Fuzzy Double Yang Transform and Its Application to Fuzzy Parabolic Volterra Integro-Differential Equation

Abstract

This article introduces a new fuzzy double integral transformation called fuzzy double Yang transformation. We review some of the main properties of the transformation and find the conditions for its existence. We prove the theorems for partial derivatives and fuzzy unitary convolution. All of the new results are applied to find an analytical solution to the fuzzy parabolic Volterra integro-differential equation (FPVIDE) with a suitably selected memory kernel. In addition, a numerical example is provided to illustrate how the proposed method might be helpful for solving FPVIDE utilizing symmetric triangular fuzzy numbers. Compared with other symmetric transforms, we conclude that our new approach is simpler and needs less calculations.

1. Introduction

Partial differential equations are used for dynamic modeling of complex processes in various fields such as physics, chemistry, fluid and quantum mechanics, biology, and economics. They are predominantly applied to the so-called instantaneous phenomena, whose behavior depends on their momentary state. A large part of the processes require the model to account for their behavior over a previous time interval. As a result, it is necessary to use partial integro-differential equations, as they show the cumulative behavior of the process. Different types of partial differential equations are related to the various types of differential and integral operators. One of them is the parabolic Volterra integro-differential equation. It has important physical applications in modeling dynamical systems, where one can explore the effects of the “memory” of the system. Such systems are developed, for example, in the compression of viscoelastic media [1], nuclear reactor dynamics [2], expansion problems [3], reaction diffusion problems [4], and thermally conductive materials with functional memory [5].

Over the last few years, we have noticed an incredible interest in fuzzy mathematics due to the many applications in various fields, especially physics [6], artificial intelligence [7], medicine [8], engineering [9], and biology [10,11]. This trend explains the need for studying fuzzy differential equations. The solution to the first-order linear fuzzy differential equation was provided in [12], using different versions of the variation of constants formula. An generalized Euler approximation method was applied to solve a first-order linear fuzzy differential equation [13]. Osman et al. [14] investigated the fuzzy Adomian decomposition method and the fuzzy variational iteration method, and applied to solve fuzzy heat-like and wave equations with variable coefficients, in the sense of gH-differentiability. In [15], the fuzzy Elzaki transform and the fuzzy Elzaki decomposition method were used to find solutions to fuzzy linear–nonlinear Schrodinger differential equations. In recent years, fuzzy double integral transformations have been used to find the exact solution to a linear fuzzy partial differential equation [16,17,18]. Analytical and numerical solutions to fuzzy integro-differential equations were obtained in [19,20,21].

Numerical solutions to the fuzzy partial Volterra integro-differential equation using the reproducing kernel Hilbert space method can be found in [22]. Recently, in order to find the exact solution to the linear fuzzy integro-differential equation, fuzzy integral transforms have been used. In [23], using the fuzzy Laplace transform, an analytical solution to FPVIDE under generalized partial Hukuhara differentiability was found. The fuzzy single and fuzzy double Sumudu transformation [24,25], as well as the fuzzy double Natural transformation [26], have been applied to the fuzzy partial Volterra integro-differential equation.

The Yang transform was introduced by Yang [27] and has been applied in the differential equation of the steady heat-transfer problem. Recently, Ullah et al. [28] proposed a fuzzy single Yang transform to find the solution to second-order fuzzy differential equations of an integer and fractional-order.

The main goal of the article is to extend the fuzzy single Yang transform to the fuzzy double Yang transform (FDYT). The fundamental properties and theorems of FDYT are presented and proven, and we compute the values of FDYT for some functions. New relations related to partial derivatives and the single convolution theorem are proven, which allows us to find the exact solution to an FPVIDE under generalized Hukuhara differentiability. More precisely, we look at the following nonhomogeneous FPVIDE with a symmetric memory kernel $k\left(t\right)$ in the infinite domain

$$g_{t,gH}^{\prime }(x,t)\oplus \underset{0}{\overset{t}{\int }}k(t-s)\odot g(x,t)dxdt=\sigma \odot g_{xx,gH}^{\prime \prime }(x,t)\oplus f(x,t),x\ge 0,t\ge 0,$$

(1)

where $\sigma$ is any positive constant, $g(x,t)$ is the unknown fuzzy function, and $f(x,t)$ is a given fuzzy function. A simple formula for the solution to the Equation (1) is obtained and applied to solve a numerical example in order to display the efficiency of this new approach.

The remainder of this work is structured as follows:

In Section 2, we briefly present some basic concepts regarding fuzzy numbers and fuzzy calculus that will be used in the paper. Section 3 introduces the single fuzzy Yang transform (FYT), some fundamental concepts, and the basic properties of this transformation.

In Section 4, an FDYT for a fuzzy function is defined as well as some properties and theorems, and several relations regarding the existence, gH-partial derivatives, and single convolution are presented. An FPVIDE with the memory kernel is defined under generalized partial Hukuhara differentiability and a solution to this equation using the FDYT method is investigated in Section 5. Moreover, a numerical example is constructed to clarify the details and efficiency of the method in Section 6. Conclusions are given in Section 7.

2. Premilinaries

The following section consists of the necessary notations, definitions, and theorems which are useful in this research.

Let $E^1$ denote the set of fuzzy subsets of the real axis, i.e., $\nu :\mathbb{R}\to [0,1]$, which possesses the following properties:

(i)

$\nu$ is upper semi-continuous on $\mathbb{R}$ for all $\nu \in E^1$;

(ii)

$\nu$ is normal for all $\nu \in E^1$;

(iii)

$\nu$ is fuzzy convex for all $\nu \in E^1$;

(iv)

$cl\{\eta \in \mathbb{R}:\nu (\eta )>0\}$ is compact, where cl denotes the closure of a subset.

Then, we say that $E^1$ is a space of fuzzy numbers. It is clear that any real number a can be interpreted as a fuzzy number $\tilde{a}=\chi \left(a\right)$; therefore, $\mathbb{R}\subset E^1$. The r-level set of the fuzzy number $\nu$ is denoted as

$${\left[\nu \right]}^r=\left\{\begin{array}{cc}\{\eta \in \mathbb{R}:\nu (\eta )\ge r\},\hfill & 0<r\le 1,\hfill \\ cl\{\eta \in \mathbb{R}:\nu (\eta )>0\},\hfill & r=0.\hfill \end{array}\right.$$

Then, from (i) to (iv), it follows that for each $0\le r\le 1$, the r-level sets of fuzzy number $\nu$ are nonempty closed intervals of the form

$${\left[\nu \right]}^r=[\underline{\nu }\left(r\right),\overline{\nu }\left(r\right)].$$

A triangular fuzzy number $\nu$ is defined as an ordered triple $\nu =({\nu }_1,{\nu }_2,{\nu }_3)$, where ${\nu }_1\le {\nu }_2\le {\nu }_3$ has r-cuts

$${\left[\nu \right]}^r=[{\nu }_1+({\nu }_2-{\nu }_1)r,{\nu }_3-({\nu }_3-{\nu }_2)r],0\le r\le 1.$$

Let $\mu$ and $\nu$ be two fuzzy numbers and $k\in \mathbb{R}$. Then, the addition $\mu \oplus \nu \in E^1$ and the scalar multiplication $k\odot \mu \in E^1$ are defined as having level cuts

$${[\mu \oplus \nu ]}^r={\left[\mu \right]}^r+{\left[\nu \right]}^r=\left\{\xi +\eta :\xi \in {\left[\mu \right]}^r,\eta \in {\left[\nu \right]}^r\right\}$$

$${[k\odot \mu ]}^r=k.{\left[\mu \right]}^r=\left\{k\xi :\xi \in {\left[\mu \right]}^r\right\},{\left[0\right]}^r=\left\{0\right\}forall0\le r\le 1.$$

Denote ${\mathbb{R}}_{+}=[0,+\infty )$.

