{L_{{\tilde{f}}_{1}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{h} \\ \frac{\underline{L_{{\tilde{f}}_{o}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \underline{L_{{\tilde{f}}_{o}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{h} \\ \frac{m_{f} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - m_{\tilde{f}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{h} \\ \frac{\underline{R_{{\tilde{f}}_{o}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \underline{R_{{\tilde{f}}_{o}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{h} \\ \frac{\bar{R_{{\tilde{f}}_{o}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \bar{R_{{\tilde{f}}_{o}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{h} \end{matrix}] \end{matrix}$

i.e. $\begin{matrix} \frac{\partial}{\partial x^{i}} \tilde{f} (X) = (\begin{matrix} \frac{\partial}{\partial x^{i}} \bar{L_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} L_{{\tilde{f}}_{1}} (X), \frac{\partial}{\partial x^{i}} \underline{L_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} m_{\tilde{f}} (X), \frac{\partial}{\partial x^{i}} \underline{R_{{\tilde{f}}_{0}}} (X), \\ \frac{\partial}{\partial x^{i}} R_{{\tilde{f}}_{1}} (X), \frac{\partial}{\partial x^{i}} \bar{R_{{\tilde{f}}_{0}}} (X) \end{matrix}) \end{matrix}$

Let $\tilde{f} (X)$ is H₂-differentiable in the second form, then $\begin{matrix} \lim_{h \to 0} [\frac{\tilde{f} (x^{1}, x^{2}, \dots, x^{i} + h, \dots, x^{n}) -^{H_{2}} \tilde{f} (x^{1}, x^{2}, \dots x^{i}, \dots, x^{n})}{- h}] \\ = \lim_{h \to 0} [\begin{matrix} \frac{\bar{R_{{\tilde{f}}_{0}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \bar{R_{{\tilde{f}}_{0}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{- h}, \\ \frac{\underline{R_{{\tilde{f}}_{0}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \underline{R_{{\tilde{f}}_{0}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{- h}, \\ \frac{m_{\tilde{f}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - m_{\tilde{f}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{h}, \\ \frac{\underline{L_{{\tilde{f}}_{0}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \underline{L_{{\tilde{f}}_{0}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{- h}, \\ \frac{\underline{L_{{\tilde{f}}_{1}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \underline{L_{{\tilde{f}}_{1}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{h}, \\ \frac{\bar{L_{{\tilde{f}}_{0}}} (x^{1}, \dots, x^{i} + h, \dots, x^{n}) - \bar{L_{{\tilde{f}}_{0}}} (x^{1}, \dots, x^{i}, \dots, x^{n})}{- h} \end{matrix}] \end{matrix}$

i.e. $\begin{matrix} \frac{\partial}{\partial x^{i}} \tilde{f} (X) = (\begin{matrix} \frac{\partial}{\partial x^{i}} \bar{R_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} R_{{\tilde{f}}_{1}} (X), \frac{\partial}{\partial x^{i}} \underline{R_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} m_{\tilde{f}} (X), \\ \frac{\partial}{\partial x^{i}} \underline{R_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} \underline{L_{{\tilde{f}}_{0}}} (X), \frac{\partial}{\partial x^{i}} \bar{L_{{\tilde{f}}_{0}}} (X) \end{matrix}) . \end{matrix}$

Hence the theorem is proved.

For example, suppose $\tilde{f} (t, x) = (t + 3 x, 2 t + 4 x, 3 t + 5 x, 4 t + 6 x, 5 t + 7 x, 6 t + 8 x, 7 t + 9 x)$ represents national consumption function where the consumption is about six billion multiplied by x dollars i.e. $\tilde{6}$ when the disposable income t is zero. In this case $\tilde{f} (t, x)$ is H₂-differentiable in the first form, then its derivative is $\frac{\partial \tilde{f}}{\partial t} = (1, 2, 3, 4, 5, 6, 7), \frac{\partial \tilde{f}}{\partial x} = (3, 4, 5, 6, 7, 8, 9)$ which can mean the marginal propensity to consume is about 4 billion dollars. Another case in point: suppose $\tilde{f} (t, x) = (e^{- kt} + k^{- x}, 2 e^{- kt} + k^{- 2 x}, 3 e^{- kt} + k^{- 3 x}, 4 e^{- kt} + k^{- 4 x}, 5 e^{- kt} + k^{- 5 x}, 6 e^{- kt} + k^{- 6 x}, 7 e^{- kt} + k^{- 7 x}), k > 1$ is H₂-differentiable in the second form, then its partial derivatives are

