A study on the fuzzy parabolic Volterra partial integro-differential equations

Abstract

This paper is concerned with aspects of the analytical fuzzy solutions of the parabolic Volterra partial integro-differential equations under generalized Hukuhara partial differentiability and it consists of two parts. The first part of this paper deals with aspects of background knowledge in fuzzy mathematics, with emphasis on the generalized Hukuhara partial differentiability. The existence and uniqueness of the solutions of the fuzzy Volterra partial integro-differential equations by considering the type of [gH - p]-differentiability of solutions are proved in this part. The second part is concerned with the central themes of this paper, using the fuzzy Laplace transform method for solving the fuzzy parabolic Volterra partial integro-differential equations with emphasis on the type of [gH - p]-differentiability of solution. We test the effectiveness of method by solving some fuzzy Volterra partial integro-differential equations of parabolic type.

Keywords

Fuzzy laplace transform generalized hukuhara partial differentiable fuzzy parabolic volterra partial integro-differential equation fuzzy triangular functions

1 Introduction

This paper is concerned with the aspects of the analytical solution of fuzzy parabolic Volterra partial integro-differential equations. The problem may be defined as follows. Suppose that Ω be a bounded domain in $ℝ$ and let ψ (η, ξ) is the unknown fuzzy function of reference position η and time ξ ≥ 0. A fuzzy linear Volterra partial integro-differential equation of parabolic type is

$\begin{matrix} \partial_{ξ ξ_{gH}} ψ (η, ξ) = \partial_{η_{gH}} ψ (η, ξ) \oplus c ψ (η, ξ) \oplus ϖ (η, ξ) \\ \oplus \int_{0}^{ξ} k (η, ξ - τ) ψ (η, τ) d τ \end{matrix}$ with prescribed fuzzy conditions. Here ∂_{ξ
_gH}ψ (η, ξ) is the generalized Hukuhara partial derivative with respect to ξ of fuzzy function ψ (η, ξ) [7] defined by $\begin{matrix} \partial_{ξ_{gH}} ψ (η, ξ) = lim_{h \to 0} \frac{ψ (η, ξ + h) ⊝_{gH} ψ (η, ξ)}{h}, \end{matrix}$ where $\underset{gH}{⊖}$ is the generalized Hukuhara difference of two fuzzy number [11]. It is assumed that ϖ (η, ξ) is a fuzzy function, k (η, ξ - τ) is a given real-valued function and sufficiently smooth functions. Moreover the coefficient c can be a fuzzy or real constant.

The concepts of fuzzy partial differentiability based on Hukurara difference for the fuzzy multivariable functions are examined by Buckley and Feuring [9]. The fuzzy partial differential equations is used to modeling hydrogeological systems in [12]. Maria Bertone [8] arrived at the fuzzy solutions of heat, wave and Poisson’ equations utilizing fuzzification of the deterministic solutions. Allahviranloo [7] adapted the fuzzy heat equation for the fuzzy partial differential equation through pertinent fuzzy initial conditions, found the corresponding analytical solution, as well substantiated exclusiveness of the solution to this equation and Gouyandeh et al. [13] introduced the fuzzy Fourier transform and investigated the analytical solution of fuzzy heat equation under generalized Hukuhara partial differentiability based on this transform. Of late in [3–5 , 14] a novel approach has been presented for linear fuzzy partial differential equations and fuzzy partial integro-differential equations. Qiu [21] introduced a new characterization of the generalized Hukuhara differentiability of interval-valued functions. The Fuzzy Laplace transform have been introduced in [6, 24]. They used this concept to solve the fuzzy fractional differential equation defined by the concept of Hukuhrara differentiability or the strongly generalized differentiability for fractional derivative. But, the fuzzy differential equations expressed by these concepts of differentiability do not have a unique solution. Moreover, they converted fuzzy differential equation to two crisp equations and then used crisp Laplace transform for each crisp differential equation. In this study we use the full fuzzy Laplace transform for obtain the fuzzy solution for the fuzzy Volterra partial integro-differential equation without converting this equation to two crisp equations.

We now give a brief outline of the main parts of the paper and state the aims and objective of each part. Section 2 deals with aspects of background knowledge in fuzzy mathematics and confirmation of some new theorems and lemmas to be used in the major part of the article, with emphasis on the generalized Hukuhara partial differentiability. The two-variables fuzzy Laplace transform is presented Section 3 transform and some new properties and theorems are proved for this fuzzy transform. Section 4 Uniqueness of Fuzzy Solution study the existence and uniqueness of the fuzzy solutions to the fuzzy Volterra partial integro-differential equation by considering the type of generalized Hukuhara partial differentiability. Section 5 is the central part of this paper, in which we are concerned with aspects of the fuzzy analytical solution of parabolic Volterra partial integro-differential equations using the fuzzy Laplace transform. In addition, some examples are provided to shed light on the details in Section 6 and the efficacy of the method and conclusions are given in Section 7.

2 Preliminaries

In the following, we focus on the basic definitions and the necessary notation which will be used throughout the paper. Let $E$ is the set of fuzzy numbers and $A, B \in E$ . The generalized Hukuhara difference for these fuzzy numbers is defined as follows [11] $\begin{matrix} A ⊝_{gH} B = C \Leftrightarrow {\begin{matrix} (i) . A = B + C; \\ or (ii) . B = A + (- 1) C . \end{matrix} \end{matrix}$ In terms of α-levels we have [A_⊝gHB] ^α = [min {a^- (α) - b^- (α) , a⁺ (α) - b⁺ (α)} , max {a^- (α) - b^- (α) , a⁺ (α) - b⁺ (α)} ], where [A] ^α = [a^- (α) , a⁺ (α)] and [B] ^α = [b^- (α) , b⁺ (α)]. The necessary conditions for the existence of $C = A ⊝_{gH} B \in E$ are $\begin{matrix} case (i) {\begin{matrix} c^{-} (α) = a^{-} (α) - b^{-} (α) and c^{+} (α) = a^{+} (α) - b^{+} (α) \forall α \in [0, 1]; \\ with c^{-} (α) increasing, c^{+} (α) decreasing, c^{-} (α) \leq c^{+} (α) . \end{matrix} \\ case (ii) {\begin{matrix} c^{-} (α) = a^{+} (α) - b^{+} (α) and c^{+} (α) = a^{-} (α) - b^{-} (α) \forall α \in [0, 1]; \\ with c^{-} (α) increasing, c^{+} (α) decreasing, c^{-} (α) \leq c^{+} (α) . \end{matrix} \end{matrix}$ It is easy to show that (i) and (ii) are both valid if and only if C is a crisp number [23].

If A = (a₁, a₂, a₃) And B = (b₁, b₂, b₃) are two triangular fuzzy numbers, then the generalized Hukuhara difference, $A \underset{gH}{⊖} B$ is

$\begin{matrix} A ⊝_{gH} B = C \Leftrightarrow (i) . C = (a_{1} - b_{1}, a_{2} - b_{2}, a_{3} - b_{3}), \\ or, (ii) . C = (a_{3} - b_{3}, a_{2} - b_{2}, a_{1} - b_{1}) . \end{matrix}$

Actually $\begin{matrix} A ⊝_{gH} B = (min {a_{1} - b_{1}, a_{3} - b_{3}}, a_{2} - b_{2}, \\ max {a_{1} - b_{1}, a_{3} - b_{3}}) . \end{matrix}$

Remark 1. Throughout the rest of this paper, we assume that $A ⊝_{gH} B \in E$ .

Let us to assume that $ψ (η, ξ) : ℝ \times [0, \infty) \to E$ be a fuzzy valued function of two independent variables which ψ (η, ξ) = (ψ₁ (η, ξ) , ψ₂ (η, ξ) , ψ₃ (η, ξ)), where ψ_i (η, ξ), i = 1, 2, 3 are real-valued functions. In the following, we will state and prove several important theorems and lemmas.

Lemma 2.1. Assume that {ψ_n} be a fuzzy sequence. If there is a fuzzy positive numbers M_n such that $\begin{matrix} \forall n \geq 1, \forall η, ξ \in ℝ, \forall ψ_{n} \in A \\ D (ψ_{n} (η, ξ), 0) \leq M_{n}, \sum_{n = 1}^{\infty} M_{n} ≺ \infty . \end{matrix}$ Then the fuzzy series $\sum_{n = 1}^{\infty} ψ_{n} (η, ξ)$ converges absolutely and uniformly on A.

Proof. Let {ψ_n} be a fuzzy sequence [15]. To show that series Ψ (η, ξ) is convergence, just enough to show that the fuzzy partial sums is convergence. Suppose that $Ψ (η, ξ) = \sum_{n = 1}^{\infty} ψ_{n} (η, ξ)$ and $S_{m} (η, ξ) = \sum_{n = 1}^{m} ψ_{n} (η, ξ)$ such that S_m (η, ξ) is a uniform convergence to Ψ (η, ξ), then $\begin{matrix} D (Ψ (η, ξ), S_{m} (η, ξ)) \\ = D (\sum_{n = 1}^{\infty} ψ_{n} (η, ξ), \sum_{n = 1}^{m} ψ_{n} (η, ξ)) \\ = D (\sum_{n = m + 1}^{\infty} ψ_{n} (η, ξ), 0) \\ \leq \sum_{n = m + 1}^{\infty} D (ψ_{n} (η, ξ), 0) \leq \sum_{n = m + 1}^{\infty} M_{n} . \end{matrix}$ So the desired result was achieved.□

Proposition 2.1. Let λ₁ and λ₂ be two real constants such that λ₁, λ₂ ≥ 0 (or λ₁, λ₂ ≤ 0). If ψ (η, ξ) is a fuzzy valued function, then $λ_{1} ⊙ ψ (η, ξ) \underset{gH}{⊖} λ_{2} ⊙ ψ (η, ξ) = (λ_{1} - λ_{2}) ⊙ ψ (η, ξ) .$

Proof. Suppose that λ₁ and λ₂ are real positive constants, then by using the definition of Hukuhara difference λ₁ ⊙ ψ (η, ξ) = (λ₁ψ₁ (η, ξ) , λ₁ψ₂ (η, ξ) , λ₁ψ₃ (η, ξ) ) , and λ₂ ⊙ ψ (η, ξ) = (λ₂ψ₁ (η, ξ) , λ₂ψ₂ (η, ξ) , λ₂ψ₃ (η, ξ) ) . Now, we have two cases

