A New Double Fuzzy Integral Transform for Solving an Advection–Diffusion Equation

Abstract

This article presents a new approach to solving fuzzy advection–diffusion equations using double fuzzy transforms, called the double fuzzy Yang–General transform. This unique double fuzzy transformation is a combination of single fuzzy Yang and General transforms. Some of the basic properties of this new transform include existence and linearity and how they relate to partial derivatives. A solution framework for the linear fuzzy advection–diffusion equation is developed to show the application of the double fuzzy Yang–General transform. To illustrate the proposed method for solving these equations, we have included a solution to a numerical problem.

1. Introduction

Many researchers have studied the use of fuzzy partial differential and integral equations. These equations are an excellent tool for modeling vagueness and misinterpretation of knowledge-based systems [1], control systems [2], image processing [3], industrial automation, power engineering, robotics [4], artificial intelligence [5], and consumer electronics. Buckley and Feuring developed the first definition of fuzzy partial differential equations [6]. The difference method for solving fuzzy partial differential equations is proposed in [7]. Nemati and Matinfar [8] constructed an implicit finite difference approach to solving complex fuzzy parabolic differential equations. An explicit numerical solution to the fuzzy hyperbolic and parabolic equation has been introduced in [9]. Recently, Arqub et al. [10] applied the reproducing kernel algorithm to find a solution to fuzzy boundary value problems. In [11], the fuzzy Fredholm–Volterra integro-differential equations are solved by using the adaptation of the reproducing kernel algorithm. Various numerical methods have been used to address fuzzy problems [12,13,14,15].

Integral transforms have been used in solving many types of equations, increasing their importance and need for research in their application. The fuzzy version of classical General transform is introduced by Rashid et al. [16]. Ullah et al. [17] proposed the fuzzy Yang transformation as a method to find a solution to second-order fuzzy differential equations of integer and fractional order.

Recently, some researchers have introduced double fuzzy transforms for solving fuzzy partial differential equations. Stabestari and Ezzati constructed a double fuzzy Laplace transform that is applicable to solve the fuzzy wave equation [18]. The double fuzzy Elzaki transform is used to find the solution of the fuzzy Poission’s equation and the fuzzy Telegraph Equation [19]. A fuzzy solution to the Telegraph equation is obtained in [20] by applying the fuzzy Sawi transform. In work [21], by use of a double fuzzy Aboodh transform, we derive the solution of a fuzzy partial differential equation of first order.

In our study, the aim was to introduce the use of a new double fuzzy integral transform, called the double fuzzy Yang–General transform (DFY-GT). To achieve this, we prove some of the basic properties of DFY-GT and compute values of DFY-GT for some functions. New theorems related to partial gH-derivatives are established. The originality of this paper comes from using a combination between the single fuzzy transforms of Yang and General, where the new double fuzzy Yang–General transformation has the advantages of two transforms.

The advection–diffusion equation is the model that can be used to simulate natural processes. The advection is due to the movement of materials from one region to another and diffusion is due to the movement of materials from higher concentration to low concentration. Different papers have appeared to solve and use this equation [22,23,24,25].

In this research, we are implemented DFY-GT to solve the following linear fuzzy advection–diffusion equation

$$u_{t,gH}^{\prime }(x,t)={\kappa }_1\odot u_{xx,gH}^{″}(x,t){\ominus }_{gH}{\kappa }_2\odot u_{x,gH}^{\prime }(x,t),x\ge 0t\ge 0$$

where ${\kappa }_1>0$ is a constant diffusion coefficient and ${\kappa }_2$ is constant advection velocity. A simple formula for the solution of the above equation is obtained and applied to solve a numerical example in order to display the efficiency of this new approach.

The rest of this paper is organized as follows: In Section 2, we present the basic concepts that will be used in the main part of the paper. In Section 3, we discuss the fundamental facts and properties of single fuzzy Yang and General transforms. In Section 4, we introduce a new integral transformation, the DFY-GT, which combines the fuzzy Yang transform and the General fuzzy transform, and we define some properties of this transform. In Section 5, we apply a DFY-GT to the fuzzy linear advection–diffusion equation, and we find a formula for the exact solution. Section 6 provides a numerical experiment that demonstrates the effectiveness of the proposed method. Finally, the concluding remarks and comments are contained in Section 7.