Definition 1

([29]). The Hausdorff distance between fuzzy numbers is given by $D:E^1\times E^1\to {\mathbb{R}}_{+}$ as

$$D(\mu ,\nu )=\underset{0\le r\le 1}{sup}max\left\{\right|\underline{\mu }\left(r\right)-\underline{\nu }\left(r\right)|,|\overline{\mu }\left(r\right)-\overline{\nu }\left(r\right)\left|\right\},$$

where ${\left[\mu \right]}^r=[\underline{\mu }\left(r\right),\overline{\mu }\left(r\right)]$ and ${\left[\nu \right]}^r=[\underline{\nu }\left(r\right),\overline{\nu }\left(r\right)]$ .

The metric space $(E^1,D)$ is complete separable and locally compact, and the following properties of the metric D are well known:

(i)

$D(\lambda \oplus \nu ,\mu \oplus \nu )=D(\lambda ,\mu )$ for all $\lambda ,\mu ,\nu \in E^1$;

(ii)

$D(k\odot \mu ,k\odot \nu )=\left|k\right|D(\mu ,\nu )$ for all $\mu ,\nu \in E^1$ and $k\in \mathbb{R}$ ;

(iii)

$D(\lambda \oplus \mu ,\nu \oplus \kappa )\le D(\lambda ,\nu )+D(\mu ,\kappa )$ for all $\lambda ,\mu ,\nu ,\kappa \in E^1$.

Definition 2

([29]). Let $\mu ,\nu \in E^1$. If there exists a fuzzy number λ, such that $\mu =\nu \oplus \lambda$, then λ is called the Hukuhara difference (H-difference) of μ and ν, and it is denoted by $\mu {\ominus }_H\nu$.

The r-cuts of H-difference are

$${\left[\mu {\ominus }_H\nu \right]}^r=[\underline{\mu }\left(r\right)-\underline{\nu }\left(r\right),\overline{\mu }\left(r\right)-\overline{\nu }\left(r\right)],$$

where ${\left[\mu \right]}^r=[\underline{\mu }\left(r\right),\overline{\mu }\left(r\right)]$ and ${\left[\nu \right]}^r=[\underline{\nu }\left(r\right),\overline{\nu }\left(r\right)]$.

Clearly, $\mu {\ominus }_H\mu =\left\{\tilde{0}\right\}$; if $\mu {\ominus }_H\nu$ exists, it is unique.

Definition 3

([29]). Given $\mu ,\nu \in E^1$, the generalized Hukuhara difference (gH-difference) is the fuzzy quantity $\lambda \in E^1$, if it exists, such that

$$\mu {\ominus }_{gH}\nu =\lambda \iff \left\{\begin{array}{cc}\left(i\right)\hfill & \mu =\nu \oplus \lambda ,\hfill \\ or\left(ii\right)\hfill & \nu =\mu \oplus (-1)\odot \lambda .\hfill \end{array}\right.$$

(2)

It is easy to show that $\left(i\right)$ and $\left(ii\right)$ are valid if and only if $\lambda$ is a crisp number.

In terms of r-cuts, we have

$${\left[\mu {\ominus }_{gH}\nu \right]}^r=[min\{\underline{\mu }\left(r\right)-\underline{\nu }\left(r\right),\overline{\mu }\left(r\right)-\overline{\nu }\left(r\right)\},max\{\underline{\mu }\left(r\right)-\underline{\nu }\left(r\right),\overline{\mu }\left(r\right)-\overline{\nu }\left(r\right)\}]$$

and if the H-difference exists, then $\mu {\ominus }_H\nu =\mu {\ominus }_{gH}\nu$. The conditions for the existence of $\mu {\ominus }_{gH}\nu =w\in E^1$ are given in [30].

Proposition 1

([29]). Let $\mu ,\nu \in E^1$, then

$$D(\mu {\ominus }_{gH}\nu ,\tilde{0})=D(\mu ,\nu ).$$

Proposition 2

([30]). Let $\mu ,\nu \in E^1$. If $\mu {\ominus }_{gH}\nu$ exists, it is unique and has the following properties

 (i) 

$\mu {\ominus }_{gH}\mu =\tilde{0}$;

 (ii) 

$(\mu \oplus \nu ){\ominus }_{gH}\nu =\mu ,\mu {\ominus }_{gH}\left(\mu {\ominus }_H\nu \right)=\nu$;

 (iii) 

if $\mu {\ominus }_{gH}\nu$ exists, then $(-\nu ){\ominus }_{gH}(-\mu )$ also does and $\tilde{0}{\ominus }_{gH}\left(\mu {\ominus }_{gH}\nu \right)=(-\nu ){\ominus }_{gH}(-\mu )$;

 (iv) 

$\mu {\ominus }_{gH}\nu =\nu {\ominus }_{gH}\mu =\lambda$ if and only if $\lambda =-\lambda$; furthermore, $\lambda =\tilde{0}$ if and only if $\mu =\nu$;

 (v) 

If $\nu {\ominus }_{gH}\mu$ exists, then either $\mu \oplus \left(\nu {\ominus }_{gH}\mu \right)=\mu$ or $\nu {\ominus }_H\left(\nu {\ominus }_{gH}\mu \right)=\mu$, and if both equalities hold then $\nu {\ominus }_{gH}\mu$ is a crisp set.

2.1. The One-Variable Fuzzy Calculus

In this section, we present basic definitions and theorems for a fuzzy-valued function of one-variable that will be used throughout the paper.

A function $g:[c,d]\subset \mathbb{R}\to E^1$ is called a fuzzy-valued function. The r-level representation of this fuzzy function g is given by $g(t,r)=[\underline{g}(t,r),\overline{g}(t,r)]$, $t\in [c,d]$ for all $0\le r\le 1.$

Definition 4

([31]). We say that fuzzy-valued function $g:[c,d]\to E^1$ is continuous at $t_0\in [c,d]$, if

$$\underset{t\to t_0}{lim}D(g\left(t\right),g\left(t_0\right))=0$$

provided that limits exist.

The function g is fuzzy continuous on $[c,d]$ if g is continuous in each $t_0\in [c,d]$.

Definition 5

([30]). Let $t_0\in (c,d)$ and k be such that $t_0+k\in (c,d)$. Then, the generalized Hukuhara derivative (gH-derivative) of a function $g:(c,d)\to E^1$ at $t_0$ is calle the fuzzy number $g_{gH}^{\prime }\left(t_0\right)$, which is defined as

$$g_{gH}^{\prime }\left(t_0\right)=\underset{k\to 0}{lim}{\displaystyle \frac{1}{k}}\left[g(t_0+k){\ominus }_{gH}g\left(t_0\right)\right],$$

(3)

if the limit exists.

Theorem 1.

Let $g:(c,d)\to E^1$ be gH-differentiable at $t_0\in [c,d)$. Then, g is fuzzy continuous at $t_0$.

Proof. 

Using properties of distance D, along with gH-differentiability of g and Proposition 1, we have

$$\begin{array}{cc}\underset{t\to t_0}{lim}D(g\left(t\right),g\left(t_0\right))\hfill & =\underset{t\to t_0}{lim}D(g\left(t\right){\ominus }_{gH}g\left(t_0\right),\tilde{0})\hfill \\ & =\underset{t\to t_0}{lim}D({\displaystyle \frac{g\left(t\right){\ominus }_{gH}g\left(t_0\right)}{t-t_0}}(t-t_0),\tilde{0})\hfill \\ & =|t-t_0|\underset{t\to t_0}{lim}D({\displaystyle \frac{g\left(t\right){\ominus }_{gH}g\left(t_0\right)}{t-t_0}},\tilde{0})=0.D(g_{gH}^{\prime }\left(t_0\right),\tilde{0})=0.\hfill \end{array}$$

□

The next theorem gives the expression of the fuzzy gH-derivative in terms of the derivatives of the endpoints of the level sets.