[1.] If λ₁ ≥ λ₂, we have $\begin{matrix} λ_{1} ⊙ ψ (η, ξ) \underset{gH}{⊖} λ_{2} ⊙ ψ (η, ξ) \\ = ((λ_{1} - λ_{2}) ψ_{1} (η, ξ), (λ_{1} - λ_{2}) ψ_{2} (η, ξ), \\ (λ_{1} - λ_{2}) ψ_{3} (η, ξ)) = (λ_{1} - λ_{2}) ⊙ ψ (η, ξ) . \end{matrix}$

[2.] If λ₁ ≤ λ₂, therefore λ₁ - λ₂ ≤ 0, then $\begin{matrix} λ_{1} ⊙ ψ (η, ξ) \underset{gH}{⊖} λ_{2} ⊙ ψ (η, ξ) \\ = (λ_{1} ψ_{1} (η, ξ), λ_{1} ψ_{2} (η, ξ), λ_{1} ψ_{3} (η, ξ)) \\ \underset{gH}{⊖} (λ_{2} ψ_{1} (η, ξ), λ_{2} ψ_{2} (η, ξ), λ_{2} ψ_{3} (η, ξ)) \\ = (\min {λ_{1} ψ_{1} (η, ξ) - λ_{2} ψ_{1} (η, ξ), λ_{1} ψ_{3} \\ (η, ξ) - λ_{2} ψ_{3} (η, ξ)}, λ_{1} ψ_{2} (η, ξ) - λ_{2} ψ_{2} \\ (η, ξ), \max {λ_{1} ψ_{1} (η, ξ) - λ_{2} ψ_{1} (η, ξ), λ_{1} ψ_{3} \\ (η, ξ) - λ_{2} ψ_{3} (η, ξ)}) \\ = (\min {(λ_{1} - λ_{2}) ψ_{1} (η, ξ), (λ_{1} - λ_{2}) ψ_{3} \\ (η, ξ)}, (λ_{1} - λ_{2}) ψ_{2} (η, ξ), \max {(λ_{1} - λ_{2}) \\ ψ_{1} (η, ξ), (λ_{1} - λ_{2}) ψ_{3} (η, ξ)}) \\ = ((λ_{1} - λ_{2}) ψ_{3} (η, ξ), (λ_{1} - λ_{2}) ψ_{2} (η, ξ), \\ (λ_{1} - λ_{2}) ψ_{1} (η, ξ)) \\ = (λ_{1} - λ_{2}) ⊙ ψ (η, ξ) . \end{matrix}$

If λ₁ and λ₂ are real negative constants, the procedure is similar and we omit the details. ▭

Theorem 2.2. Assume that $D \subseteq ℝ \times [0, \infty)$ and ψ (η, ξ) be a [gH - p]-differentiable fuzzy function with respect to ξ such that the type of [gH - p]-differentiability does not change in $D$ . Moreover g (ξ) is a differentiable real-valued function, then

g (ξ) >0, g′ (ξ) >0

If ψ (η, ξ) is [i - p]-differentiable, then ∂_{ξ
_i.gH} (g (ξ) ⊙ ψ (η, ξ) ) = g′ (ξ) ⊙ ψ (η, ξ) ⊕ g (ξ) ⊙ ∂_{ξ
_gH}ψ (η, ξ) ,

If ψ (η, ξ) is [ii - p]-differentiable, then $\partial_{ξ_{i . gH}} (g (ξ) ⊙ ψ (η, ξ)) = g^{'} (ξ) ⊙ ψ (η, ξ) \underset{gH}{⊖} (- 1) g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ),$

If ψ (η, ξ) is [ii - p]-differentiable, then $\partial_{ξ_{ii . gH}} (g (ξ) ⊙ ψ (η, ξ)) = g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ) \underset{gH}{⊖} (- 1) g^{'} (ξ) ⊙ ψ (η, ξ) .$

g (ξ) >0, g′ (ξ) <0

If ψ (η, ξ) is [ii - p]-differentiable, then ∂_{ξ
_ii.gH} (g (ξ) ⊙ ψ (η, ξ) ) = g′ (ξ) ⊙ ψ (η, ξ) ⊕ g (ξ) ⊙ ∂_{ξ
_gH}ψ (η, ξ) ,

If ψ (η, ξ) is [i - p]-differentiable, then $\partial_{ξ_{ii . gH}} (g (ξ) ⊙ ψ (η, ξ)) = g^{'} (ξ) ⊙ ψ (η, ξ) \underset{gH}{⊖} (- 1) g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ),$

If ψ (η, ξ) is [i - p]-differentiable, then $\partial_{ξ_{i . gH}} (g (ξ) ⊙ ψ (η, ξ)) = g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ) \underset{gH}{⊖} (- 1) g^{'} (ξ) ⊙ ψ (η, ξ) .$

g (ξ) <0, g′ (ξ) >0

If ψ (η, ξ) is [ii - p]-differentiable, then ∂_{ξ
_i.gH} (g (ξ) ⊙ ψ (η, ξ) ) = g′ (ξ) ⊙ ψ (η, ξ) ⊕ g (ξ) ⊙ ∂_{ξ
_gH}ψ (η, ξ) ,

If ψ (η, ξ) is [i - p]-differentiable, then $\partial_{ξ_{i . gH}} (g (ξ) ⊙ ψ (η, ξ)) = g^{'} (ξ) ⊙ ψ (η, ξ) \underset{gH}{⊖} (- 1) g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ),$

If ψ (η, ξ) is [i - p]-differentiable, then $\partial_{ξ_{ii . gH}} (g (ξ) ⊙ ψ (η, ξ)) = g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ) \underset{gH}{⊖} (- 1) g^{'} (ξ) ⊙ ψ (η, ξ) .$

g (ξ) <0, g′ (ξ) <0

If ψ (η, ξ) is [i - p]-differentiable, then ∂_{ξ
_ii.gH} (g (ξ) ⊙ ψ (η, ξ) ) = g′ (ξ) ⊙ ψ (η, ξ) ⊕ g (ξ) ⊙ ∂_{ξ
_gH}ψ (η, ξ) ,

If ψ (η, ξ) is [ii - p]-differentiable, then $\partial_{ξ_{ii . gH}} (g (ξ) ⊙ ψ (η, ξ)) = g^{'} (ξ) ⊙ ψ (η, ξ) \underset{gH}{⊖} (- 1) g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ),$

If ψ (η, ξ) is [ii - p]-differentiable, then $\partial_{ξ_{i . gH}} (g (ξ) ⊙ ψ (η, ξ)) = g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ) \underset{gH}{⊖} (- 1) g^{'} (ξ) ⊙ ψ (η, ξ) .$

Proof. We are going to prove case 3 - 3.

3-3 Let g (ξ) <0, g′ (ξ) >0 and ψ (η, ξ) is [i - p]-differentiable, hence $\begin{matrix} g (ξ) ⊙ \partial_{ξ_{gH}} ψ (η, ξ) \underset{gH}{⊖} (- 1) g^{'} (ξ) ⊙ ψ (η, ξ) = \\ = g (ξ) ⊙ (\partial_{ξ} ψ_{1} (η, ξ), \partial_{ξ} ψ_{2} (η, ξ), \partial_{ξ} ψ_{3} (η, ξ)) \\ \underset{gH}{⊖} (- 1) g^{'} (ξ) ⊙ (ψ_{1} (η, ξ), ψ_{2} (η, ξ), ψ_{3} (η, ξ)) \\ = (g (ξ) \partial_{ξ} ψ_{3} (η, ξ), g (ξ) \partial_{ξ} ψ_{2} (η, ξ), g (ξ) \partial_{ξ} ψ_{1} (η, ξ)) \\ \underset{gH}{⊖} (- 1) (g^{'} (ξ) ψ_{1} (η, ξ), g^{'} (ξ) ψ_{2} (η, ξ), g^{'} (ξ) ψ_{3} (η, ξ)) \\ = (g (ξ) \partial_{ξ} ψ_{3} (η, ξ), g (ξ) \partial_{ξ} ψ_{2} (η, ξ), g (ξ) \partial_{ξ} ψ_{1} (η, ξ)) \\ \underset{gH}{⊖} (- g^{'} (ξ) ψ_{3} (η, ξ), - g^{'} (ξ) ψ_{2} (η, ξ), - g^{'} (ξ) ψ_{1} (η, ξ)) \\ = (g (ξ) \partial_{ξ} ψ_{3} (η, ξ) + g^{'} (ξ) ψ_{3} (η, ξ), g (ξ) \partial_{ξ} ψ_{2} (η, ξ) \\ + g^{'} (ξ) ψ_{2} (η, ξ), g (ξ) \partial_{ξ} ψ_{1} (η, ξ) + g^{'} (ξ) ψ_{1} (η, ξ)) \\ = \partial_{ξ_{ii . gH}} (g (ξ) ⊙ ψ (η, ξ)) . \end{matrix}$ □

Lemma 2.2. Suppose that ψ (η, ξ) is a fuzzy function and [gH - p]-differentiable respect to ξ such that the type of [gH - p]-differentiability does not change in $D$ , then

1. If ψ (η, ξ) is [i - p]-differentiable, then $\int_{a}^{ξ} \partial_{ξ_{i . gH}} ψ (η, ξ) d ξ = ψ (η, ξ) ⊖ ψ (η, a) .$

2. If ψ (η, ξ) is [ii - p]-differentiable, then $\int_{a}^{ξ} \partial_{ξ_{ii . gH}} ψ (η, ξ) d ξ = (- 1) ψ (η, a) ⊖ (- 1) ψ (η, ξ) .$

Proof. We have $\int_{a}^{ξ} \partial_{ξ_{gH}} ψ (η, ξ) d ξ = ψ (η, ξ) \underset{gH}{⊖} ψ (η, a)$ [10]. Using the definition of Hukuhara difference [11]

$\begin{matrix} ψ (η, ξ) & = ψ (η, a) \oplus \int_{a}^{ξ} \partial_{ξ_{i . gH}} ψ (η, ξ) d ξ \\ \Rightarrow \int_{a}^{ξ} \partial_{ξ_{i . gH}} ψ (η, ξ) d ξ = ψ (η, ξ) ⊖ ψ (η, a) . \\ Or, \\ ψ (η, a) & = ψ (η, ξ) \oplus (- 1) \int_{a}^{ξ} \partial_{ξ_{ii . gH}} ψ (η, ξ) d ξ \\ \Rightarrow \int_{a}^{ξ} \partial_{ξ_{ii . gH}} ψ (η, ξ) d ξ = (- 1) ψ (η, a) \\ ⊖ (- 1) ψ (η, ξ) . \end{matrix}$ □

Theorem 2.3. (See [13]) (Differentiation under the integral sign) Assume that ψ (η, ξ) and ∂_{η
_gH}ψ (η, ξ), are fuzzy continuous in [a, b] × [c, ∞). Assume that the fuzzy integral