$\begin{matrix} \frac{\partial \tilde{f}}{\partial t} = (\begin{matrix} - 7 {ke}^{- kt}, - 6 {ke}^{- kt}, - 5 {ke}^{- kt}, - 4 {ke}^{- kt}, \\ - 3 {ke}^{- kt}, - 2 {ke}^{- kt}, - {ke}^{- kt} \end{matrix}), \\ \frac{\partial \tilde{f}}{\partial x} = (\begin{matrix} - 7 k^{- 7 x}, - 6 k^{- 6 x}, - 5 k^{- 5 x}, - 4 k^{- 4 x}, \\ - 3 k^{- 3 x}, - 2 k^{- 2 x}, - k^{- x} \end{matrix}) log k . \end{matrix}$

Definition 2.2.6 (Type 2 fuzzy boundary value problem). The fuzzy boundary value problem for type-2 fuzzy system could be written as $F (D) \tilde{X} = \tilde{f} (\tilde{X})$ , with initial conditions $\tilde{X} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) = {\tilde{I}}_{i} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n})$ and boundary conditions $\tilde{X} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}),$ and $\tilde{X} (x^{1}, \dots, x_{0}^{+ r}, \dots, x^{n}) = \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), r = 1, 2, \dots, n$ . Where $\tilde{f} : E_{2}^{n} \to E_{2}^{n}$ is a continuous fuzzy mapping, F (D) is partial differential operator $\tilde{I_{i}}$ is a type 2 fuzzy initial condition function and $[\tilde{B_{r}^{+}}, \tilde{B_{r}^{+}}], r = 1, 2, \dots, n$ are type 2 fuzzy boundary condition function. In terms of $\tilde{α}$ -cut this could be written as ${[\tilde{I_{i}}]}_{\tilde{α}} = [{\underline{I_{i}}}_{\tilde{α}}, {\bar{I_{i}}}_{\tilde{α}}]$ and ${\tilde{f}}_{\tilde{α}} (X) = [\underline{f_{\tilde{α}}} (X), \bar{f_{\tilde{α}}} (X)]$ .

Case I. If we consider $F (D) \tilde{u} (X)$ by using the derivative in the first form, then we have $F (D) \tilde{u} {(X)}_{\tilde{α}} = [\underline{F (D) \tilde{u} {(X)}_{\tilde{α}}} (X), \bar{F (D) \tilde{u} {(X)}_{\tilde{α}}} (X)]$ . Now this could be written as follows: ${\begin{matrix} \underline{F (D) \tilde{u} (X)_{\tilde{α}}} (X) = \underline{f_{\tilde{α}}} (X), \bar{F (D) \tilde{u} (X)_{\tilde{α}}} (X) = \bar{f_{\tilde{α}}} (X) \\ \underline{u} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) = \underline{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}), \\ \bar{u} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) = \bar{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \underline{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \underline{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \bar{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \bar{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \underline{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \underline{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \bar{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \bar{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \end{matrix}$

Case II. If we consider $F (D) \tilde{u} (X)$ by using the derivative in the second form, then we have $F (D) \tilde{u} {(X)}_{\tilde{α}} = [\bar{F (D) \tilde{u} {(X)}_{\tilde{α}}} (X), \underline{F (D) \tilde{u} {(X)}_{\tilde{α}}} (X)]$ . Now this could be written as follows: ${\begin{matrix} \underline{F (D) \tilde{u} (X)_{\tilde{α}}} (X) = \bar{f_{\tilde{α}}} (X), \bar{F (D) \tilde{u} (X)_{\tilde{α}}} (X) = \underline{f_{\tilde{α}}} (X) \\ \underline{u} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) = \underline{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}), \\ \bar{u} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) = \bar{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \underline{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \underline{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \bar{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \bar{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \underline{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \underline{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \bar{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) = \bar{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \end{matrix}$

From the definition of $\tilde{α}$ -plane of type-2 fuzzy number it’s clear that for any particular $\tilde{α}$ -cut a type-2 fuzzy variable is basically an interval valued fuzzy variable i.e. interval type-2 fuzzy variable. Thus for each $\tilde{α}$ -plane the variables and parameter values of a type-2 fuzzy differential equation can be expressed as two intervals.

Definition 2.2.7 (Type-2 fuzzy differential inclusion). The initial value problem (IVP) for type-2 fuzzy differential equation (T2FDE) could be defined based on a family of differential inclusions at each $\tilde{α}$ -plane $0 \leq \tilde{α} \leq 1$ , $\tilde{u} (X) \in {[\tilde{f} (X)]}_{\tilde{α}}, \tilde{x} (X_{0}) \in {[x_{0}]}_{\tilde{α}}$ . Where ${[\tilde{f} (X)]}_{\tilde{α}} : [0, T] \times E_{2}^{n} \to E_{2}^{n}$ and $\overset{´}{x} (t)$ is the usual derivative of the crisp differentiable function x (t) with respect to t. Where $E_{2}^{n}$ is the family of all nonempty type-2 fuzzy compact subsets of Rⁿ.