Theorem 2.

Let $g:[c,d]\to E^1$ be a fuzzy-valued function with r-levels

$g(t,r)=[\underline{g}(t,r),\overline{g}(t,r)]$ and the real-valued functions $\underline{g}(.,r)$ and $\overline{g}(.,r)$ be differentiable at $t_0$ for all $0\le r\le 1$. Then, the function $g\left(t\right)$ is gH-differentiable at $t_0\in (c,d)$, if and only if one of the following two cases holds:

 (i) 

$\underline{g}(t_0,r)$ is increasing, $\overline{g}(t_0,r)$ is decreasing, and $\underline{g}(t_0,r)\le \overline{g}(t_0,r)$;

 (ii) 

$\underline{g}(t_0,r)$ is decreasing, $\overline{g}(t_0,r)$ is increasing, and $\overline{g}(t_0,r)\le \underline{g}(t_0,r)$.

Moreover, we have

$$g_{gH}^{\prime }(t_0,r)=[min\{\underline{g}(t_0,r),\overline{g}(t_0,r)\},max\{\underline{g}(t_0,r),\overline{g}(t_0,r)\}]$$

for all $0\le r\le 1$.

Proof. 

See Theorem 24 in [30]. □

If $\underline{g}(.,r)$ and $\overline{g}(.,r)$ are both differentiable, according to Theorem 2, for the definition of gH-differentiability, we distinguish two cases corresponding to $\left(i\right)$ and $\left(ii\right)$ of Equation (3).

Definition 6

([30]). Let $g:[c,d]\to E^1$ and $t_0\in (c,d)$, with $\underline{g}(.,r)$ and $\overline{g}(.,r)$ both be differentiable at $t_0$. The fuzzy-valued function g is called:

1. 

(i)-gH-differentiable at $t_0$ if

$$g_{gH}^{\prime }(t_0,r)=\left[{\underline{g}}^{\prime }(t_0,r),{\overline{g}}^{\prime }(t_0,r)\right]forallr\in [0,1];$$

(4)

2. 

(ii)-gH-differentiable at $t_0$ if

$$g_{gH}^{\prime }(t_0,r)=\left[{\overline{g}}^{\prime }(t_0,r),{\underline{g}}^{\prime }(t_0,r)\right]forallr\in [0,1].$$

(5)

Theorem 3.

Let $f,g:(c,d)\to E^1$ be gH-differentiable. Then, $f\left(t\right)\oplus g\left(t\right)$ is gH-differentiable and

$${(f\oplus g)}_{gH}^{\prime }\left(t\right)=f_{gH}^{\prime }\left(t\right)\oplus g_{gH}^{\prime }\left(t\right).$$

Proof. 

Suppose that f and g are both $\left(i\right)$-gH-differentiable. Then, for every $r\in [0,1]$ we have,

$$f_{gH}^{\prime }(t_0,r)=\left[{\underline{f}}^{\prime }(t_0,r),{\overline{f}}^{\prime }(t_0,r)\right]$$

and

$$g_{gH}^{\prime }(t_0,r)=\left[{\underline{g}}^{\prime }(t_0,r),{\overline{g}}^{\prime }(t_0,r)\right].$$

Then

$$\begin{array}{cc}{f}_{gH}^{\prime }(t_0,r)\oplus g_{gH}^{\prime }(t_0,r)\hfill & =\left[{\underline{f}}^{\prime }(t_0,r),{\overline{f}}^{\prime }(t_0,r)\right]+\left[{\underline{g}}^{\prime }(t_0,r),{\overline{g}}^{\prime }(t_0,r)\right]\hfill \\ & =\left[{\underline{f}}^{\prime }(t_0,r)+{\underline{g}}^{\prime }(t_0,r),{\overline{f}}^{\prime }(t_0,r)+{\overline{f}}^{\prime }(t_0,r)\right]\hfill \\ & ={(f\oplus g)}_{gH}^{\prime }(t_0,r).\hfill \end{array}$$

□

Theorem 4.

Let $f:[c,d]\to E^1$ and $g:[c,d]\to \mathbb{R}$ be two differentiable functions. Then,

$$\underset{c}{\overset{d}{\int }}g\left(t\right)\odot f_{gH}^{\prime }\left(t\right)dt=\left(g\left(d\right)\odot f\left(d\right)\right){\ominus }_{gH}\left(g\left(c\right)\odot f\left(c\right)\right){\ominus }_{gH}\underset{c}{\overset{d}{\int }}{g}^{\prime }\left(t\right)\odot f\left(t\right)dt.$$

Proof. 

See [29]. □

Theorem 5.

Let $g:\mathbb{R}\to E^1$ be a fuzzy-valued function with r-levels

$g(t,r)=[\underline{g}(t,r),\overline{g}(t,r)]$. Suppose that the functions $\underline{g}(t,r)$ and $\overline{g}(t,r)$ are Riemann integrable on $\mathbb{R}$ for all $0\le r\le 1$. Then, $g\left(t\right)$ is improper fuzzy Riemann-integrable in $\mathbb{R}$. Moreover, we have

$${\left[\underset{-\infty }{\overset{\infty }{\int }}g\left(t\right)dt\right]}^r=\left[\underset{-\infty }{\overset{\infty }{\int }}\underline{g}(t,r)dt,\underset{-\infty }{\overset{\infty }{\int }}\overline{g}(t,r)dt\right]$$

for all $0\le r\le 1$.

Proof. 

See [29]. □

2.2. The Two-Variable Fuzzy Calculus

Let $g:Q\subset \mathbb{R}\times \mathbb{R}\to E^1$ be a fuzzy-valued function of two variables with r-levels $g(x,t,r)=[\underline{g}(x,t,r),\overline{g}(x,t,r)]$ for all $(x,t)\in Q$ and $0\le r\le 1$.

Definition 7

([32]). For $(x_0,t_0)\in Q$, let the constants h and k be such that $(x_0+h,t_0)\in Q$ and $(x_0,t_0+k)\in Q$. Then, the first generalized Hukuhara partial derivatives (gH-p-derivative) of a fuzzy-valued function $g:Q\to E^1$ at $(x_0,t_0)\in Q$ with respect to x and t are called the fuzzy numbers $g_x^{\prime }(x_0,t_0)$ and $g_t^{\prime }(x_0,t_0)$, and are defined as

$$g_{x,gH}^{\prime }(x_0,t_0)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h}}\left[g(x_0+h,t_0){\ominus }_{gH}g(x_0,t_0)\right],$$

$$g_{t,gH}^{\prime }(x_0,t_0)=\underset{k\to 0}{lim}{\displaystyle \frac{1}{k}}\left[g(x_0,t_0+k){\ominus }_{gH}g(x_0,t_0)\right].$$

Definition 8

([32]). Let $g:Q\to E^1$ be fuzzy-valued function, and $(x_0,t_0)\in Q$. Suppose that the functions $\underline{g}(x,t,r)$ and $\overline{g}(x,t,r)$ are partially differentiable in $(x_0,t_0)$ with respect to variable t. Also, we say that the function $g(x,t)$ is:

 1. 