$Ψ (η) = \int_{c}^{\infty} ψ (η, ξ) d ξ,$ (1) converges for $η \in ℝ$ and also let the fuzzy integral $\int_{c}^{\infty} \partial_{η_{gH}} ψ (η, ξ) d ξ$ converges uniformly on [a, b]. So Ψ is gH-differentiable on [a, b] and $\begin{matrix} \frac{d Ψ}{d η} = \int_{c}^{\infty} \partial_{η_{gH}} ψ (η, ξ) d ξ . \end{matrix}$

Theorem 2.4. Suppose that ψ (η, ξ) be a fuzzy function. If ψ (η, ξ) and ∂_gHψ (η, ξ) are [gH - p]-differentiable respect to ξ such that the type of [gH - p]-differentiability does not change in $D$ , then

If ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [i - p]-differentiable, then $\begin{matrix} \int_{0}^{ξ} \int_{0}^{ξ} \partial_{ξ ξ_{gH}} ψ (η, ξ) d ξ d ξ \\ = ψ (η, ξ) ⊖ ψ (η, 0) ⊖ ξ \partial_{ξ_{gH}} ψ (η, 0) . \end{matrix}$

If ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [ii - p]-differentiable, then $\begin{matrix} \int_{0}^{ξ} \int_{0}^{ξ} \partial_{ξ ξ_{gH}} ψ (η, ξ) d ξ d ξ \\ = ψ (η, ξ) ⊖ ψ (η, 0) \oplus (- 1) ξ \partial_{ξ_{gH}} ψ (η, 0) . \end{matrix}$

If ψ (η, ξ) is [i - p]-differentiable and ∂_{ξ
_gH}ψ (η, ξ) is [ii - p]-differentiable, then $\begin{matrix} \int_{0}^{ξ} \int_{0}^{ξ} \partial_{ξ ξ_{gH}} ψ (η, ξ) d ξ d ξ = ⊖ (- 1) ψ (η, ξ) \\ \oplus (- 1) ψ (η, 0) \oplus (- 1) ξ \partial_{ξ_{gH}} ψ (η, 0) . \end{matrix}$

If ψ (η, ξ) is [ii - p]-differentiable and ∂_{ξ
_gH}ψ (η, ξ) is [i - p]-differentiable, then $\begin{matrix} \int_{0}^{ξ} \int_{0}^{ξ} \partial_{ξ ξ_{gH}} ψ (η, ξ) d ξ d ξ = ⊖ (- 1) ψ (η, ξ) \\ \oplus (- 1) ψ (η, 0) ⊖ ξ \partial_{ξ_{gH}} ψ (η, 0) . \end{matrix}$

Proof. We are going to prove case 3. Let ψ (η, ξ) is [i - p]-differentiable and ∂_{ξ
_gH}ψ (η, ξ) is [ii - p]-differentiable, by applying Lemma 2.2 we get

$\begin{matrix} \int_{0}^{ξ} \int_{0}^{ξ} \partial_{ξ ξ_{gH}} ψ (η, ξ) d ξ d ξ \\ = \int_{0}^{ξ} ((- 1) \partial_{ξ_{gH}} ψ (η, 0) ⊖ (- 1) \partial_{ξ_{gH}} ψ (η, ξ)) d ξ \\ = (- 1) ξ \partial_{ξ_{gH}} ψ (η, 0) ⊖ (- 1) \int_{0}^{ξ} \partial_{ξ_{gH}} ψ (η, ξ) d ξ \\ = (- 1) ξ \partial_{ξ_{gH}} ψ (η, 0) ⊖ (- 1) (ψ (η, ξ) ⊖ ψ (η, 0)) \\ = ⊖ (- 1) ψ (η, ξ) \oplus (- 1) ψ (η, 0) \oplus (- 1) ξ \partial_{ξ_{gH}} ψ (η, 0) . \end{matrix}$ □

Theorem 2.5. (Fuzzy integration by parts under [gH - p]-differentiability) Assume that ψ (η, ξ) be a fuzzy function and [gH - p]-differentiable respect to ξ and also let the type of [gH - p]-differentiability does not change in $D$ . If g (ξ) be a differentiable real-valued function, then

g (ξ) >0, g′ (ξ) >0

If ψ (η, ξ) is [i - p]-differentiable, then $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = (ψ (η, b) ⊙ g (b)) ⊖ (ψ (η, a) ⊙ g (a)) \underset{gH}{⊖} \int_{a}^{b} ψ (η, ξ) g^{'} (ξ) d ξ,$

If ψ (η, ξ) is [ii - p]-differentiable, then $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = (- 1) (ψ (η, a) ⊙ g (a)) ⊖ (- 1) (ψ (η, b) ⊙ g (b)) \oplus (- 1) \int_{a}^{b} ψ (η, ξ) g^{'} (ξ) d ξ .$

g (ξ) >0, g′ (ξ) <0

If ψ (η, ξ) is [i - p]-differentiable, then $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = (ψ (η, b) ⊙ g (b)) ⊖ (ψ (η, a) ⊙ g (a)) \oplus (- 1) \int_{a}^{b} ψ (η, ξ) g^{'} (ξ) d ξ .$

If ψ (η, ξ) is [ii - p]-differentiable, then $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = (- 1) (ψ (η, a) ⊙ g (a)) ⊖ (- 1) (ψ (η, b) ⊙ g (b)) \underset{gH}{⊖} \int_{a}^{b} ψ (η, ξ) g^{'} (ξ) d ξ .$

g (ξ) <0, g′ (ξ) >0

If ψ (η, ξ) is [i - p]-differentiable, then $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = (- 1) (ψ (η, a) ⊙ g (a)) ⊖ (- 1) (ψ (η, b) ⊙ g (b)) \oplus (- 1) \int_{a}^{b} ψ (η, ξ) g^{'} (ξ) d ξ .$

If ψ (η, ξ) is [ii - p]-differentiable, then $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = (ψ (η, b) ⊙ g (b)) ⊖ (ψ (η, a) ⊙ g (a)) \underset{gH}{⊖} \int_{a}^{b} ψ (η, ξ) g^{'} (ξ) d ξ .$

g (ξ) <0, g′ (ξ) <0

If ψ (η, ξ) is [i - p]-differentiable, then $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = (- 1) (ψ (η, a) ⊙ g (a)) ⊖ (- 1) (ψ (η, b) ⊙ g (b)) \underset{gH}{⊖} \int_{a}^{b} ψ (η, ξ) g^{'} (ξ) d ξ .$

If ψ (η, ξ) is [ii - p]-differentiable, then $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = (ψ (η, b) ⊙ g (b)) ⊖ (ψ (η, a) ⊙ g (a)) \oplus (- 1) \int_{a}^{b} ψ (η, ξ) g^{'} (ξ) d ξ .$

Proof. Since the proofs of all cases are similar, here we will prove case (2 -2), assume that ψ (η, ξ) is [ii - p]-differentiable so, by applying Theorem 2.2 (2-2) we have $\begin{matrix} \partial_{ξ_{ii . gH}} (ψ (η, ξ) ⊙ g (ξ)) = ψ (η, ξ) ⊙ g^{'} (ξ) \\ \underset{gH}{⊖} (- 1) \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) . \end{matrix}$ (2) Then by integration (Lemma 2.2 (2)) of bout Eq. (2) $\begin{matrix} (- 1) (ψ (η, a) ⊙ g (a)) ⊖ (- 1) (ψ (η, b) ⊙ g (b)) \\ = \int_{a}^{b} ψ (η, ξ) ⊙ g^{'} (ξ) d ξ \underset{gH}{⊖} (- 1) \\ \int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ, \end{matrix}$ so $\int_{a}^{b} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ g (ξ) d ξ = ⊖ (- 1) (ψ (η, b) ⊙ g (b)) \oplus (- 1) (ψ (η, a) ⊙ g (a)) \underset{gH}{⊖} \int_{a}^{b} ψ (η, ξ) ⊙ g^{'} (ξ) d ξ .$ □

Theorem 2.6. (Fuzzy Leibniz Rule under generalized Hukuhara difference) Suppose that ψ (η, ξ) be a fuzzy valued function and aso let ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are fuzzy continuous, defines on a (ξ) ≤ η ≤ b (ξ) and ξ₀ ≤ ξ ≤ ξ₁. Also, consider a (ξ) and b (ξ) are continuously differentiable for ξ₀ ≤ ξ ≤ ξ₁. Therefore

$\begin{matrix} \partial_{ξ_{gH}} (\int_{a (ξ)}^{b (ξ)} ψ (η, ξ) d η) = ψ (b (ξ), ξ) ⊙ b^{'} (ξ) \\ \underset{gH}{⊖} ψ (a (ξ), ξ) ⊙ a^{'} (ξ) \oplus \int_{a (ξ)}^{b (ξ)} \partial_{ξ_{gH}} ψ (η, ξ) d η . \end{matrix}$ (3)