Similarly the boundary value problem (BVP) for type-2 fuzzy partial differential equation (T2FPDE) could be defined based on a family of differential inclusions at each $\tilde{α}$ -plane $0 \leq \tilde{α} \leq 1$ .

$F (D) \tilde{X} \in {[\tilde{f} (\tilde{X})]}_{\tilde{α}},$ with initial conditions $\tilde{X_{\tilde{α}}} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \in [\underline{I_{i}} (x^{1}, \dots, x^{n}), \bar{I_{i}} (x^{1}, \dots, x^{n})]_{\tilde{α}}$ and boundary conditions $\tilde{X} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n}) \in [\underline{B_{r}^{-}} (x^{1}, \dots, x^{n}), \bar{B_{r}^{-}} (x^{1}, \dots, x^{n})]_{\tilde{α}}$ and $\tilde{X} (x^{1}, \dots, x_{0}^{+ r}, \dots, x^{n}) \in [\underline{B_{r}^{+}} (x^{1}, \dots, x^{n}), \bar{B_{r}^{+}} (x^{1}, \dots, x^{n})]_{\tilde{α}},$ r = 1, 2, …, n. where $\tilde{f_{\tilde{α}}} (X) : Φ \times E_{2}^{n} \to E_{2}^{n}$ and $E_{2}^{n}$ is the family of all nonempty type-2 fuzzy compact subsets of Rⁿ. $F (D) \tilde{X}$ is any partial differential operator where $\frac{\partial}{\partial x^{i}} Y$ is the usual crisp partial derivative of the crisp differentiable function Y with respect to xⁱ.

Definition 2.2.8. Any type-1 fuzzy PDE system $F (D) \tilde{X} = \tilde{f} (\tilde{X})$ , $\begin{matrix} \tilde{X} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ = \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \tilde{X} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \tilde{X} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n \end{matrix}$ is said to be embedded within the type-2 fuzzy PDE system $[F (D) {\tilde{X]}}_{\tilde{α}} = {[\tilde{f} (\tilde{X})]}_{\tilde{α}}$ , $\begin{matrix} {[\tilde{X} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n})]}_{\tilde{α}} \\ {[\tilde{X} (x^{1}, \dots, x_{-}^{r}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ {[\tilde{X} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ r = 1, 2, \dots, n \end{matrix}$ if and only if $\begin{matrix} \tilde{f} (\tilde{X}) \in {[\tilde{f} (\tilde{X})]}_{\tilde{α}}, \\ \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \in {[\tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \\ \in {[\tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \\ \in {[\tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}} . \end{matrix}$

Definition 2.2.9. Any ordinary (crisp) PDE system F (D) X = f (X), $\begin{matrix} X (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ = I_{i} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ X (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ = B_{r}^{-} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ X (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = B_{r}^{+} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n \end{matrix}$ is said to be embedded within the type-1 fuzzy PDE system $F (D) \tilde{X} = \tilde{f} (\tilde{X})$ , $\begin{matrix} \tilde{X} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ = \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \tilde{X} (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \tilde{X} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n \end{matrix}$ if and only if $f (X) \in \tilde{f} (\tilde{X})$ , $I_{i} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \in \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n})$ , $\begin{matrix} B_{r}^{-} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \\ \in \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ B_{r}^{+} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \\ \in \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \end{matrix}$

Theorem 2.2.6. Any solution of the type-1 fuzzy embedded PDE system is also a solution of the type-2 fuzzy PDE system.

Proof. Let us consider a type-2 fuzzy boundary value problem as $[F (D) {\tilde{X]}}_{\tilde{α}} = {[\tilde{f} (\tilde{X})]}_{\tilde{α}}$ ,

$\begin{matrix} {[\tilde{X} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n})]}_{\tilde{α}} \\ {[\tilde{X} (x^{1}, \dots, x_{-}^{r}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ {[\tilde{X} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ r = 1, 2, \dots, n; \end{matrix}$ where ${[\tilde{f}]}_{\tilde{α}} : E_{2}^{n} \to E_{2}^{n}$ is a continuous type-2 fuzzy mapping, F (D) is partial differential operator ${[\tilde{I_{i}}]}_{\tilde{α}}$ is a type 2 fuzzy initial condition function and ${[\tilde{B_{r}^{-}}]}_{\tilde{α}}, {[\tilde{B_{r}^{+}}]}_{\tilde{α}}$ r = 1, 2, …, n are type 2 fuzzy boundary condition functions.