(i)-p-gH-differentiable at $(x_0,t_0)$ with respect to t if

$$g_{t,gH}^{\prime }(x_0,t_0,r)=\left[{\underline{g}}_t^{\prime }(x_0,t_0,r),{\overline{g}}_t^{\prime }(x_0,t_0,r)\right]forallr\in [0,1],$$

(6)

 2. 

(ii)-p-gH-differentiable at $(x_0,y_0)$ with respect to variable x if

$$g_{t,gH}^{\prime }(x_0,t_0,r)=\left[{\overline{g}}_t^{\prime }(x_0,t_0,r),{\underline{g}}_t^{\prime }(x_0,t_0,r)\right]forallr\in [0,1].$$

(7)

Theorem 6.

Let $g:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to E^1$ be a fuzzy-valued function. Assume that $\underset{0}{\overset{\infty }{\int }}g(x,t)dt$ is convergent for each ${\mathbb{R}}_{+}$ and $\underset{0}{\overset{\infty }{\int }}g(x,t)dx$, as function t is convergent on ${\mathbb{R}}_{+}$. Then

$$\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}g(x,t)dtdx=\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}g(x,t)dxdt.$$

Proof. 

See [33]. □

3. Fuzzy Yang Transform

In this section, we present the definition and basic properties of FYT [28].

Definition 9.

The FYT for a fuzzy function $g\left(t\right)$ is defined as

$$Y_t\left[g\left(t\right)\right]=G\left(\beta \right)=\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g\left(t\right)dt,$$

(8)

provided that the improper fuzzy integral exists and where t and β are transform variables.

Definition 10.

The inverse fuzzy Yang transform is given by

$$Y_{\beta }^{-1}\left[G\left(\beta \right)\right]=g\left(t\right)={\displaystyle \frac{1}{2\pi i}}\underset{b-i\infty }{\overset{b+i\infty }{\int }}{e}^{{\displaystyle \frac{t}{\beta }}}\odot G\left(\beta \right)d\beta ,$$

(9)

where the function $G\left(\beta \right)$ is analytic for all β, such that $Re\beta >b$.

Definition 11.

A fuzzy-valued function $g:{\mathbb{R}}_{+}\to E^1$ is said to be of exponential order $e^{dt}$ as $t\to \infty$, if there exists a positive constant L, such that for all $t>T$, we have

$$D(g\left(t\right),\tilde{0})\le L{e}^{dt}.$$

Theorem 7.

If $g\left(t\right)$ is a continuous fuzzy function in every finite interval $0\le t\le T$ and $g\left(t\right)$ is of exponential order $e^{dt}$.

Then, the FYT of $g\left(t\right)$ exists for all β, such that $Re\left({\displaystyle \frac{1}{\beta }}\right)>d$.

Proof. 

Using Definition 9, we obtain

$$D(Y_t\left[g\left(t\right)\right],\tilde{0})=D(G\left(\beta \right),\tilde{0})=D\left(\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g\left(t\right)dt,\tilde{0}\right).$$

Using the property of the improper fuzzy integral, we obtain

$$\begin{array}{cc}D(G\left(\beta \right),\tilde{0})\hfill & =D(\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g\left(t\right)dt,\tilde{0})\le \underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}D(g\left(t\right),\tilde{0})dt\hfill \\ & \le L\underset{0}{\overset{\infty }{\int }}{e}^{-({\displaystyle \frac{1}{\beta }}-d)t}dt={\displaystyle \frac{L\alpha }{1-d\beta }}.\hfill \end{array}$$

Thus, the improper fuzzy integral converges for all $Re\left({\displaystyle \frac{1}{\beta }}\right)>d$, and $Y_t\left[g\left(t\right)\right]$ exists. □

The classical Yang transform is applied to some special functions in [27].

(i)

$Y_t\left[1\right]=\beta$;

(ii)

$Y_t\left[t^n\right]=\left\{\begin{array}{cc}{\beta }^{n+1}n!\hfill & n=1,2,3,\dots \hfill \\ {\beta }^{n+1}\mathsf{\Gamma }(n+1)\hfill & n>0\hfill \end{array}\right.$

(iii)

$Y_t\left[e^{dt}\right]={\displaystyle \frac{\beta }{1-d\beta }}$  for all $d\in \mathbb{R}$;

(iv)

$Y_t[sindt]={\displaystyle \frac{d\beta }{1+d^2{\beta }^2}}$  for all $d\in \mathbb{R}$;

(v)

$Y_t[cosdt]={\displaystyle \frac{{\beta }^2}{1+d^2{\beta }^2}}$  for all $d\in \mathbb{R}$;

(vi)

$Y_t[sinhdt]={\displaystyle \frac{d{\beta }^2}{1-d^2{\beta }^2}}$  for all $d\in \mathbb{R}$;

(vii)

$Y_t[coshdt]={\displaystyle \frac{\beta }{1-d^2{\beta }^2}}$ for all $d\in \mathbb{R}$.

We will give some of the basic properties of FYT.

Theorem 8

(Linearity). If $G_1\left(\beta \right)=Y_t\left[g_1\left(t\right)\right]$ and $G_2\left(\beta \right)=Y_t\left[g_2\left(t\right)\right]$. Then

$$Y_t[b_1\odot g_1\left(t\right)\oplus b_2\odot g_2\left(t\right)]=b_1\odot Y_t\left[g_1\left(t\right)\right]\oplus b_2\odot Y_t\left[g_2\left(t\right)\right],$$

where $b_1,b_2\in \mathbb{R}$, such that $b_1,b_2\ge 0$ or $b_1,b_2\le 0$.

Proof. 

Using Definition 9 and the property of the improper fuzzy integral, we obtain

$$\begin{array}{cc}{Y}_t[b_1\odot g_1\left(t\right)\oplus b_2\odot g_2\left(t\right)]\hfill & =\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\left(b_1\odot g_1\left(t\right)\oplus b_2\odot g_2\left(t\right)\right)dt\hfill \\ & =\underset{0}{\overset{\infty }{\int }}{b}_1{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g_1\left(t\right)dt\oplus \underset{0}{\overset{\infty }{\int }}{b}_2{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g_2\left(t\right))dt\hfill \\ & =b_1\odot Y_t\left[g_1\left(t\right)\right]\oplus b_2\odot Y_t\left[g_2\left(t\right)\right].\hfill \end{array}$$

□

Remark 1.

Using the Definition 10, we can show that $Y_t^{-1}$ a linear transformation, i.e.,

$$Y_t^{-1}\left[b_1\odot G_1\left(\beta \right)\oplus b_2\odot G_2\left(\beta \right)\right]=b_1\odot Y_t^{-1}\left[G_1\left(\beta \right)\right]\oplus b_2\odot Y_t^{-1}\left[G_2\left(\beta \right)\right]$$

Theorem 9

(Change of Scale). If $G\left(\beta \right)=Y_t\left[g\left(t\right)\right]$, then for some constant b, it follows

$$Y_t\left[g\left(bt\right)\right]={\displaystyle \frac{1}{b}}\odot G\left({\displaystyle \frac{1}{b}}\beta \right).$$

Proof. 

Using Definition 9, we have

$$Y_t\left[g\left(bt\right)\right]=\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g\left(bt\right)dt.$$

Put $bt=t_1$ and $dt={\displaystyle \frac{1}{b}}d{t}_1$ in the above equation, we have

$$Y_t\left[g\left(bt\right)\right]=\underset{0}{\overset{\infty }{\int }}{\displaystyle \frac{1}{b}}{e}^{-{\displaystyle \frac{t_1}{b\beta }}}\odot g\left(t_1\right)d\left(t_1\right)={\displaystyle \frac{1}{b}}\odot G\left({\displaystyle \frac{1}{b}}\beta \right).$$

□

Theorem 10

(Duality). If $Y_t\left[g\left(t\right)\right]=G\left(\beta \right)$ is FYT and $L_t\left[g\left(t\right)\right]=F\left(\beta \right)$ is the fuzzy Laplace transform of $g\left(t\right)$, then

$$G\left(\beta \right)=F\left({\displaystyle \frac{1}{\beta }}\right)andF\left(\beta \right)=G\left({\displaystyle \frac{1}{\beta }}\right)$$

Proof. 