Proof. Consider $φ = \int_{a}^{b} ψ (η, ξ) d η$ where a : = a (ξ) and b : = b (ξ) are two real functions. Using Lemma $\int_{a}^{b} ψ (η, ξ) d ξ = ⊖ \int_{b}^{a} ψ (η, ξ) d ξ$ , we can write $\begin{matrix} Δ φ & = φ (ξ + Δ ξ) \underset{gH}{⊖} φ (ξ) \\ = \int_{a + Δ ξ}^{b + Δ ξ} ψ (η, ξ + Δ ξ) d η \underset{gH}{⊖} \int_{a}^{b} ψ (η, ξ) d η \\ = \int_{a + Δ a}^{a} ψ (η, ξ + Δ ξ) d η \oplus \int_{a}^{b} ψ (η, ξ + Δ ξ) d η \\ \oplus \int_{b}^{b + Δ ξ} ψ (η, ξ + Δ ξ) d η \underset{gH}{⊖} \int_{a}^{b} ψ (η, ξ) d η \\ = ⊖ \int_{a}^{a + Δ a} ψ (η, ξ + Δ ξ) d η \\ \oplus \int_{a}^{b} (ψ (η, ξ + Δ ξ) \underset{gH}{⊖} ψ (η, ξ)) d η \\ \oplus \int_{b}^{b + Δ b} ψ (η, ξ + Δ ξ) d η . \end{matrix}$ From mean value Theorem [2], there exist ζ₁ ∈ [a, a + Δa] and ζ₂ ∈ [b, b + Δb] such that $\begin{matrix} \int_{a}^{a + Δ a} ψ (η, ξ + Δ ξ) d η = Δ a ⊙ ψ (ζ_{1}, ξ + Δ ξ), \\ \int_{b}^{b + Δ b} ψ (η, ξ + Δ ξ) d η = Δ b ⊙ ψ (ζ_{2}, ξ + Δ ξ) . \end{matrix}$ Hence $Δ φ = ⊖ Δ a ⊙ ψ (ζ_{1}, ξ + Δ ξ) \oplus \int_{a}^{b} (ψ (η, ξ + Δ ξ) \underset{gH}{⊖} ψ (η, ξ)) d η \oplus Δ b ⊙ ψ (ζ_{2}, ξ + Δ ξ) .$ Dividing by Δξ and letting Δξ → 0 and notice that ζ₁ → a and ζ₂ → b, conclude that $\begin{matrix} lim_{Δ ξ \to 0} ⊖ \frac{Δ a}{Δ ξ} ⊙ ψ (ζ_{1}, ξ + Δ ξ) \\ = ⊖ \frac{da}{d ξ} ⊙ ψ (a, ξ) = ⊖ ψ (a (ξ), ξ) ⊙ a^{'} (ξ) . \\ lim_{Δ ξ \to 0} \frac{Δ b}{Δ ξ} ⊙ ψ (ζ_{2}, ξ + Δ ξ) \\ = \frac{db}{d ξ} ⊙ ψ (b, ξ) = ψ (b (ξ), ξ) ⊙ b^{'} (ξ) . \\ lim_{Δ ξ \to 0} \int_{a}^{b} \frac{(ψ (η, ξ + Δ ξ) ⊖_{gH} ψ (η, ξ))}{Δ ξ} d η \\ = \int_{a (ξ)}^{b (ξ)} \partial_{ξ_{gH}} ψ (η, ξ) d η . \end{matrix}$ We have established (3). □

3 Fuzzy Laplace transform

Fuzzy Laplace transform is a strong tool for solving different types of equations. In this section, we will briefly outline the fuzzy Laplace transform for fuzzy function of two variables and prove some important properties for this transform.

Definition 3.1. A fuzzy valued function ψ (η, ξ) is said to be of exponential order r > 0 respect to ξ if there exists a positive constant M and Re (ϱ) ≥ r such that for all T > 0 $\begin{matrix} D (ψ (η, ξ), 0) \leq M e^{r ξ}, ξ > T . \end{matrix}$

Definition 3.2. The convolution of two functions g (η, ξ) and ψ (η, ξ) is given by $\begin{matrix} (ψ * g) (η, ξ) = \int_{0}^{ξ} ψ (η, τ) ⊙ g (η, ξ - τ) d τ, \end{matrix}$ where g (η, ξ) is a real-valued piecewise continuous function and ψ (η, ξ) is a triangular fuzzy piecewise continuous function. Substituting u = ξ - τ gives $\begin{matrix} (ψ * g) (η, ξ) \\ = \int_{0}^{ξ} g (η, u) ⊙ ψ (η, ξ - u) du = (g * ψ) (η, ξ), \end{matrix}$ so, the convolution is commutative.

Definition 3.3. Suppose that ψ (η, ξ) be a fuzzy valued function and ϱ is a real parameter. The fuzzy Laplace transform ψ (η, ξ) with respect to ξ shown by Ψ (η, ϱ), is defined as the following equation, whenever the limits exist and is a fuzzy number $\begin{matrix} L_{ξ} [ψ (η, ξ)] & = \int_{0}^{\infty} e^{- ϱ ξ} ⊙ ψ (η, ξ) d ξ \\ = lim_{τ \to \infty} \int_{0}^{τ} e^{- ϱ ξ} ⊙ ψ (η, ξ) d ξ = Ψ (η, ϱ) . \end{matrix}$

Definition 3.4. Laplace transform (See [20]) If $L_{ξ} [ψ (η, ξ)] = Ψ (η, ϱ)$ , then $L_{ξ}^{- 1} [Ψ (η, ϱ)] = ψ (η, ξ), ξ \geq 0$ such that which maps the fuzzy Laplace transform of a fuzzy valued function Ψ (η, ϱ) back to original fuzzy valued function ψ (η, ξ).

Theorem 3.1. If ψ (η, ξ) be a fuzzy piecewise continuous function on [0, ∞], and of exponential order r, then the fuzzy Laplace transform $L_{ξ} [ψ (η, ξ)] = Ψ (η, ϱ)$ exists for all ϱ > r.

Proof. According to Definition 3.3, we have $Ψ (η, ϱ) = \int_{0}^{\infty} e^{- ϱ ξ} ⊙ ψ (η, ξ) d ξ .$ There is a T ∈ [0, ∞) such that [17] $\begin{matrix} \int_{0}^{\infty} e^{- ϱ ξ} ⊙ ψ (η, ξ) d ξ = \int_{0}^{T} e^{- ϱ ξ} ⊙ ψ (η, ξ) d ξ \\ \oplus \int_{T}^{\infty} e^{- ϱ ξ} ⊙ ψ (η, ξ) d ξ = Ψ_{1} (η, ϱ) \oplus Ψ_{2} (η, ϱ) . \end{matrix}$ Since ψ (η, ξ) is a fuzzy piecewise continuous, so the first part of the integral is exists. Now let’s just show that the second part of the integral is also exists, then $\begin{matrix} D (Ψ_{2} (η, ϱ), 0) & = & D (\int_{T}^{\infty} e^{- ϱ ξ} ⊙ ψ (η, ξ) d ξ, 0) \\ \leq & \int_{T}^{\infty} e^{- ϱ ξ} D (ψ (η, ξ), 0) d ξ \\ \leq & \int_{0}^{\infty} e^{- ϱ ξ} D (ψ (η, ξ), 0) d ξ \\ \leq & M \int_{0}^{\infty} e^{- (ϱ - r) ξ} d ξ = \frac{M}{ϱ - r} . \end{matrix}$ Thus, Ψ₂ (η, ϱ) is exists for ϱ > r.□

Proposition 3.2. Suppose that ψ (η, ξ) be a fuzzy continuous on [0, ∞] and exponential order r, then $lim_{τ \to \infty} (ψ (η, τ) ⊙ e^{- τ}) = 0 .$

Proof. By using Definition 3.1, D ( (ψ (η, τ) ⊙ e^-τ) , 0) ≤ M ⊙ e^rξ-τ . Thus $lim_{τ \to \infty}$ , the result is obtain.□

Theorem 3.3. Suppose that ψ (η, ξ) be a fuzzy continuous on [0, ∞] and ∂_{ξ
_gH}ψ (η, ξ) be a fuzzy piecewise continuous valued function on [0, ∞], then

[1.] If ψ (η, ξ) is [i - p]-differentiable, then $L_{ξ} [\partial_{ξ_{gH}} ψ (η, ξ)] = ϱ Ψ (η, ϱ) ⊖ ψ (η, 0) .$

[2.] If ψ (η, ξ) is [ii - p]-differentiable, then $L_{ξ} [\partial_{ξ_{gH}} ψ (η, ξ)] = (- 1) ψ (η, 0) \underset{gH}{⊖} (- 1) ϱ Ψ (η, ϱ) .$

Proof. Suppose that ψ (η, ξ) is a [ii - p]-differentiable, then by Theorem 2.5 (2-2), we get

$\begin{matrix} L_{ξ} [\partial_{ξ_{gH}} ψ (η, ξ)] \\ = lim_{τ \to \infty} \int_{0}^{τ} e^{- ϱ ξ} ⊙ \partial_{ξ_{gH}} ψ (η, ξ) d ξ \\ = lim_{τ \to \infty} [(- 1) (ψ (η, 0) ⊙ e^{0}) ⊖ (- 1) (ψ (η, τ) ⊙ e^{τ}) \\ \underset{gH}{⊖} \int_{0}^{τ} ψ (η, ξ) ⊙ ((- 1) ϱ e^{- ϱ ξ}) d ξ] \\ = (- 1) ψ (η, 0) \underset{gH}{⊖} (- 1) ϱ lim_{τ \to \infty} \int_{0}^{\infty} ψ (η, ξ) ⊙ e^{- ϱ ξ} d ξ \\ = (- 1) ψ (η, 0) \underset{gH}{⊖} (- 1) ϱ Ψ (η, ϱ) . \end{matrix}$

The proof of part (1) according to Theorem 2.5 (2-1) and Proposition 3.2 is easily obtain.□

Theorem 3.4. If ψ (η, ξ) be a fuzzy continuous on [0, ∞] and ∂_{ξξ
_gH}ψ (η, ξ) be a fuzzy piecewise continuous valued function on [0, ∞], then

[1.] If ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [i - p]-differentiable, then $\begin{matrix} L_{ξ} [\partial_{ξ ξ_{gH}} ψ (η, ξ)] \\ = ϱ^{2} Ψ (η, ϱ) ⊖ ϱ ψ (η, 0) ⊖ \partial_{ξ_{gH}} ψ (η, 0), \end{matrix}$

[2.] If ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [ii - p]-differentiable, then $\begin{matrix} L_{ξ} [\partial_{ξ ξ_{gH}} ψ (η, ξ)] \\ = ϱ^{2} Ψ (η, ϱ) \underset{gH}{⊖} ϱ ψ (η, 0) \oplus (- 1) \partial_{ξ_{gH}} ψ (η, 0), \end{matrix}$

[3.] If ψ (η, ξ) is [i - p]-differentiable and ∂_{ξ
_gH}ψ (η, ξ) is [ii - p]-differentiable, then $\begin{matrix} L_{ξ} [\partial_{ξ ξ_{gH}} ψ (η, ξ)] = (- 1) ϱ ψ (η, 0) \oplus (- 1) \\ \partial_{ξ_{gH}} ψ (η, 0) \underset{gH}{⊖} (- 1) ϱ^{2} Ψ (η, ϱ), \end{matrix}$

[4.]If ψ (η, ξ) is [ii - p]-differentiable and ∂_{ξ
_gH}ψ (η, ξ) is [i - p]-differentiable, then $\begin{matrix} L_{ξ} [\partial_{ξ ξ_{gH}} ψ (η, ξ)] = (- 1) ϱ ψ (η, 0) \\ ⊖ \partial_{ξ_{gH}} ψ (η, 0) \underset{gH}{⊖} (- 1) ϱ^{2} Ψ (η, ϱ) . \end{matrix}$

Proof. We will prove the case 1, the rest of the other cases can be proved similarly. Suppose that ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [i - p]-differentiable then by Theorems 2.5 (2-1), 3.3 (1) and Proposition 3.2 we have