The corresponding type-1 fuzzy embedded system can be written as $F (D) \tilde{X} = \tilde{f} (\tilde{X})$ ,

$\begin{matrix} \tilde{X} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ = \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \tilde{X} (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \tilde{X} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n . \end{matrix}$

Where $\tilde{f}$ , ${\tilde{I}}_{i}$ , ${\tilde{B^{+}}}_{r}$ , ${\tilde{B^{-}}}_{r}$ are some type-1 fuzzy function such that $\tilde{f} (\tilde{X}) \in [\tilde{f} (\tilde{x})]_{\tilde{α}}$ ,

$\begin{matrix} {\tilde{I}}_{i} (x^{1}, \dots, x^{i - 1}, x^{i - 1}, \dots, x^{n}) \\ \in {[{\tilde{I}}_{i} (x^{1}, \dots, x^{i - 1}, x^{i - 1}, \dots, x^{n})]}_{\tilde{α}}, \\ {\tilde{B^{-}}}_{r} (x^{1}, \dots, x^{r - 1}, x^{r - 1}, \dots, x^{n}) \\ \in {[{\tilde{B^{-}}}_{r} (x^{1}, \dots, x^{r - 1}, x^{r - 1}, \dots, x^{n})]}_{\tilde{α}}, \\ {\tilde{B^{+}}}_{r} (x^{1}, \dots, x^{r - 1}, x^{r - 1}, \dots, x^{n}) \\ \in {[{\tilde{B^{+}}}_{r} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}} \end{matrix}$

Let $\tilde{Y}$ be any type-1 fuzzy solution to the embedded system. Then $F (D) \tilde{Y} = \tilde{f} (\tilde{Y})$ , $\begin{matrix} \tilde{Y} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ = \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \tilde{Y} (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \tilde{Y} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n . \end{matrix}$

That is $F (D) \tilde{Y} \in {[\tilde{f} (\tilde{Y})]}_{\tilde{α}}$ , $\begin{matrix} \tilde{Y} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ \in {[\tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n})]}_{\tilde{α}} \\ \tilde{Y} (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ \in {[\tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ \tilde{Y} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ \in {[\tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ r = 1, 2, \dots, n \end{matrix}$

As type-2 fuzzy number is a generalization to type-1 fuzzy numbers therefore we could consider $\tilde{Y}$ to be a type-2 fuzzy number and this shows that $\tilde{Y}$ is a solution to the type-2 fuzzy PDE. $\tilde{Y}$ is eventually any footprint of uncertainty of the solution of corresponding type-2 PDE system. Since $\tilde{Y}$ is any arbitrary solution of the type-1 fuzzy embedded system therefore any solution of the type-1 fuzzy embedded system is a solution to the original type-2 fuzzy PDE system. This completes the proof.

Theorem 2.2.7. Any solution of the ordinary (crisp) embedded PDE system is also a solution of the type-1 fuzzy PDE system.

Proof. Let us consider a type-1 fuzzy boundary value problem as $F (D) \tilde{X} = \tilde{f} (\tilde{X})$ , $\begin{matrix} \tilde{X} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ = \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \tilde{X} (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ \tilde{X} (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n . \end{matrix}$ where $\tilde{f} : E_{1}^{n} \to E_{1}^{n}$ is a continuous type-2 fuzzy mapping, F (D) is partial differential operator $\tilde{I_{i}}$ is a type-1 fuzzy initial condition function and ${\tilde{B^{-}}}_{r}, {\tilde{B^{+}}}_{r}; r = 1, 2, \dots, n$ are type-1 fuzzy boundary condition functions.

The corresponding ordinary embedded system can be written as F (D) X = f (X), $\begin{matrix} X (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ = I_{i} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ X (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ = B_{r}^{-} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ X (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = B_{r}^{+} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n \end{matrix}$

Where f, I_i, $B_{r}^{-}, B_{r}^{+}$ are some ordinary function such that $f (X) \in \tilde{f} (\tilde{X}),$ $\begin{matrix} I_{i} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ \in \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}), \\ B_{r}^{-} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \\ \in \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ B_{r}^{+} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) \\ \in \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}) . \end{matrix}$

Let Y be any type-1 fuzzy solution to the embedded system. Then F (D) Y = f (Y), $\begin{matrix} Y (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ = I_{i} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ Y (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ = B_{r}^{-} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ Y (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ = B_{r}^{+} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n \end{matrix}$