Using Definition 9, we have

$$G\left(\beta \right)=Y_t\left[g\left(t\right)\right]=\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g\left(t\right)dt=\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{1}{\beta }}t}\odot g\left(t\right)dt=F\left({\displaystyle \frac{1}{\beta }}\right).$$

and

$$F\left(\beta \right)=\underset{0}{\overset{\infty }{\int }}{e}^{-\beta t}\odot g\left(t\right)dt=\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\frac{1}{\beta }}}}\odot g\left(t\right)dt=G\left({\displaystyle \frac{1}{\beta }}\right).$$

□

Theorem 11.

Let us consider

 (i) 

$g\left(t\right)$ is a continuous fuzzy function for all $t\ge 0$;

 (ii) 

$g\left(t\right)$ is of exponential order $e^{dt}$, i.e.,

$$D(g\left(t\right),\tilde{0})\le L{e}^{dt},t\in [0,T],L>0;$$

 (iii) 

$g_{gH}^{\prime }\left(t\right)$ is continuous in every finite closed interval $0\le t\le T$.

Then,

 1. 

$Y_t\left[g_{gH}^{\prime }\left(t\right)\right]=(-1)\odot g\left(0\right){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot Y_t\left[g\left(t\right)\right]$;

 2. 

$Y_t\left[g_{gH}^{\prime \prime }\left(t\right)\right]=(-1)\odot g_{gH}^{\prime }\left(0\right){\ominus }_{gH}\left({\displaystyle \frac{1}{\beta }}\odot g\left(0\right){\ominus }_{gH}{\displaystyle \frac{1}{{\beta }^2}}\odot Y_t\left[g\left(t\right)\right]\right)$,

for all $Re\left({\displaystyle \frac{1}{\beta }}\right)>d$.

Proof. 

We prove case 1. Using definition of an improper fuzzy integral and Theorem 4, we obtain

$$\begin{array}{c}{Y}_t\left[g_{gH}^{\prime }\left(t\right)\right]=\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g_{gH}^{\prime }\left(t\right)dt=\underset{\xi \to \infty }{lim}\underset{0}{\overset{\xi }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g_{gH}^{\prime }\left(t\right)dt\hfill \\ =\underset{\xi \to \infty }{lim}(e^{-{\displaystyle \frac{x}{\alpha }}}\odot g\left(t\right){|}_0^{\xi }){\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\underset{\xi \to \infty }{lim}\underset{0}{\overset{\xi }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g\left(t\right)dt\hfill \\ =\underset{\xi \to \infty }{lim}(e^{-{\displaystyle \frac{\xi }{\beta }}}\odot g\left(\xi \right)){\ominus }_{gH}g\left(0\right){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{t}{\beta }}}\odot g\left(t\right)dt.\hfill \end{array}$$

From condition $\left(ii\right)$, we obtain

$$\underset{\xi \to \infty }{lim}D(e^{-{\displaystyle \frac{\xi }{\beta }}}\odot g\left(\xi \right),\tilde{0})=\underset{\xi \to \infty }{lim}{e}^{-{\displaystyle \frac{\xi }{\beta }}}D(g\left(\xi \right),\tilde{0})\le \underset{\xi \to \infty }{lim}L{e}^{-({\displaystyle \frac{1}{\beta }}-d)\xi }=0.$$

(10)

Hence, using Proposition 1 and Equation (10), we have

$$Y_t\left[g_{gH}^{\prime }\left(t\right)\right]=(-1)\odot g\left(0\right){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot Y_t\left[g\left(t\right)\right].$$

(11)

Similarly, from Equation (11) and Definition 9, we obtain

$$\begin{array}{cc}{Y}_t\left[g_{gH}^{\prime \prime }\left(t\right)\right]\hfill & =(-1)\odot g_{gH}^{\prime }\left(0\right){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot Y_t\left[g_{gH}^{\prime }\left(t\right)\right]\hfill \\ & =(-1)\odot g_{gH}^{\prime }\left(0\right){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\left((-1)\odot g\left(0\right){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot Y_t\left[g\left(t\right)\right]\right)\hfill \\ & =(-1)\odot g_{gH}^{\prime }\left(0\right){\ominus }_{gH}\left({\displaystyle \frac{1}{\beta }}\odot g\left(0\right){\ominus }_{gH}{\displaystyle \frac{1}{{\beta }^2}}\odot Y_t\left[g\left(t\right)\right]\right).\hfill \end{array}$$

□

Corollary 1.

Let $g(x,t)$ be a fuzzy function of two variables. Then, we have

 (i) 

$Y_t\left[g_{t,gH}^{\prime }(x,t)\right]=(-1)\odot g(x,0){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot Y_t\left[g(x,t)\right]$;

 (ii) 

$Y_t\left[g_{tt,gH}^{\prime \prime }(x,t)\right]=(-1)\odot g_{t,gH}^{\prime }(x,0){\ominus }_{gH}\left({\displaystyle \frac{1}{\beta }}\odot g(x,0){\ominus }_{gH}{\displaystyle \frac{1}{{\beta }^2}}\odot Y_t\left[g(x,t)\right]\right)$.

4. Fuzzy Double Yang Transform

In the following section, we introduce FDYT, that is, two fuzzy Yang transforms of order one. We give the fundamental properties and theorems related to the existence and fuzzy partial derivatives. Moreover, the fuzzy single convolution theorem is illustrated.

Definition 12.

The FDYT of a fuzzy function $g:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to E^1$ is defined by

$$G(\alpha ,\beta )=Y_2\left[g(x,t)\right]=Y_x\left[Y_t\left[g(x,t)\right]\right]=\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{t}{\beta }}}\odot g(x,t)dtdx,$$

(12)

provided that the improper fuzzy double integral exists. Here, α and β are complex numbers.

Definition 13.

The inverse FDYT is given by

$$Y_2^{-1}\left[G(\alpha ,\beta )\right]=g(x,t)={{\displaystyle \frac{1}{2\pi i}}}^2\underset{a-i\infty }{\overset{a+i\infty }{\int }}\underset{b-i\infty }{\overset{b+i\infty }{\int }}{e}^{{\displaystyle \frac{x}{\alpha }}}{e}^{{\displaystyle \frac{t}{\beta }}}\odot G(\alpha ,\beta )d\alpha d\beta ,$$

(13)

where the function $G(\alpha ,\beta )$ is analytic for all α and β, such that $Re\alpha >a$ and $Re\beta >b$.

Definition 14.

A fuzzy-valued function $g:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to E^1$ is said to be of exponential order $e^{cx+dt}$ as $x\to \infty$ and $t\to \infty$, if there exists a positive constant L, such that for all $x>X$ and $t>T$, we have

$$D(g(x,t),\tilde{0})\le L{e}^{cx+dt}.$$

Theorem 12.

Let $g(x,t)$ be a continuous fuzzy function in $(0,X)\times (0,T)$ and $g(x,t)$ be of exponential order $e^{cx+dt}$.

Then, the FDYT of the function $g(x,t)$ exists for all α and β such that $Re\left({\displaystyle \frac{1}{\alpha }}\right)>c$ and $Re\left({\displaystyle \frac{1}{\beta }}\right)>d$.

Proof. 