$\begin{matrix} L_{ξ} [\partial_{ξ ξ_{gH}} ψ (η, ξ)] \\ = lim_{τ \to \infty} \int_{0}^{τ} e^{- ϱ ξ} ⊙ \partial_{ξ ξ_{gH}} ψ (η, ξ) d ξ \\ = lim_{τ \to \infty} [(\partial_{ξ_{gH}} ψ (η, τ) ⊙ e^{- τ}) ⊖ (\partial_{ξ_{gH}} ψ (η, 0) ⊙ e^{0}) \\ \oplus (- 1) \int_{0}^{τ} \partial_{ξ_{gH}} ψ (η, ξ) ⊙ ((- 1) ϱ e^{- ϱ ξ}) d ξ] \\ = 0 ⊖ \partial_{ξ_{gH}} ψ (η, 0) \oplus ϱ L_{ξ} [\partial_{ξ_{gH}} ψ (η, ϱ)] \\ = ϱ L_{ξ} [\partial_{ξ_{gH}} ψ (η, ϱ)] ⊖ \partial_{ξ_{gH}} ψ (η, 0) \\ = ϱ (ϱ Ψ (η, ϱ) ⊖ ψ (η, 0)) ⊖ \partial_{ξ_{gH}} ψ (η, 0) \\ = ϱ^{2} Ψ (η, ϱ) ⊖ ϱ ψ (η, 0) ⊖ \partial_{ξ_{gH}} ψ (η, 0) . \end{matrix}$ □

Theorem 3.5. (Fuzzy convolution Theorem) Theorem Suppose that ψ (η, ξ) be a piecewise continuous fuzzy valued function on [0, ∞] and g (η, ξ) be a piecewise continuous real-valued function on [0, ∞], and of exponential order r. If Ψ (η, ϱ) be the fuzzy Laplace transforms with respect to ξ of ψ (η, ξ) and the Laplace transform of real-valued function g (η, ξ) denotes by G (η, ϱ), then $\begin{matrix} L_{ξ} [(ψ * g) (η, ξ)] = Ψ (η, ϱ) ⊙ G (η, ϱ) . \end{matrix}$ Proof. Let $\begin{matrix} Ψ (η, ϱ) = \int_{0}^{\infty} e^{- ϱ σ} ⊙ ψ (η, σ) d σ, \\ G (η, ϱ) = \int_{0}^{\infty} e^{- ϱ τ} g (η, τ) d τ, \end{matrix}$ then

$\begin{matrix} L_{ξ} [ψ (η, ξ)] ⊙ L_{ξ} [g (η, ξ)] \\ = \int_{0}^{\infty} [\int_{0}^{\infty} e^{- ϱ (σ + τ)} ⊙ ψ (η, σ) d σ] ⊙ g (η, τ) d τ . \end{matrix}$ (4) Let τ is fixed and ξ = σ + τ, dξ = dσ, so we can rewrite the Eq. (4) $\begin{matrix} L_{ξ} [ψ (η, ξ)] ⊙ L_{ξ} [g (η, ξ)] \\ = \int_{0}^{\infty} (\int_{τ}^{\infty} e^{- ϱ ξ} ⊙ ψ (η, ξ - τ) d ξ) ⊙ g (η, τ) d τ \\ = (\int_{0}^{\infty} \int_{τ}^{\infty} e^{- ϱ ξ} ψ_{1} (η, ξ - τ) g (η, τ) d ξ d τ, \\ \int_{0}^{\infty} \int_{τ}^{\infty} e^{- ϱ ξ} ψ_{2} (η, ξ - τ) g (η, τ) d ξ d τ, \\ \int_{0}^{\infty} \int_{τ}^{\infty} e^{- ϱ ξ} ψ_{3} (η, ξ - τ) g (η, τ) d ξ d τ) . \end{matrix}$ We can reverse the order of integration, so $\begin{matrix} L_{ξ} [ψ (η, ξ)] ⊙ L_{ξ} [g (η, ξ)] \\ = (\int_{0}^{\infty} \int_{0}^{ξ} e^{- ϱ ξ} ψ_{1} (η, ξ - τ) g (η, τ) d τ d ξ, \\ \int_{0}^{\infty} \int_{0}^{ξ} e^{- ϱ ξ} ψ_{2} (η, ξ - τ) g (η, τ) d τ d ξ, \\ \int_{0}^{\infty} \int_{0}^{ξ} e^{- ϱ ξ} ψ_{3} (η, ξ - τ) g (η, τ) d τ d ξ) \\ = (\int_{0}^{\infty} e^{- ϱ ξ} (\int_{0}^{ξ} ψ_{1} (η, ξ - τ) g (η, τ) d τ) d ξ, \\ \int_{0}^{\infty} e^{- ϱ ξ} (\int_{0}^{ξ} ψ_{2} (η, ξ - τ) g (η, τ) d τ) d ξ, \\ \int_{0}^{\infty} e^{- ϱ ξ} (\int_{0}^{ξ} ψ_{3} (η, ξ - τ) g (η, τ) d τ) d ξ) . \end{matrix}$ Hence $\begin{matrix} L_{ξ} [ψ (η, ξ)] ⊙ L_{ξ} [g (η, ξ)] \\ = & \int_{0}^{\infty} e^{- ϱ ξ} (\int_{0}^{ξ} ψ (η, ξ - τ) g (η, τ) d τ) d ξ \\ = & L_{ξ} [(ψ * g) (η, ξ)] . \end{matrix}$ □

4 Existence and uniqueness of fuzzy solution for Volterra partial integro-differential equation

Consider the following Volterra integro-diffe-rential equation.

$\begin{matrix} \sum_{j = 1}^{2} \frac{\partial^{j} ψ (η, ξ)}{\partial ξ^{j}} = ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, ξ - τ) ψ (η, τ) d τ, \\ η \in j_{1} : = [0, b], ξ \in j_{2} : = [0, d], \end{matrix}$ (5) with prescribed fuzzy conditions. Here, the operator $\partial_{ξ_{gH}} ψ (η, ξ) : = \frac{\partial ψ (η, ξ)}{\partial t}$ denotes the generalized Hukuhara partial derivative w.r.t ξ of fuzzy function ψ (η, ξ).

In this section, we are going to prove that the equation (5) has a unique fuzzy solution by considering the type of [gH - p]-differentiability w.r.t ξ. For this purpose, consider the following fuzzy Volterra integral equation

$\begin{matrix} ψ (η, ξ) = ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ, \\ η \in j_{1} : = [0, b], ξ \in j_{2} : = [0, d], \end{matrix}$ (6) where ϖ is a given fuzzy continuous function and $k : j_{1} \times j_{2} \times E \to E$ is a fuzzy continuous function and there exist real positive function L independent of η, ξ, ψ₁ and ψ₂ such that k satisfies the following Lipschitz condition

$\begin{matrix} D (k (η, τ, ψ_{1} (η, τ)), k (η, τ, ψ_{2} (η, τ))) \\ \leq L D (ψ_{1} (η, τ), ψ_{2} (η, τ)) . \end{matrix}$ (7)

Lemma 4.1. Assume that $k : j_{1} \times j_{2} \times E \to E$ be a bonded continuous function such that satisfies (7). Then the fuzzy problem (6) has a unique continuous fuzzy solution.

Proof. We define the following recursive equations

${\begin{matrix} ψ_{0} (η, ξ) = ϖ (η, ξ), \\ ψ_{n} (η, ξ) = ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ_{n - 1} (η, τ)) d τ, & n = 1, 2, . . . . \end{matrix}$ (8) Using gH-difference leads to

$\begin{matrix} ψ_{n} (η, ξ) \underset{gH}{⊖} ψ_{n - 1} (η, ξ) \\ = \int_{0}^{ξ} [k (η, τ, ψ_{n - 1} (η, τ)) \underset{gH}{⊖} k (η, τ, ψ_{n - 2} (η, τ))] d τ \end{matrix}$ (9)

We therefore may define the following fuzzy functions $\begin{matrix} {\begin{matrix} φ_{0} (η, ξ) = ψ_{0} (η, ξ) \\ φ_{n} (η, ξ) = ψ_{n} (η, ξ) \underset{gH}{⊖} ψ_{n - 1} (η, ξ), & n = 1, 2, . . . . \end{matrix} \end{matrix}$ Then $ψ_{n} (η, ξ) = \sum_{i = 0}^{n} φ_{i} (η, ξ)$ . By using the properties of Hausdorff distance, we have

$\begin{matrix} D (φ_{n} (η, ξ), 0) \\ = D (ψ_{n} (η, ξ) \underset{gH}{⊖} ψ_{n - 1} (η, ξ), 0) \\ = D (\int_{0}^{ξ} k (η, τ, ψ_{n - 1} (η, τ)) d τ, \int_{0}^{ξ} k (η, τ, ψ_{n - 2} (η, τ)) d τ) \\ \leq \int_{0}^{ξ} D (k (η, τ, ψ_{n - 1} (η, τ), k (η, τ, ψ_{n - 2} (η, τ))) d τ \\ \leq L \int_{0}^{ξ} D (ψ_{n - 1} (η, τ), ψ_{n - 2} (η, τ)) d τ \\ \leq L^{2} \int_{0}^{ξ} \int_{0}^{ξ} D (ψ_{n - 2} (η, τ), ψ_{n - 3} (η, τ)) d τ d τ \\ \leq \dots \leq L^{n} \underset{n}{\underset{︸}{\int_{0}^{ξ} \dots \int_{0}^{ξ}}} D (ψ_{0} (η, τ), 0) \underset{n}{\underset{︸}{d τ \dots d τ}} \\ = {\frac{(L ξ)}{(n!)^{2}}}^{n} M_{0} \leq {\frac{(Ld)}{(n!)^{2}}}^{n} M_{0} . \end{matrix}$

Where $M_{0} = max_{0 \leq ξ < d} D (ψ_{0} (η, ξ), 0)$ , then according Lemma 2.1, the fuzzy sequence ψ_n (η, ξ) is convergence and $ψ (η, ξ) = \sum_{n = 0}^{\infty} ψ_{n} (η, ξ)$ as a fuzzy continuous function.