That is $F (D) Y \in \tilde{f} (\tilde{X})$ , $\begin{matrix} Y (x^{1}, \dots, x_{0}^{i}, \dots, x^{n}) \\ \in \tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n}) \\ Y (x^{1}, \dots, x_{-}^{r}, \dots, x^{n}) \\ \in \tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ Y (x^{1}, \dots, x_{+}^{r}, \dots, x^{n}) \\ \in \tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n}), \\ r = 1, 2, \dots, n \end{matrix}$

As type-1 fuzzy number is a generalization to ordinary crisp numbers therefore we could consider Y to be a type-1 fuzzy number and this shows that Y is a solution to the type-1 fuzzy PDE. Y is eventually any footprint of uncertainty of the solution of the corresponding type-1 PDE system. Since Y is any arbitrary solution of the ordinary (crisp) embedded system therefore any solution of the ordinary (crisp) embedded system is a solution to the type-1 fuzzy PDE system. This completes the proof.

3 Numerical solution of type-2 fuzzy partial differential equation

To solve the fuzzy arbitrary order differential systems, it might be seem intuitive to apply the interval mathematics directly to the numerical algorithms for deterministic systems. However the straightforward use of interval calculus for the resolution of fuzzy fractional differential equation can produce incorrect results for even type 1 fuzzy systems. As a consequence, spurious values are introduced into the solution and the evolution of the system may reach to a region where no numerical solution exists [23].

Consider a system of PDEs in the vector form where variable and parameter values can be expressed as intervals. Let n is the order of the system, and k is the number of interval-valued parameters. Then $\begin{matrix} {[F (D) \tilde{u} (X)]}_{\tilde{α}} = {[\tilde{f} (X)]}_{\tilde{α}} \\ {[\tilde{u} (x^{1}, \dots, x_{0}^{i}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{I_{i}} (x^{1}, \dots, x^{i - 1}, x^{i + 1}, \dots, x^{n})]}_{\tilde{α}} \\ {[\tilde{u} (x^{1}, \dots, x_{0}^{- r}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{B_{r}^{-}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}} \\ {[\tilde{u} (x^{1}, \dots, x_{0}^{+ r}, \dots, x^{n})]}_{\tilde{α}} \\ = {[\tilde{B_{r}^{+}} (x^{1}, \dots, x^{r - 1}, x^{r + 1}, \dots, x^{n})]}_{\tilde{α}}, \\ r = 1, 2, \dots, n . \end{matrix}$

Where I_i ; i = 1, 2, …… is the vector of interval initial conditions, B_r ; r = 1, 2, …, n is the interval boundary condition and $\tilde{P_{\tilde{α}}} = {[\tilde{P_{1}}, \tilde{P_{2}}, \dots, \tilde{P_{k}}]}_{\tilde{α}}$ be the parameter vector where the parameters are generally fuzzy numbers. Then similarly N = n + k interval valued entities would generate the region of uncertainty and the foot print of uncertainty of the solution of fuzzy partial dynamical systems thus is the union of the solution of the corresponding ordinary partial dynamical systems with any initial condition within the region of uncertainty.

Now for each $\tilde{α} -$ plane the region of uncertainty is constituted by ${[\tilde{u} (X)]}_{\tilde{α}}$ and ${[\tilde{P_{1}}, \tilde{P_{2}}, \dots, \tilde{P_{k}}]}_{\tilde{α}}$ i.e. $[\underline{u_{\tilde{α}}} (X), \bar{u_{\tilde{α}}} (X)]$ and $[\underline{P_{\tilde{α} i}}, \bar{P_{\tilde{α} i}}], i = 1, 2, \dots, k$ ; which could be characterized by $[\underline{u_{l \tilde{α}}} (X), \underline{u_{u \tilde{α}}} (X), \bar{u_{l \tilde{α}}} (X), \bar{u_{u \tilde{α}}} (X)]$ and $[\underline{P_{l \tilde{α}}}, \underline{P_{u \tilde{α}}}, \bar{P_{l \tilde{α}}}, \bar{P_{u \tilde{α}}}], i = 1, 2, \dots, k$ ; where $\underline{w_{l \tilde{α}}}, \underline{w_{u \tilde{α}}}, \bar{w_{l \tilde{α}}}$ and $\bar{w_{u \tilde{α}}}$ are respectively represent the lower and bound of the lower interval and of the upper interval of the type 2 fuzzy variable $\tilde{w}$ respectively for a particular $\tilde{α} -$ plane. The evolution of surface of uncertainty is carried out by taking the initial condition, boundary conditions and parameter values as any arbitrary footprint of uncertainty of the fuzzy variables as follows: $\begin{matrix} I \in [\underline{I_{l α}}, \underline{I_{u α}}, \bar{I_{l α}}, \bar{I_{u α}}], \\ B_{r}^{-} \in [\underline{B_{l α}^{- r}}, \underline{B_{u α}^{- r}}, \bar{B_{l α}^{- r}}, \bar{B_{u α}^{- r}}], \\ B_{r}^{+} \in [\underline{B_{l α}^{+ r}}, \underline{B_{u α}^{+ r}}, \bar{B_{l α}^{+ r}}, \bar{B_{u α}^{+ r}}], \\ r = 1, 2, \dots, n \end{matrix}$ and $P_{i} \in [\underline{P_{l α}^{i}}, \underline{P_{u α}^{i}}, \bar{P_{l α}^{i}}, \bar{P_{u α}^{i}}];$