Using Definition 12 and the property of improper fuzzy double integral, we obtain

$$\begin{array}{cc}D(Y_2\left[g(x,t)\right],\tilde{0})\hfill & =D(G(\alpha ,\beta ),\tilde{0})=D(\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{t}{\beta }}}\odot g(x,t)dxdt,\tilde{0})\hfill \\ & \le \underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{t}{\beta }}}D(g(x,t),\tilde{0})dxdt\hfill \\ & \le L\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-({\displaystyle \frac{1}{\alpha }}-c)x}{e}^{-({\displaystyle \frac{1}{\beta }}-d)t}dxdt={\displaystyle \frac{L\alpha \beta }{(1-c\beta )(1-d\beta )}}.\hfill \end{array}$$

Thus, the improper fuzzy double integral converges for all $Re({\displaystyle \frac{1}{\alpha }})>c$ and $Re({\displaystyle \frac{1}{\beta }})>d$, and $Y_2\left[g(x,t)\right]$ exist. □

Double Yang transform of some important functions.

(i)

$Y_2\left[1\right]=\alpha \beta$;

(ii)

$Y_2\left[x^m{t}^n\right]=\left\{\begin{array}{cc}{\alpha }^{m+1}{\beta }^{n+1}m!n!\hfill & m,n=1,2,3,...\hfill \\ {\alpha }^{m+1}{\beta }^{n+1}\mathsf{\Gamma }(m+1)\mathsf{\Gamma }(n+1)\hfill & m>0,n>0\hfill \end{array}\right.$

(iii)

$Y_2\left[e^{cx+dt}\right]={\displaystyle \frac{\alpha \beta }{(1-c\alpha )(1-d\beta )}}$  for all $c,d\in \mathbb{R}$;

(iv)

$Y_2[sin(cx+dt)]={\displaystyle \frac{\alpha \beta }{(1+c^2{\alpha }^2)(1+d^2{\beta }^2)}}$  for all $c,d\in \mathbb{R}$;

(v)

$Y_2[cos(cx+dt)]={\displaystyle \frac{cd{\alpha }^2{\beta }^2}{(1+c^2{\alpha }^2)(1+d^2{\beta }^2)}}$  for all $c,d\in \mathbb{R}$.

Now, we present some properties for FDYT.

Remark 2.

According to Theorem 8, we can prove that if $g_1(x,t)$ and $g_2(x,t)$ are fuzzy functions, then

$$Y_2[{\gamma }_1\odot g_1(x,t)\oplus {\gamma }_2\odot g_2(x,t)]={\gamma }_1\odot Y_2\left[g(x,t)\right]\oplus {\gamma }_2\odot Y_2\left[g(x,t)\right],$$

where ${\gamma }_1,{\gamma }_2\in \mathbb{R}$, such that ${\gamma }_1,{\gamma }_2\ge 0$ or ${\gamma }_1,{\gamma }_2\le 0$.

Theorem 13

(Shifting). Let c and d be any constants, and $g(x,t)$ be a continuous fuzzy function of two variables x and t. Then,

$$Y_2[e^{-(cx+dt)}\odot g(x,t)]=G\left({\displaystyle \frac{\alpha }{1+c\alpha }},{\displaystyle \frac{\beta }{1+b\beta }}\right).$$

(14)

Proof. 

Using Definition 12, we have

$$\begin{array}{cc}{Y}_2[e^{-(cx+dt)}\odot g(x,t)]\hfill & =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-(cx+dt)}{e}^{-({\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{t}{\beta }})}\odot g(x,t)dxdt\hfill \\ & =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-x({\displaystyle \frac{1}{\alpha }}+c)-t({\displaystyle \frac{1}{\beta }}+d)}\odot g(x,t)dxdt\hfill \\ & =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-x({\displaystyle \frac{1+c\alpha }{\alpha }})-t({\displaystyle \frac{1+b\beta }{\beta }})}\odot g(x,t)dxdt\hfill \\ & =G\left({\displaystyle \frac{\alpha }{1+c\alpha }},{\displaystyle \frac{\beta }{1+b\beta }}\right).\hfill \end{array}$$

□

Theorem 14

(Heaviside Function). Let $g(x,t)$ be a continuous fuzzy function and

$$H(x-\delta ,t-\epsilon )=\left\{\begin{array}{cc}1,\hfill & x>\delta ,t>\epsilon \hfill \\ 0,\hfill & x<\delta ,t<\epsilon ,\hfill \end{array}\right.$$

where $H(x-\delta ,t-\epsilon )$ is the Heaviside function and $\delta ,\epsilon \in \mathbb{R}$. If $Y_2\left[g(x,t)\right]=G(\alpha ,\beta )$, then

$$Y_2[H(x-\delta ,t-\epsilon )\odot g(x-\delta ,t-\epsilon )]=e^{-{\displaystyle \frac{\delta }{\alpha }}-{\displaystyle \frac{\epsilon }{\beta }}}\odot G(\alpha ,\beta ).$$

Proof. 

Using Definition 12, we find

$$\begin{array}{c}{Y}_2[H(x-\delta ,t-\epsilon )\odot g(x-\delta ,t-\epsilon )]\hfill \\ =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{t}{\beta }}}H(x-\delta ,t-\epsilon )\odot g(x-\delta ,t-\epsilon )dxdt\hfill \\ =\underset{\epsilon }{\overset{\infty }{\int }}\underset{\delta }{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{t}{\beta }}}\odot g(x-\delta ,t-\epsilon )dxdt.\hfill \end{array}$$

We make a change of variable

$$\nu =x-\delta ,\mu =t-\epsilon .$$

Then

$$x=\nu +\delta ,t=\mu +\epsilon ,d\nu =dx,d\mu =dt.$$

Hence

$$\begin{array}{c}{Y}_2[H(x-\delta ,t-\epsilon )\odot g(x-\delta ,t-\epsilon )]=\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{\nu +\delta }{\alpha }}-{\displaystyle \frac{\mu +\epsilon }{\beta }}}\odot g(\nu ,\mu )d\nu d\mu \hfill \\ =e^{-{\displaystyle \frac{\delta }{\alpha }}-{\displaystyle \frac{\epsilon }{\beta }}}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{\nu }{\alpha }}-{\displaystyle \frac{\mu }{\beta }}}\odot g(\nu ,\mu )d\nu d\mu =e^{-{\displaystyle \frac{\delta }{\alpha }}-{\displaystyle \frac{\epsilon }{\beta }}}\odot G(\alpha ,\beta ).\hfill \end{array}$$

□

Definition 15

([26]). If $k\left(t\right)$ and $g(x,t)$ are fuzzy Riemann integrable functions defined for all $x,t\ge 0$, then fuzzy convolution of $k\left(t\right)$ and $g(x,t)$, with respect to t, is given by

$$(k*g)(x,t)=\underset{0}{\overset{t}{\int }}k(t-s)g(x,s)ds$$

and the symbol * denotes the fuzzy convolution with respect to t.

Theorem 15

(Convolution Theorem). Let $k:{\mathbb{R}}_{+}\to E^1$ and $g:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to E^1$ be continuous fuzzy functions. Then, the FDYT of the convolution of these two functions is as

$$Y_2\left[(k*g)(x,t)\right]=Y_t\left[k\left(t\right)\right]Y_2\left[g(x,t)\right].$$

(15)

Proof. 