Now, we want to show that $ψ (η, ξ) = \sum_{n = 0}^{\infty} ψ_{n} (η, ξ)$ is satisfies in to Eq. (6). To do this, suppose that

$ψ (η, ξ) = ψ_{n} (η, ξ) \oplus Δ_{n} (η, ξ) .$ (10) Then $ψ_{n} (η, ξ) = ψ (η, ξ) \underset{gH}{⊖} Δ_{n} (η, ξ)$ . By setting this equation in Eq. (8), we can write $\begin{matrix} ψ (η, ξ) \underset{gH}{⊖} Δ_{n} (η, ξ) \\ = ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ) \underset{gH}{⊖} Δ_{n - 1} (η, τ)) d τ . \end{matrix}$ Thus $\begin{matrix} ψ (η, ξ) \underset{gH}{⊖} ϖ (η, ξ) \underset{gH}{⊖} \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ \\ = Δ_{n} (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ) \underset{gH}{⊖} Δ_{n - 1} (η, τ)) d τ \\ \underset{gH}{⊖} \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ \end{matrix}$ Using the Lipschitz condition (7), we conclude that

$\begin{matrix} D (ψ (η, ξ) \underset{gH}{⊖} ϖ (η, ξ) \underset{gH}{⊖} \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ, 0) \\ = D (Δ_{n} (η, ξ), 0) + D (\int_{0}^{ξ} k (η, τ, ψ (η, τ) \underset{gH}{⊖} \\ Δ_{n - 1} (η, τ)) d τ \underset{gH}{⊖} \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ, 0) \\ = D (Δ_{n} (η, ξ), 0) + D ((\int_{0}^{ξ} k (η, τ, ψ (η, τ) \underset{gH}{⊖} \\ Δ_{n - 1} (η, τ)) d τ, \int_{0}^{ξ} k (x, τ, u (x, τ)) d τ) \\ \leq D (Δ_{n} (η, ξ), 0) + L \int_{0}^{ξ} D (ψ (η, τ) \underset{gH}{⊖} \\ Δ_{n - 1} (η, τ), ψ (η, τ)) d τ \\ \leq D (Δ_{n} (η, ξ), 0) + L \int_{0}^{ξ} (D (ψ (η, τ), ψ (η, τ)) \\ + D (Δ_{n - 1} (η, τ), 0)) d τ \\ = D (Δ_{n} (η, ξ), 0) + L ξ D (Δ_{n - 1} (η, ξ), 0) . \end{matrix}$

But using Eq. (10), $lim_{n \to \infty} D (Δ_{n} (η, ξ), 0) = 0$ , so when n→ ∞, ψ (η, ξ) is a fuzzy solution of the Eq. (6) and we have $\begin{matrix} ψ (η, ξ) = ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ . \end{matrix}$ (11) Now, to proof that the fuzzy solution is unique, we consider φ (η, ξ) is another fuzzy solution that holds in to Eq. (6), so

$\begin{matrix} ψ (η, ξ) \underset{gH}{⊖} φ (η, ξ) & = & \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ \underset{gH}{⊖} \\ \int_{0}^{ξ} k (η, τ, φ (η, τ)) d τ . \end{matrix}$ (12) and by the Lipschitz condition, we get

$\begin{matrix} D (ψ (η, ξ), φ (η, ξ)) \\ = D (\int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ, \int_{0}^{ξ} k (η, τ, φ (η, τ)) d τ) \\ \leq \int_{0}^{ξ} D (k (η, τ, ψ (η, τ)), k (η, τ, φ (η, τ))) d τ \\ \leq L \int_{0}^{ξ} D (ψ (η, τ), φ (η, τ)) d τ \\ \leq L \int_{0}^{ξ} (ψ_{n} (η, τ) \oplus Δ_{n_{1}} (η, τ), φ_{n} (η, τ) \oplus Δ_{n_{2}} (η, τ)) d τ \\ \leq L \int_{0}^{ξ} D (ψ_{n} (η, τ), φ_{n} (η, τ)) d τ \\ + L \int_{0}^{ξ} D (Δ_{n_{1}} (η, τ), Δ_{n_{2}} (η, τ)) d τ \\ ⋮ \\ \leq L^{n} \underset{n}{\underset{︸}{\int_{0}^{ξ} \dots \int_{0}^{ξ}}} D (ψ_{0} (η, τ), φ_{0} (η, τ)) \underset{n}{\underset{︸}{d τ \dots d τ}} \\ + L^{n} \underset{n}{\underset{︸}{\int_{0}^{ξ} \dots \int_{0}^{ξ}}} D (Δ_{0_{1}} (η, τ), Δ_{0_{2}} (η, τ)) \underset{n}{\underset{︸}{d τ \dots d τ}} \\ \leq {\frac{(Ld)}{(n!)^{2}}}^{n} M_{1} + {\frac{(Ld)}{(n!)^{2}}}^{n} M_{2} . \end{matrix}$

Where $\begin{matrix} M_{1} = max_{0 \leq η < b, 0 \leq ξ < d} D (ψ_{0} (η, ξ), φ_{0} (η, ξ)) and \\ M_{2} = max_{0 \leq η < b, 0 \leq ξ < d} D (Δ_{0_{1}} (η, ξ), Δ_{0_{2}} (η, ξ)) . \end{matrix}$ Therefore when n→ ∞, the desired result is obtained.□

Theorem 4.1. Let ϖ be a fuzzy continuous function and k is a bonded continuous function which satisfies in the Lipschitz condition (7), then the following parabolic Volterra partial integro-differential equation has a unique solution by considering the type of [gH - p]-differentiability. ${\begin{matrix} \partial_{ξ_{gH}} ψ (η, ξ) = ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ, 0 \leq η \leq b, 0 < ξ \leq d, \\ ψ (η, 0) = δ_{1} (η), \end{matrix}$ (13) where δ₁ (η) is a given fuzzy function.

Proof. Let ψ (η, ξ) be a [i - p]-differentiable then by integration from two side of Eq (13) with respect to ξ. Lemma 2.2 gives the results $\begin{matrix} ψ (η, ξ) ⊖ ψ (η, 0) \\ = \int_{0}^{ξ} (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ) d τ . \end{matrix}$ Therefore by considering the initial value of problem (13) we get $\begin{matrix} ψ (η, ξ) \\ = δ_{1} (η) \oplus \int_{0}^{ξ} (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ) d τ . \end{matrix}$ (14) Let $K (η, τ, ψ) = ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ$ is the the kernel of Eq. (14). Now, it is enough to show that K satisfy in Lipschitz condition. For this purpose

$\begin{matrix} D (K_{1} (η, τ, ψ_{1} (η, τ)), K_{2} (η, τ, ψ_{2} (η, τ))) \\ \leq D (\int_{0}^{ξ} k (η, τ, ψ_{1} (η, τ)) d τ, \int_{0}^{ξ} k (η, τ, ψ_{2} (η, τ) d τ) \\ \leq (L ξ) D (ψ_{1} (η, ξ), ψ_{2} (η, ξ)) \\ \leq (Ld) D (ψ_{1} (η, ξ), ψ_{2} (η, ξ)) . \end{matrix}$

In this case, the kernel K satisfies in Lipschitz condition. Then by Lemma 4.1, the fuzzy parabolic Volttera partial integro-differential equation (13) has a unique [i - p]-differentiable solution. Now, let ψ (η, ξ) is [ii - p]-differentiable and by using Lemma 2.2 we have $\begin{matrix} (- 1) ψ (η, 0) ⊖ (- 1) ψ (η, ξ) \\ = \int_{0}^{ξ} (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ) d ξ, \end{matrix}$ or equivalently

$\begin{matrix} ψ (η, ξ) = δ_{1} (η) ⊖ (- 1) \\ \int_{0}^{ξ} (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ) d ξ . \end{matrix}$ (15)

Now let $K (η, τ, ψ) = ⊖ (- 1) (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ)$ is the kernel of Eq. (15). Applying the properties of the Hausdorff distance and the method which is discussed in detail in the previous case, it is easy to show that this kernel satisfies in Lipschitz condition. Hence, the [ii - p]-differentiable fuzzy solution of Eq. (13) is unique.□

Theorem 4.2. Consider the following fuzzy Volterra partial integro-differential equation ${\begin{matrix} \partial_{ξ ξ_{gH}} ψ (η, ξ) = ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ, \\ ψ (η, 0) = δ_{1} (η) \in E, \partial_{ξ_{gH}} ψ (η, 0) = δ_{2} (η) \in E . \end{matrix}$ (16) Where $ψ, ϖ : j_{1} \times j_{2} \to E$ , $k : j_{1} \times j_{2} \times E \to E$ and ψ, ϖ, k are continuous fuzzy function. If k satisfies the Lipschitz condition then by considering the type of [gH - p]-differentiability, the Eq. (16) has a unique continuous fuzzy solution.

Proof. This equation should be considered in the following cases

Case 1. ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [i - p]-differentiable w.r.t ξ.

Case 2. ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [ii - p]-differentiable w.r.t ξ.

Case 3. ψ (η, ξ) is [i - p]-differentiable respect to ξ and ∂_{ξ
_gH}ψ (η, ξ) is [ii - p]-differentiable w.r.t ξ

Case 4. ψ (η, ξ) is [ii - p]-differentiable respect to ξ and ∂_{ξ
_gH}ψ (η, ξ) is [i - p]-differentiable w.r.t ξ

Here we will prove case 3 and the other cases can be proved in a similar way. Let ψ (η, ξ) is [i - p]-differentiable w.r.t ξ and ∂_{ξ
_gH}ψ (η, ξ) is [ii - p]-differentiable w.r.t ξ, then by twice integration of both sides of Eq. (16). By applying Theorem 2.4 and by attention to the initial values of Eq. (16), we get

$\begin{matrix} ψ (η, ξ) = δ_{1} (η) \oplus ξ δ_{2} (η) ⊖ (- 1) \\ \int_{0}^{ξ} \int_{0}^{ξ} (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ) d ξ d ξ . \end{matrix}$ (17) Now, let $K (η, τ, ψ) = ⊖ (- 1) \int_{0}^{ξ} (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ (η, τ)) d τ) d ξ$ . It is enough to show that K (η, τ, ψ) satisfies in the Lipschitz condition.