Then the above fuzzy partial dynamical systems is converted into sets of ordinary partial dynamical systems with any arbitrary ordinary functions $f (X), I, B_{r}^{+}, B_{r}^{-}$ and parameters P as follows: $\begin{matrix} F (D) u (X) = f (X), \\ u (x^{1}, \dots, x_{0}^{i} \dots, x^{n}) \\ = I (x^{1}, \dots, x^{i - 1}, x^{i + 1} \dots, x^{n}) \end{matrix}$ and $\begin{matrix} u (x^{1}, \dots, x_{r}^{-} \dots, x^{n}) \\ = B_{r}^{-} (x^{1}, \dots, x^{r - 1}, x^{r + 1} \dots, x^{n}), \\ u (x^{1}, \dots, x_{r}^{+} \dots, x^{n}) \\ = B_{r}^{+} (x^{1}, \dots, x^{r - 1}, x^{r + 1} \dots, x^{n}), \\ r = 1, \dots \dots, i - 1, i + 1, \dots, n . \end{matrix}$

Now in the consequence if ${\tilde{v_{\tilde{α}}} : \tilde{v_{\tilde{α}}} (X) \in [\underline{v_{l α}}, \underline{v_{u α}}, \bar{v_{l α}}, \bar{v_{u α}}]}$ , where the initial condition and boundary conditions f (X) , I and B_r satisfies the above conditions; then any such function $\tilde{v_{\tilde{α}}} (X)$ is a solution to the given fuzzy partial dynamical systems. From the definition of fuzzy differential inclusion it is clear that this solution to the fuzzy partial dynamical systems is a fuzzy differential inclusion. We numerically simulate the solution of the PDE and find the corresponding $\tilde{u_{\tilde{α}}} (X)$ .

Finally, find the maximum and minimum value of each state variable from all the above solutions to obtain $[\underline{u_{l α} (X)}, \underline{u_{u α} (X)}, \bar{u_{l α} (X)}, \bar{u_{u α} (X)}]$ .

Thus the solution of a type-2 fuzzy PDE is obtained by solving the corresponding embedded system.

4 Illustrative examples of solving type-2 fuzzy partial differential equation

In this section we have considered three different PDE, namely heat equation, advection diffusion equation and wave equation. These are some well-known mathematical models of parabolic, elliptic and hyperbolic PDEs respectively. The state variables and the initial condition of each is considered to be imprecise and represented as triangular T2FNs. Hence we extend those PDE systems into type-2 fuzzy PDE systems. In this section each model example for FPDE is solved by the above mentioned algorithm in Section 3 based on corresponding embedded system and fuzzy differential inclusion. The results of each FPDE system are described graphically with respect to different independent variables. For different $\tilde{α} -$ plane the distribution of state variables are represented in the form $[\underline{u_{l α} (X)}, \underline{u_{u α} (X)}, \bar{u_{l α} (X)}, \bar{u_{u α} (X)}]$ as described in Section 3 and the findings are described in the Result and discussion section.

4.1 PDE with one spatial variable

4.1.1 Heat Equation in one spatial dimension

Let us suppose the function u describes the temperature at a given location. This function will change over time as heat spreads throughout space. The heat equation is used to determine the change in the function u over time. The rate of change of u is proportional to the “curvature” of u. Over time, the tendency is for peaks to be eroded, and valleys to be filled in. If u is linear in space (or has a constant gradient) at a given point, then u has reached a steady-state and is unchanging at this point (assuming a constant thermal conductivity).

An interesting property of heat equation is that even if u has a discontinuity at an initial time t = t₀, the temperature becomes smooth as soon as t > t₀. For example, if a bar of metal has temperature t₁ and another has temperature t₂ (t₁ > t₂) and they are stuck together end to end, then very quickly the temperature at the point of connection will become t₃, t₁ > t₃ > t₂ and the graph of the temperature will run smoothly from t₁ to t₂.