Using the definition of fuzzy double Yang transform and convolution, we find

$$\begin{array}{cc}{Y}_2\left[(k*g)(x,t)\right]\hfill & =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{t}{\beta }}}\odot (k*g)(x,t)dxdt\hfill \\ & =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{t}{\beta }}}\odot \left(\underset{0}{\overset{t}{\int }}k(t-s)g(x,s)ds\right)dxdt.\hfill \end{array}$$

Let $\xi =t-s,d\xi =dt$. Then

$$\begin{array}{cc}{Y}_2\left[(k*g)(x,t)\right]\hfill & =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{\xi +s}{\beta }}}\odot \left(\underset{0}{\overset{\infty }{\int }}k\left(\xi \right)g(x,s)ds\right)dxd\xi \hfill \\ & =\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{\xi }{\beta }}}\odot k\left(\xi \right)d\xi \underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{{\displaystyle \frac{x}{\alpha }}-{\displaystyle \frac{s}{\beta }}}\odot g(x,s)dxds\hfill \\ & =Y_t\left[k\left(t\right)\right]Y_2\left[g(x,t)\right].\hfill \end{array}$$

□

Theorem 16.

Let $g(x,t)$ be a continuous fuzzy function and $Y_2\left[g(x,t)\right]=G(\alpha ,\beta )$, then

 (i) 

$Y_2\left[g_{x,gH}^{\prime }(x,t)\right]=(-1)\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot G(\alpha ,\beta )$;

 (ii) 

$Y_2\left[g_{t,gH}^{\prime }(x,t)\right]=(-1)\odot Y_x\left[g(x,0)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot G(\alpha ,\beta )$;

 (iii) 

$Y_2\left[g_{xx,gH}^{\prime \prime }(x,t)\right]=(-1)\odot Y_t\left[g_x^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot G(\alpha ,\beta )\right);$

 (iv) 

$Y_2\left[g_{tt,gH}^{\prime \prime }(x,t)\right]=(-1)\odot Y_x\left[g_t^{\prime }(x,0)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\beta }}\odot Y_x\left[g(x,0)\right]\ominus {\displaystyle \frac{1}{{\beta }^2}}\odot G(\alpha ,\beta )\right)$;

 (v) 

$Y_2\left[g_{xt,gH}^{\prime \prime }(x,y)\right]=(-1)\odot Y_x\left[g_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\beta }}\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{\alpha \beta }}\odot G(\alpha ,\beta )\right)$;

 (vi) 

$S_2\left[g_{tx,gH}^{\prime \prime }(x,y)\right]=(-1)\odot Y_t\left[g_{t,gH}^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot Y_x\left[g(x,0)\right]{\ominus }_{gH}{\displaystyle \frac{1}{\alpha \beta }}\odot G(\alpha ,\beta )\right).$

Proof. 

Using Theorem 11, we find

$$\begin{array}{cc}{Y}_2\left[g_{x,gH}^{\prime }(x,t)\right]\hfill & =Y_t\left[Y_x\left[g_{x,gH}^{\prime }(x,t)\right]\right]=Y_t\left[(-1)\odot g(0,t){\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot Y_x\left[g(x,t)\right]\right]\hfill \\ & =(-1)\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot Y_t\left[Y_x\left[g(x,t)\right]\right]\hfill \\ & =(-1)\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot G(\alpha ,\beta ).\hfill \end{array}$$

In the same manner, we can obtain the case $\left(iii\right)$.

$$\begin{array}{c}{Y}_2\left[g_{xx,gH}^{\prime \prime }(x,t)\right]=Y_t\left[Y_x\left[g_{xx,gH}^{\prime \prime }(x,t)\right]\right]\hfill \\ =Y_t\left[(-1)\odot g_{x,gH}^{\prime }(0,t){\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot g(0,t){\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot Y_x\left[g(x,t)\right]\right)\right]\hfill \\ =(-1)\odot Y_t\left[g_{x,gH}^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot Y_t\left[Y_x\left[g(x,t)\right]\right]\right)\hfill \\ =(-1)\odot Y_t\left[g_{x,gH}^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot G(\alpha ,\beta )\right).\hfill \end{array}$$

The proof of case $\left(v\right)$ is analogous to the proof of case $\left(iii\right)$.

$$\begin{array}{c}{Y}_2\left[g_{xt,gH}^{\prime \prime }(x,t)\right]=Y_x\left[Y_t\left[g_{xt,gH}^{\prime \prime }(x,t)\right]\right]\hfill \\ =Y_x\left[(-1)\odot g_{x,gH}^{\prime }(x,0){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot Y_t\left[g_{x,gH}^{\prime }(x,t)\right]\right]\hfill \\ =(-1)\odot Y_x\left[g_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot Y_t\left[Y_x\left[g_{x,gH}^{\prime }(x,t)\right]\right]\hfill \\ =(-1)\odot Y_x\left[g_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot Y_t\left[(-1)\odot g(0,t){\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot Y_x\left[g(x,y)\right]\right]\hfill \\ =(-1)\odot Y_x\left[g_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\beta }}\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{\alpha \beta }}\odot S_y\left[S_x\left[f(x,y)\right]\right]\right)\hfill \\ =(-1)\odot Y_x\left[g_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\beta }}\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{\alpha \beta }}\odot G(\alpha ,\beta )\right).\hfill \end{array}$$

□

5. Method of Fuzzy Double Yang Transform

To illustrate the use of FDYT, we solve FPVIDE with a memory kernel $k:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$. This equation is defined as

$$g_{t,gH}^{\prime }(x,t)\oplus \underset{0}{\overset{t}{\int }}k(t-s)\odot g(x,t)dxdt=\sigma \odot g_{xx,gH}^{\prime \prime }(x,t)\oplus f(x,t),x\ge 0,t\ge 0,$$

(16)

where $\sigma$ is any positive constant, $g(x,t)$ is the unknown fuzzy function, and $f(x,t)$ is a given fuzzy function. Furthermore, we assume that the fuzzy Yang transformation of the kernel $k\left(t\right)$ satisfies the condition

$${\alpha }^2+{\alpha }^2\beta K\left(\beta \right)-\sigma \beta \ne 0,$$

where $K\left(\beta \right)=Y_t\left[k\left(t\right)\right]$.

Assume initial conditions of

$$g(x,0)={\psi }_0\left(x\right)$$

(17)

and boundary conditions of

$$g(0,t)={\phi }_0\left(t\right),g_x^{\prime }(0,t)={\phi }_1\left(t\right).$$

(18)

We will provide an algorithm for solving Equations (16)–(18).

(i)

We apply the fuzzy double Yang transform to both sides of Equation (16);

(ii)

We apply Equation (15) of Theorem 15;

(iii)

We use the derivative properties of FDYT-case (i) and (iii) of Theorem 16;

(iv)

We apply FYT to the initial condition (17) and boundary conditions (18);

(v)

We use Proposition 1 and we obtain $G(\alpha ,\beta )=Y_2\left[g(x,t)\right]$;

(vi)

We apply the inverse FDYT and we find to the analytical solution of Equations (16)–(18).