$\begin{matrix} D (K_{1} (η, τ, ψ (η, τ)), K_{2} (η, τ, ψ_{2} (η, τ))) = \\ = D (⊖ (- 1) \int_{0}^{ξ} (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ_{1} (η, τ)) d τ) d ξ, \\ ⊖ (- 1) \int_{0}^{ξ} (ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, τ, ψ_{2} (η, τ)) d τ) d ξ) \\ \leq D (\int_{0}^{ξ} ϖ (η, ξ) d ξ, \int_{0}^{ξ} ϖ (η, ξ) d ξ) + D (\int_{0}^{ξ} \int_{0}^{ξ} k (η, τ, \\ ψ_{1} (η, τ)) d τ d ξ, \int_{0}^{ξ} \int_{0}^{ξ} k (η, τ, ψ_{2} (η, τ)) d τ d ξ) \\ \leq \int_{0}^{ξ} \int_{0}^{ξ} D (k (η, τ, ψ_{1} (η, τ)), k (η, τ, ψ_{1} (η, τ))) d τ d ξ \\ \leq L \int_{0}^{ξ} \int_{0}^{ξ} D (ψ_{1} (η, τ)), ψ_{1} (η, τ))) d τ d ξ \\ \leq (\frac{L d^{2}}{2}) D (ψ_{1} (η, ξ), ψ_{2} (η, ξ)) . \end{matrix}$

So K (η, τ, ψ) satisfies in the Lipschitz condition. Hence by Lemma 4.1, the parabolic Volttera partial integro-differential equation (16) has a unique solution, ψ (η, ξ), such that ψ (η, ξ) is [i - p]-differentiable and ∂_{ξ
_gH}ψ (η, ξ) is [ii - p]-differentiable w.r.t ξ. □

5 Analytical solutions of parabolic Volterra partial integro-differential equations

In this section, we give some implementation details for the fuzzy Laplace method for solving the following fuzzy parabolic Volterra partial integro-differential equation (FPVPID).

${\begin{matrix} \partial_{ξ_{gH}} ψ (η, ξ) = \partial_{η_{gH}} ψ (η, ξ) \oplus ϖ (η, ξ) \oplus \int_{0}^{ξ} k (η, ξ - τ) ψ (η, τ) d τ, η \in Ω \subseteq ℝ, ξ \in ℝ, \\ ψ (η, 0) = δ (η) . \end{matrix}$ (18) First, let ψ (η, ξ) is a [(i) - gH]-differentiable function with respect to ξ. Applying the fuzzy Laplace transform to Eq. (5). Making use Theorem 3.3 and taking the initial condition ψ (η, 0) = δ (η) into account, we have

$\begin{matrix} ϱ Ψ (η, ϱ) ⊖ δ (η) & = L_{ξ} [\partial_{η_{gH}} ψ (η, ξ)] \oplus L_{ξ} [ϖ (η, ξ)] \\ \oplus L_{ξ} [\int_{0}^{ξ} k (η, ξ - τ) ψ (η, τ) d τ] . \end{matrix}$ (19) Moreover, by Theorem 2.3 and Theorem 3.5, we get $\begin{matrix} L_{ξ} [\partial_{η_{gH}} ψ (η, ξ)] = \frac{d Ψ (η, ϱ)}{d η} . \\ L_{ξ} [\int_{0}^{ξ} k (η, ξ - τ) ⊙ ψ (η, τ) d τ] \\ = L_{ξ} [(k * ψ) (η, ξ)] = L_{ξ} [k (η, ξ)] ⊙ L_{ξ} [ψ (η, ξ)] . \end{matrix}$ Set $L_{ξ} [ϖ (η, ξ)] = W (η, ϱ)$ and $L_{ξ} [k (η, ξ)] = K (η, ϱ)$ . Therefore we obtain

$\begin{matrix} \frac{d Ψ (η, ϱ)}{d η} \oplus K (η, ϱ) Ψ (η, ϱ) \\ \underset{gH}{⊖} ϱ Ψ (η, ϱ) \oplus F (η, ϱ) = 0 . \end{matrix}$ (20) Where $F (η, ϱ) = W (η, ϱ) \oplus δ (η)$ . Equation (20) is a first order fuzzy differential equation that can be solved by the method described in [1]. Solving this fuzzy first order differential equation and taking fuzzy inverse Laplace transform of Ψ (η, ϱ), we will obtain the [(i) - gH]-differentiable solution of problem (18). Now, let ψ (η, ξ) is a [(ii) - gH]-differentiable function w.r.t ξ. The fuzzy Laplace transform is applied to equation (5), Theorem 3.3 and the initial condition ψ (η, 0) = δ (η) conclude that

$\begin{matrix} (- 1) δ (η) \underset{gH}{⊖} (- 1) ϱ Ψ (η, ϱ) = L_{ξ} [\partial_{η_{gH}} ψ (η, ξ)] \\ \oplus L_{ξ} [ϖ (η, ξ)] \oplus L_{ξ} [\int_{0}^{ξ} k (η, ξ - τ) ψ (η, τ) d τ] . \end{matrix}$ (21) Making use the method which is discussed in detail in the previous case, Eq. (21) can easily be reformulated as the following first order fuzzy differential equation. $\begin{matrix} \frac{d Ψ (η, ϱ)}{d η} \oplus K (η, ϱ) Ψ (η, ϱ) \\ \oplus (- 1) ϱ Ψ (η, ϱ) \oplus F (η, ϱ) = 0, \end{matrix}$ where $F (η, ϱ) = W (η, ϱ) \underset{gH}{⊖} (- 1) δ (η)$ . Now, let the following FPVPID

${\begin{matrix} \partial_{ξ ξ_{gH}} ψ (η, ξ) = \partial_{η_{gH}} ψ (η, ξ) \oplus ϖ (η, ξ) \oplus \int_{0}^{ξ} κ (η, ξ - τ) ψ (η, τ) d τ, η \in Ω \subseteq ℝ, ξ \in ℝ, \\ ψ (η, 0) = δ_{1} (η), \partial_{ξ_{gH}} ψ (η, 0) = δ_{2} (η) . \end{matrix}$ (22) For this problem, the following situations may be appeared

Case 1. ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [i - p]-differentiable respect to ξ.

Case 2. ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [ii - p]-differentiable respect to ξ.

Case 3. ψ (η, ξ) is [i - p]-differentiable respect to ξ and ∂_{ξ
_gH}ψ (η, ξ) is [ii - p]-differentiable respect to ξ

Case 4. ψ (η, ξ) is [ii - p]-differentiable respect to ξ and ∂_{ξ
_gH}ψ (η, ξ) is [i - p]-differentiable respect to ξ

Here, we show how to use the fuzzy Laplace method for solving problem (22) provided that ψ (η, ξ) and ∂_{ξ
_gH}ψ (η, ξ) are [ii - p]-differentiable respect to ξ. The rest of cases will be done in the same way.

Now applying the fuzzy Laplace transform to Eq. (22). Making use Theorem 3.4 and taking the initial conditions ψ (η, 0) = δ₁ (η) and ∂_{ξ
_gH}ψ (η, 0) = δ₂ (η) into account, we have $\begin{matrix} ϱ^{2} Ψ (η, ϱ) \underset{gH}{⊖} ϱ δ_{1} (η) \oplus (- 1) δ_{2} (η) \\ = \frac{d Ψ (η, ϱ)}{d η} \oplus W (η, ϱ) \oplus K (η, ϱ) ⊙ Ψ (η, ϱ), \end{matrix}$ or equivalently $\begin{matrix} \frac{d Ψ (η, ϱ)}{d η} \oplus K (η, ϱ) ⊙ Ψ (η, ϱ) \\ \underset{gH}{⊖} ϱ^{2} Ψ (η, ϱ) \oplus F (η, ϱ) = 0 . \end{matrix}$ Where $F (η, ϱ) = W (η, ϱ) \oplus ϱ δ_{1} (η) \underset{gH}{⊖} (- 1) δ_{2} (η)$ . This equation is a first order fuzzy differential equation. Solve this fuzzy first order differential equation by using the method described in [1]. The fuzzy solution of problem (22) is obtained by take fuzzy inverse Laplace transform of Ψ (η, ϱ),.

6 Illustrative examples

To demonstrate the efficiency and precision of the method for solving the fuzzy Volterra partial integro-differential equations, some different examples will be solved in this section. All calculations were performed on a PC running Mathematica software.

Example 6.1. Consider the FPVPID

$\begin{matrix} η \partial_{η_{gH}} ψ (η, ξ) \oplus \partial_{ξ ξ_{gH}} ψ (η, ξ) = (1, 2, 4) ⊙ η sin ξ \\ \oplus \int_{0}^{ξ} sin (ξ - s) ⊙ ψ (η, s) ds, 0 \leq η < 1, \end{matrix}$ (23) with fuzzy initial conditions and fuzzy boundary condition

$\begin{matrix} ψ (η, 0) = 0, 0 \leq ξ \leq η, \partial_{ξ_{gH}} ψ (η, 0) = (1, 2, 4) ⊙ η, \\ ψ (1, ξ) = (1, 2, 4) ⊙ ξ . \end{matrix}$

We want to find the [i - p]-differentiable solution this problem, then by using fuzzy Laplace transform with respect to ξ on both sides of Eq. (23) and by applying Theorems 3.4 and 3.5, we get $\begin{matrix} η ⊙ \frac{d Ψ}{d η} \oplus ϱ^{2} Ψ (η, ϱ) ⊖ ϱ ψ (η, 0) ⊖ \partial_{ξ_{gH}} ψ (η, 0) \\ = η ⊙ (\frac{(1, 2, 4)}{ϱ^{2} + 1}) \oplus \frac{1}{ϱ^{2} + 1} ⊙ Ψ (η, ϱ) . \end{matrix}$ By considering the fuzzy initial and boundary conditions, we obtain $\begin{matrix} η ⊙ \frac{d Ψ}{d η} \oplus ϱ^{2} Ψ (η, ϱ) \underset{gH}{⊖} \frac{1}{ϱ^{2} + 1} ⊙ Ψ (η, ϱ) \\ = (1, 2, 4) ⊙ η \oplus (1, 2, 4) ⊙ \frac{η}{ϱ^{2} + 1} . \end{matrix}$ Hence $\begin{matrix} η ⊙ \frac{d Ψ}{d η} \oplus (ϱ^{2} - \frac{1}{ϱ^{2} + 1}) ⊙ Ψ (η, ϱ) \\ = (1, 2, 4) ⊙ η (\frac{1}{ϱ^{2} + 1}) . \end{matrix}$ Then by dividing the sides of the Eq. (24) by $η \in ℝ$ , we have

$\begin{matrix} \frac{d Ψ}{d η} \oplus \frac{1}{η} (ϱ^{2} - \frac{1}{ϱ^{2} + 1}) ⊙ Ψ (η, ϱ) \\ = (1, 2, 4) ⊙ (1 + \frac{1}{ϱ^{2} + 1}) . \end{matrix}$ (24) Therefore by using to method described in [1], we get $μ = η^{(ϱ^{2} - \frac{1}{ϱ^{2} + 1})}$ as the integrating factor

$\begin{matrix} η^{(ϱ^{2} - \frac{1}{ϱ^{2} + 1})} ⊙ \frac{d Ψ}{d η} \oplus η^{(ϱ^{2} - \frac{1}{ϱ^{2} + 1})} \frac{1}{η} (ϱ^{2} - \frac{1}{ϱ^{2} + 1}) ⊙ Ψ (η, ϱ) \\ = (1, 2, 4) ⊙ (1 + \frac{1}{ϱ^{2} + 1}) η^{(ϱ^{2} - \frac{1}{ϱ^{2} + 1})} . \end{matrix}$

After performing the required operations $\begin{matrix} Ψ = (1, 2, 4) ⊙ (\frac{1}{ϱ^{2}}) ⊙ η \oplus c ⊙ η^{(ϱ^{2} - \frac{1}{ϱ^{2} + 1})} . \end{matrix}$ Setting fuzzy boundary condition ψ (1, ξ) = (1, 2, 4) ⊙ ξ yields to $ψ (1, ϱ) = (1, 2, 4) ⊙ (\frac{1}{ϱ^{2}}) \oplus c .$ If c = 0, we get

$Ψ (η, ϱ) = (1, 2, 4) ⊙ (\frac{1}{ϱ^{2}}) .$ (25) Taking fuzzy inverse Laplace transform 3.4 on both the sides of Eq. (25), we obtain $\begin{matrix} ψ (η, ξ) = (1, 2, 4) ⊙ η ξ . \end{matrix}$

Fig. 1

Plots of Example 6.1 for 0 ≤ α ≤ 1 and η = 1.