The heat equation in one spatial dimension is given below: $\frac{\partial u}{\partial t} = α \frac{\partial^{2} u}{\partial x^{2}} .$ With initial condition $u (0, x) = c (2 \sin (\frac{π x}{2}) - \sin (π x) + 4 \sin (2 π x))$ and boundary condition u (t, 0) =0 and u (t, L) =0. Here α is considered to be 0.3.

The distribution of u is provided graphically for different α-cuts (α = 0, 0.2, 0.4, 0.6, 0.8, 1). The upper value of upper interval are represented by red, lower value of upper interval by black, upper value of lower interval by blue and lower value of lower interval by green.

4.1.2 Wave equation in one spatial dimension

The wave equation is a hyperbolic partial differential equation. It typically concerns a time variable t, one or more spatial variables x₁, x₂, …, x_n, and a scalar function u = u (x₁, x₂, …, x_n, t). The wave equation for u is solutions of this equation describe propagation of disturbances out from the region at a fixed speed in one or all spatial directions, as do physical waves from plane or localized sources; the constant α is identified with the propagation speed of the wave which is a linear equation. Therefore, the sum of any two solutions is again a solution: in physics this property is called the superposition principle. The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments. The wave equation in one spatial dimension is given below: $\frac{\partial^{2} u}{\partial t^{2}} = α \frac{\partial^{2} u}{\partial x^{2}}$

With initial condition $u (0, x) = c (\cos (x)) and \frac{\partial}{\partial t} u (0, x) = 0$ and boundary condition u (t, 0) =0 and u (t, L) =0, Here α is considered to be 0.09.

4.2 PDE with two spatial variables

4.2.1 Advection-diffusion equation in two spatial dimension

The advection–diffusion equation is a combination of the diffusion and advection equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Where u is the variable of interest which might be species concentration for mass transfer or temperature for heat transfer. d is the diffusivity (also called diffusion coefficient), such as mass diffusivity for particle motion or thermal diffusivity for heat transport, $\vec{v}$ is the average velocity with which the quantity is moving. For example, in advection, c might be the concentration of salt in a river, and then $\vec{v}$ would be the velocity of the water flow. As another example, c might be the concentration of small bubbles in a calm lake, and then $\vec{v}$ would be the average velocity of bubbles rising towards the surface by buoyancy (see below). For multiphase flows and flows in porous media, $\vec{v}$ is the (hypothetical) superficial velocity. λ is the advection coefficient. S describes “sources” or “sinks” of the quantity c. For example, for a chemical species, R > 0 means that a chemical reaction is creating more of the species, and S < 0 means that a chemical reaction is destroying the species. For heat transport, S > 0 might occur if thermal energy is generated by friction. The advection diffusion equation in two spatial dimension is given below: $\frac{\partial u}{\partial t} = d (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}) - v_{x} \frac{\partial u}{\partial x} - v_{y} \frac{\partial u}{\partial y} - λ u + S$

With initial condition $u (0, x) = \tilde{c_{\tilde{α}}} e^{(- \frac{{(\sqrt{x^{2} + y^{2}} + 0.5)}^{2}}{0.00125})}$ and boundary condition $\begin{matrix} u (t, x, 0) = 0.025 \frac{e^{(- \frac{{(0.5 - t)}^{2}}{0.00125 + 0.4 t})}}{\sqrt{0.000625 + 0.02 t}}, \\ u (t, x, L_{y}) = 0.025 \frac{e^{(- \frac{{(2.5 - t)}^{2}}{0.00125 + 0.4 t})}}{\sqrt{0.000625 + 0.02 t}}, \end{matrix}$

$\begin{matrix} u (t, 0, y) = 0.025 \frac{e^{(- \frac{{(0.5 - t)}^{2}}{0.00125 + 0.4 t})}}{\sqrt{0.000625 + 0.02 t}} and \\ u (t, L_{x}, y) = 0.025 \frac{e^{(- \frac{{(1.5 - t)}^{2}}{0.00125 + 0.4 t})}}{\sqrt{0.000625 + 0.02 t}} \end{matrix}$

Here we consider the parameter values as follows d = 0.01, v_x = 1.1, v_y = 0.9, λ = 0.06, S = 0;

5 Results and discussion

After solving different one dimensional and two dimensional PDE systems numerically we have obtained some results that are shown with corresponding graphical representation.