First, we apply FDYT to (16) as

$$Y_2\left[g_{t,gH}^{\prime }(x,t)\right]\oplus Y_2\left[\underset{0}{\overset{t}{\int }}k(t-s)\odot g(x,t)dxdt\right]=Y_2\left[\sigma \odot g_{xx,gH}^{\prime \prime }(x,t)\right]\oplus Y_2\left[f(x,t)\right].$$

Using the convolution theorem, we have

$$Y_2\left[g_{t,gH}^{\prime }(x,t)\right]\oplus Y_t\left[k\left(t\right)\right]\odot Y_2\left[g(x,t)\right]=\sigma \odot Y_2\left[g_{xx,gH}^{\prime \prime }(x,t)\right]\oplus Y_2\left[f(x,t)\right].$$

The derivative properties of FDYT (Theorem 16) and the above equation yield

$$\begin{array}{c}(-1)\odot Y_x\left[g(x,0)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot G(\alpha ,\beta )\oplus K\left(\beta \right)\odot G(\alpha ,\beta )\hfill \\ =\sigma \odot \left[(-1)\odot Y_t\left[g_x^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot Y_t\left[g(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot G(\alpha ,\beta )\right)\right]\oplus F(\alpha ,\beta ).\hfill \end{array}$$

(19)

where

$$K\left(\beta \right)=Y_t\left[k\left(t\right)\right],G(\alpha ,\beta )=Y_2\left[g(x,t)\right],F(\alpha ,\beta )=Y_2\left[f(x,t)\right].$$

Next, apply the FYT to the initial and boundary conditions

$${\Psi }_0\left(\alpha \right)=Y_x\left[{\psi }_0\left(x\right)\right],{\Phi }_0\left(\beta \right)=Y_t\left[{\phi }_0\left(t\right)\right],{\Phi }_1\left(\beta \right)=Y_t\left[{\phi }_1\left(t\right)\right].$$

and then put in (19) as

$$\begin{array}{c}(-1)\odot {\Psi }_0\left(\alpha \right){\ominus }_{gH}(-1){\displaystyle \frac{1}{\beta }}\odot G(\alpha ,\beta )\oplus K\left(\beta \right)\odot G(\alpha ,\beta )\hfill \\ =(-\sigma )\odot {\Phi }_1\left(\beta \right){\ominus }_{gH}\left({\displaystyle \frac{\sigma }{\alpha }}\odot {\Phi }_0\left(\beta \right){\ominus }_{gH}{\displaystyle \frac{\sigma }{{\alpha }^2}}\odot G(\alpha ,\beta )\right)\oplus F(\alpha ,\beta ).\hfill \end{array}$$

Using Proposition 1, we have

$$\begin{array}{c}\left({\displaystyle \frac{1}{\beta }}+K\left(\beta \right)-{\displaystyle \frac{\sigma }{{\alpha }^2}}\right)\odot G(\alpha ,\beta )\hfill \\ ={\Psi }_0\left(\alpha \right)\oplus (-\sigma )\odot {\Phi }_1\left(\beta \right)\oplus {\displaystyle \frac{(-\sigma )}{\alpha }}\odot {\Phi }_0\left(\beta \right)\oplus F(\alpha ,\beta ).\hfill \end{array}$$

Hence

$$G(\alpha ,\beta )=B\odot \left[{\Psi }_0\left(\alpha \right)\oplus (-\sigma )\odot {\Phi }_1\left(\beta \right)\oplus {\displaystyle \frac{(-\sigma )}{\alpha }}\odot {\Phi }_0\left(\beta \right)\oplus F(\alpha ,\beta )\right],$$

(20)

where

$$B={\displaystyle \frac{{\alpha }^2\beta }{{\alpha }^2+{\alpha }^2\beta K\left(\beta \right)-\sigma \beta }}.$$

(21)

Finally, take the inverse FDYT of (20) as

$$g(x,t)=Y_2^{-1}\left[B\odot \left({\Psi }_0\left(\alpha \right)\oplus (-\sigma )\odot {\Phi }_1\left(\beta \right)\oplus {\displaystyle \frac{(-\sigma )}{\alpha }}\odot {\Phi }_0\left(\beta \right)\oplus F(\alpha ,\beta )\right)\right].$$

(22)

6. Examples

Example 1.

Consider the following FVIDE

$$g_{t,gH}^{\prime }(x,t)\oplus \underset{0}{\overset{t}{\int }}k(t-s)\odot g(x,t)dxdt=g_{xx,gH}^{\prime \prime }(x,t)\oplus f(x,t),x\ge 0,t\ge 0$$

(23)

with initial conditions

$$g(x,0,r)=x\odot (1,2,3)$$

(24)

and boundary conditions

$$g(0,t,r)=0\odot (1,2,3),g_x^{\prime }(0,t,r)=e^{-t}\odot (1,2,3).$$

(25)

In this case

$$\sigma =1,k\left(t\right)=2e^t,f(x,t)=\left(x{e}^t-2x{e}^{-t}\right)\odot (1,2,3).$$

Then, we have

$${\psi }_0\left(x\right)=x\odot (1,2,3),{\phi }_0\left(t\right)=0\odot (1,2,3),{\phi }_1\left(t\right)=e^{-t}\odot (1,2,3).$$

Hence

$$Y_t\left[k\left(t\right)\right]=K\left(\beta \right)={\displaystyle \frac{2\beta }{1-\beta }}\odot (1,2,3),Y_x\left[{\psi }_0\left(x\right)\right]={\Psi }_0\left(\alpha \right)={\alpha }^2\odot (1,2,3),$$

$$Y_t\left[{\phi }_0\left(t\right)\right]={\Phi }_0\left(\beta \right)=0\odot (1,2,3),Y_t\left[{\phi }_1\left(t\right)\right]={\Phi }_1\left(\beta \right)={\displaystyle \frac{\beta }{1+\beta }}\odot (1,2,3),$$

$$Y_2[f(x,t)=F(\alpha ,\beta )={\alpha }^2\beta \left({\displaystyle \frac{1}{1-\beta }}+{\displaystyle \frac{2}{1+\beta }}\right)\odot (1,2,3).$$

By substituting the values of the fuzzy functions $K\left(\beta \right)$, ${\Psi }_0\left(\alpha \right)$, ${\Phi }_0\left(\beta \right)$ and ${\Phi }_1\left(\beta \right)$ in Equations (20) and (21), we obtain

$$\begin{array}{c}G(\alpha ,\beta )=B\odot \left[{\alpha }^2\odot (1,2,3)\oplus {\displaystyle \frac{(-\beta )}{1+\beta }}\odot (1,2,3)\oplus {\alpha }^2\beta \left({\displaystyle \frac{1}{1-\beta }}+{\displaystyle \frac{2}{1+\beta }}\right)\odot (1,2,3)\right],\hfill \end{array}$$

where

$$B={\displaystyle \frac{{\alpha }^2\beta (1-\beta )}{{\alpha }^2(2{\beta }^2-\beta +1)-\beta (1-\beta )}}.$$

Hence

$$G(\alpha ,\beta )=B\left[{\alpha }^2-{\displaystyle \frac{\beta }{1+\beta }}+{\displaystyle \frac{{\alpha }^2\beta }{1-\beta }}-{\displaystyle \frac{2{\alpha }^2\beta }{1+\beta }}\right]\odot (1,2,3)={\displaystyle \frac{{\alpha }^2\beta }{1+\beta }}\odot (1,2,3).$$

Taking the inverse of FDYT, we find the solution to the Equations (23)–(25) is

$$g(x,t)=Y_2^{-1}\left[G(\alpha ,\beta )\right]=Y_2^{-1}\left[{\displaystyle \frac{{\alpha }^2\beta }{1+\beta }}\odot (1,2,3)\right]=x{e}^{-t}\odot (1,2,3).$$

7. Conclusions

In this research paper, we introduce a new fuzzy integral transformation called FDYT, which is defined with the help of the fuzzy unitary Yang transform. We find conditions for its existence and establish some of its basic properties. We prove theorems about partial derivatives and fuzzy unit convolution. Using these new results, we successfully obtained the exact solution to an FVIDE with a symmetric memory kernel. We construct a numerical example to verify the application of the new method.

Fuzzy integral transforms cannot solve nonlinear problems directly unless they are combined with iteration methods. As a result, we propose that this method is further expanded in future work, so that it can be applied to the solution of various nonlinear fuzzy partial differential and integro-differential equations related to physical and engineering problems.