Example 6.2. Consider the following FPVPID

$\begin{matrix} \partial_{η_{gH}} ψ (η, ξ) = \partial_{ξ ξ_{gH}} ψ (η, ξ) \underset{gH}{⊖} 2 \int_{0}^{ξ} (ξ - s) \\ ⊙ ψ (η, s) ds \oplus 2 (1, 2.5, 4.5) ⊙ e^{η}, η \in [0, \frac{π}{2}], \end{matrix}$ (26) with ψ (η, 0) = (1, 2.5, 4.5) ⊙ e^η, ∂_{ξ
_gH}ψ (η, 0) =0, and ψ (0, ξ) = (1, 2.5, 4.5) ⊙ cos ξ . We want to find the [ii - p]-differentiable solution this problem So using the process described in Section 5, we have

$Ψ (η, ϱ) = (1, 2.5, 4.5) ⊙ (\frac{ϱ}{ϱ^{2} + 1}) e^{η} .$ (27) Taking fuzzy inverse Laplace transform 3.4 on both the sides of Eq. (27), we obtain $\begin{matrix} ψ (η, ξ) = (1, 2.5, 4.5) ⊙ e^{η} cos ξ . \end{matrix}$

Fig. 2

Plots of ψ (η, ξ) (left) and ∂_{ξ
_gH}ψ (η, ξ) (right) for $α = \frac{1}{2}$ , Example 6.2.

Fig. 3

Plots of Example 6.3 for 0 ≤ α ≤ 1 and ξ = 1.

Example 6.3. Consider the following FPVPID $\begin{matrix} {\begin{matrix} \partial_{η η_{gH}} ψ (η, ξ) = \partial_{ξ_{gH}} ψ (η, ξ) \oplus ψ (η, ξ) \oplus \int_{0}^{ξ} e^{ξ - s} ⊙ ψ (η, s) ds \underset{gH}{⊖} (0.5, 3, 5) ⊙ (η^{2} + 1) e^{ξ} \oplus 2 ⊙ (0.5, 3, 5), \\ ψ (η, 0) = (0.5, 3, 5) ⊙ η^{2}, ψ (0, ξ) = (0.5, 3, 5) ⊙ ξ, \\ \partial_{ξ_{gH}} ψ (η, 0) = (0.5, 3, 5), \partial_{η_{gH}} ψ (0, ξ) = 0 . \end{matrix} \end{matrix}$ We want to find the [i - p]-differentiable solution this problem, then by applying the method described in Section 5 we attain

$\begin{matrix} \frac{d^{2} Ψ}{d η^{2}} \underset{gH}{⊖} \frac{ϱ^{2}}{ϱ - 1} ⊙ Ψ = \underset{gH}{⊖} (0.5, 3, 5) ⊙ η^{2} \frac{ϱ}{ϱ - 1} \\ \underset{gH}{⊖} (0.5, 3, 5) ⊙ \frac{1}{ϱ - 1} \oplus (0.5, 3, 5) ⊙ \frac{2}{ϱ}, \end{matrix}$ (28)

[18] by using reduction of a fuzzy second order differential equation (28) to a system of first order equations (29), so

${\begin{matrix} y_{1_{gH}}^{'} = y_{2} \\ y_{2_{gH}}^{'} \underset{gH}{⊖} \frac{ϱ^{2}}{ϱ - 1} y_{1} = \underset{gH}{⊖} (0.5, 3, 5) ⊙ η^{2} \frac{ϱ}{ϱ - 1} \underset{gH}{⊖} (0.5, 3, 5) ⊙ \frac{1}{ϱ - 1} \oplus (0.5, 3, 5) ⊙ \frac{2}{ϱ}, \\ y_{1} (0, ϱ) = \frac{1}{ϱ^{2}} ⊙ (0.5, 3, 5), y_{2} (0, ϱ) = 0 . \end{matrix}$ (29) Therefore by taking the fuzzy Laplace transform, we obtain

${\begin{matrix} {sY}_{1} ⊖ (0.5, 3, 5) ⊙ \frac{1}{ϱ^{2}} = Y_{2} \\ \underset{gH}{⊖} \frac{ϱ^{2}}{ϱ - 1} Y_{1} \oplus {sY}_{2} = \underset{gH}{⊖} (0.5, 3, 5) ⊙ (\frac{2!}{s^{3}}) \frac{ϱ}{ϱ - 1} \underset{gH}{⊖} (0.5, 3, 5) ⊙ \frac{1}{ϱ - 1} (\frac{1}{s}) \oplus (0.5, 3, 5) ⊙ \frac{2}{ϱ} (\frac{1}{s}) . \end{matrix}$ (30) The fuzzy inverse Laplace transform is applied on both the sides of Eq. (30), so $\begin{matrix} ψ (η, ξ) = (0.5, 3, 5) ⊙ η^{2} \oplus (0.5, 3, 5) ⊙ ξ . \end{matrix}$

7 Conclusion

The fuzzy parabolic Volterra partial integro-differential equations had been inspected under generalized Hukuhara partial differentiability. the existence and uniqueness of the solution for the fuzzy Volterra partial integro-differential equations by considering the type of [gH - p]-differentiability of solution were proved. The fuzzy Laplace transform for two variables fuzzy function was introduced to obtain the solutions of the fuzzy parabolic Volterra partial integro-differential equations with emphasis on the type of [gH - p]-differentiability of solution. In addition, to demonstrate the efficiency of the method some various examples are solved.

References

Armand

and Gouyandeh

, On Fuzzy Solution for Exact Second Order Fuzzy Differential Equation, International Journal of Industrial Mathematics 9 (2017), 279–288.

Armand

, Allahviranloo

and Gouyandeh

, Some fundamental results on fuzzy calculus, Iranian Journal of Fuzzy Systems 15 (2018), 27–46.

Allahviranloo

, Uncertain Information and Linear Systems. Studies in Systems, Decision and Control 254, Springer 2020, ISBN 978-3-030-31323-4

Allahviranloo

, Amirteimoori

and Khezerloo

, A new method for solving fuzzy Volterra integro-differential equations, Australian Journal of Basic and Applied Sciences 5(4) (2011), 154–164.

Allahviranloo

, Abbasbandy

, Sedaghgatfar

and Darabi

, A new method for solving fuzzy integro-differential equation under generalized differentiability, Neural Computing and Applications 21 (2012), 191–196.

Allahviranloo

and Ahmadi

M.B.

, Fuzzy Laplace transforms, Soft Comput 14 (2010), 235–243.

Allahviranloo

, Gouyandeh

, Armand

and Hasanoglu

, On fuzzy solutions for heat equation based on generalized hukuhara differentiability, Fuzzy Sets Syst 265 (2015), 1–23.

Bertone

A.M.

, Jafelice

R.M.

, Barros

L.C.

and Bassanezi

R.C.

, On fuzzy solutions for partial differential equations, Fuzzy SetsandSystems 219 (2013), 68–80.

Buckley

J.J.

and Feuring

, Introduction to fuzzy partial differential equations, Fuzzy Sets Systems 105(2) (1999), 241–248.

10.

Bede

and Stefanini

, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems 230 (2013), 119–141.

11.

Bede

, Mathematics of fuzzy sets and fuzzy logic, Springer, London, 2013.

12.

Faybishenko

B.A.

, Introduction to modeling of hydrogeologic systems using fuzzy partial differential equation, in: M. Nikravesh, L. Zadeh, V. Korotkikh (Eds.), Fuzzy Partial Differential Equations and Relational Equations, Reservoir Characterization and Modeling, Springer, 2004.

13.

Gouyandeh

, Allahviranloo

, Abbasbandy

and Armand,

, A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform, Fuzzy Sets and System 309 (2017), 81–97.

14.

Hajighasemi

, Allahviranloo

, Khezerloo

, Khorasany

and Salahshour

, Existence and uniqueness of solutions of fuzzy volterra Integro-differential equations, Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications 2010, 491–500.

15.

Kadak

and Efe

, On uniform convergence of sequences and series of fuzzy-valued functions, , Journal of function spaces 2015 (2015), 1–11.

16.

Kaufmann

and Gupta

M.M.

, Introduction Fuzzy Arithmetic, Van Nostrand Reinhold, New York, 1985.

17.

Kaleva

, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987), 301–317.

18.

Keshavarz

, Allahviranloo

, Abbasbandy

and Modarressi

M.H.

, A study of fuzzy methods for solving system of fuzzy differential equations, New Mathematics and Natural Computation, 2020, In Press.

19.

Negoita

C.V.

and Ralescu

, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975.

20.

Salahshour

and Allahviranloo

, Applications of fuzzy Laplace transforms, Soft Comput 17 (2013), 145–158.

21.

Qiu

, The generalized Hukuhara differentiability of interval-valued function is not fully equivalent to the onezsided differentiability of its endpoint functions, Fuzzy Sets and Systems 2020, 1–11, https://doi.org/10.1016/j.fss.2020.07.012.

22.

Stefanini

and Bede

, Generalized hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal 71(34) (2009), 1311–1328.

23.

Stefanini

, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems 161 (2010), 1564–1584.

24.

Salahshour

, Allahviranloo

and Abbasbandy

, Solving fuzzy fractional differential equationsby fuzzy Laplace transforms, Nonlinear Sci Numer Simul 17 (2012), 1372–1381.