In Section 4.1 two PDEs in one spatial dimension is considered. Here the type-2 fuzzy heat equation is described in 4.1.1 and type-2 fuzzy wave equation in one spatial dimension are described in 4.1.2. In Section 4.1.1 type-2 fuzzy heat equation is solved and the plot for the temperature variable u verses time t is represented in Figs. 4.1.1.1 and 4.1.1.2 which describes the plot for the temperature variable u verses spatial variable x.

Fig.4.1.1.1

Plot of t verses u for a fixed value of x at.

Fig.4.1.1.2

Plot of x verses u for a fixed value of t.

In Section 4.1.2 type-2 fuzzy wave equation is solved and the plot for the variable u verses time t is represented in Figs. 4.1.2.1 and 4.1.2.2 which describes the plot for the temperature variable u verses spatial variable x.

Fig.4.1.2.1

Plot of t verses u for a fixed value of x.

Fig.4.1.2.2

Plot of x verses u for a fixed value of t.

In Section 4.2 the type-2 advection diffusion equation is considered in two spatial dimensions. Type-2 fuzzy advection diffusion equation is solved and the plot for the concentration variable u verses time t is represented in Figs. 4.2.1.1, 4.2.1.2 which describes the plot for the concentration variable u verses spatial variable x and Fig. 4.2.1.3 describes the plot for the concentration variable u verses spatial variable y.

Fig.4.2.1.1

Plot of t verses u for a fixed value of x.

Fig.4.2.1.2

Plot of x verses u for a fixed value of t.

Fig.4.2.1.3

Plot of y verses u for a fixed value of t.

From the above results we can notice that at α-level 1 (α = 1) the solution for all preceding PDEs converges with the type-1 fuzzy solution of the corresponding models. Also the defuzzyfication of all the models yields the result for the corresponding PDE i.e. the solution of the corresponding deterministic PDE with defuzzyfied values of parameters and/or initial conditions and/or boundary conditions are the same as the defuzzified value of the solution of T2FPDEs in all of the above case. Hence the proposed method illustrate the impact of the imprecision and uncertainty of the initial condition, boundary conditions and/or parameter values of different PDEs on the solution (state variables) of the dynamical systems. Moreover, at the absolute precision level or grade of membership is at α-level 1 (α = 1) the solution coincide with the ordinary dynamical system. Thus a T2FPDE gives a generalized solution of ordinary dynamical systems in this sense the solution obtained by solving T2FPDE is more general than the corresponding PDE.

The solution of a T2FPDE is obtained by solving the corresponding embedded system which is an embedment of ordinary PDEs with the initial condition, boundary conditions and parameter values within the uncertainty region. The proposed algorithm provides an acceptable and reliable solution of T2FPDE and could be successfully applied to other nonlinear T2FPDEs.

6 Conclusions

Studying type-2 fuzzy partial differential equations (T2FPDE) in some detail, we can conclude that for any system where the membership functions of some uncertain variables/parameters/functions and data itself are uncertain, T2FSs are one of the best alternative to model such uncertain system. The main objective of this paper is to define the type-2 fuzzy partial differential inclusion and develop a methodology to obtain numerical solution of T2FPDE. The representation of T2FPDEs is based on differentiability of the type-2 fuzzy number-valued functions, i.e., H₂-differentiability.

H₂-differentiability is a more general definition of strongly generalized differentiability since it is based on type-2 fuzzy number-valued function and type-2 Hukuhara difference. When the vertical slices of the type-2 fuzzy number-valued function are singleton, type-2 fuzzy number-valued function and type-2 Hukuhara difference are reduced to the type-1 fuzzy number-valued function and Hukuhara difference respectively. Consequently, T2FPDEs are the generalization of T1FPDEs, but the computational complexity of T2FPDEs is higher than of T1FPDEs to solve numerically. Also, the definition of type-2 fuzzy differential inclusion is developed so that if vertical slices of the type-2 fuzzy number-valued function are singleton, type-2 fuzzy differential inclusion reduces to type-1 fuzzy differential inclusion, which is verified by the plot of numerical solutions when at α-level 1, both the lower value and the upper value of the upper and lower interval of the state variables are overlapped. Also the defuzzification of the solution of all the models yields the result for the corresponding ordinary PDE i.e. the solution of original ordinary PDE with defuzzified values of parameters and/or initial conditions and/or boundary conditions is the same as the defuzzified value of the solution of the T2FPDEs in all the cases. This verifies and validates the obtained result.

In this paper, we have presented the definition of type-2 fuzzy differential inclusion and some related theorems, a methodology to solve T2FPDEs and the parametric closed form of the triangular perfect QT2FNs. In future studies, we may investigate the n^th order T2FPDEs and type-2 fuzzy based on type-2 fuzzy differential inclusion.

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