2. Basic Concepts

In this section, we introduce some definitions and theorems that will be used in most of the paper.

We will denote $E^1$ the set of fuzzy numbers. A fuzzy number is a mapping $\mu :\mathbb{R}\to [0,1]$ and satisfies the conditions

(i)

$\mu$ is normal, i.e., there exists $\xi \in \mathbb{R}$ for which $\mu \left(\xi \right)=1$;

(ii)

$\mu$ is upper semi continuous in the higher half;

(iii)

$\mu (r\xi +(1-r\left)\eta \right)\ge min\left\{\mu \right(\xi ),\mu (\eta \left)\right\}$ for all $r\in [0,1]$ and $\xi ,\eta \in \mathbb{R}$;

(iv)

$cl\left(supp\mu \right)=cl\left(\right\{\xi \in \mathbb{R}:\mu \left(\xi \right)>0\left\}\right)$ is compact.

Definition 1 

([26]). The r-level set of fuzzy number $w\in E^1$ is defined by

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

The core of the fuzzy number w is the set of elements of $\mathbb{R}$ that have membership grade 1, i.e.,

$${\left[w\right]}^1=\{\xi \in \mathbb{R}:w\left(\xi \right)=1\}.$$

A fuzzy set w is a fuzzy number if and only if the r-levels are nonempty compact intervals of the form ${\left[w\right]}^r=[\underline{w}\left(r\right),\overline{w}\left(r\right)]$.

Definition 2 

([26]). An ordered pair $w\left(r\right)=(\underline{w}\left(r\right),\overline{w}\left(r\right))$ is called a parametric form of fuzzy number w, if the functions $\underline{w}(.),\overline{w}(.):[0,1]\to \mathbb{R}$ satisfy the conditions:

(i) 

$\underline{w}\left(r\right)$, $\overline{w}\left(r\right)$ are bounded monotonic left continuous for all $r\in [0,1]$ and right continuous for $r=0$;

(ii) 

$\underline{w}\left(r\right)$, $\overline{w}\left(r\right)$ are non-decreasing and non-increasing, respectively;

(iii) 

$\underline{w}\left(r\right)\le \overline{w}\left(r\right)$ for all $r\in [0,1]$.

Definition 3 

([26]). An ordered foursome $w=(a_1,a_2,a_3,a_4)$ is called a trapezoidal fuzzy number w, if $a_1\le a_2\le a_3\le a_4$ and $a_1,a_2,a_3{a}_4\in \mathbb{R}$. The r-levels of this fuzzy number are $[a_1+(a_2-a_1)r,a_4-(a_4-a_3)r]$. A triangular fuzzy number is obtained if $a_2=a_3$.

Let $v,w\in E^1$, where $v\left(r\right)=(\underline{v}\left(r\right),\overline{v}\left(r\right))$ and $w\left(r\right)=(\underline{w}\left(r\right),\overline{w}\left(r\right))$ and $\alpha \in \mathbb{R}$. The addition $v\oplus w$ and the scalar multiplication $\alpha \odot v$ are defined

$$(v\oplus w)\left(r\right)=v\left(r\right)+w\left(r\right)=(\underline{v}\left(r\right)+\underline{w}\left(r\right),\overline{v}\left(r\right)+\overline{w}\left(r\right)),$$

$$(\alpha \odot v)\left(r\right)=\alpha .v\left(r\right)=\left\{\begin{array}{c}(\alpha \underline{v}\left(r\right),\alpha \overline{v}\left(r\right)),\alpha \ge 0\hfill \\ (\alpha \overline{v}\left(r\right),\alpha \underline{v}\left(r\right)),\alpha <0.\hfill \end{array}\right.$$

The subtraction of fuzzy numbers v and w is defined as addition, that is,

$$(v-w)\left(r\right)=(v\oplus ((-1)\odot w))\left(r\right)=(\underline{v}\left(r\right)-\overline{w}\left(r\right),\overline{v}\left(r\right)-\underline{w}\left(r\right)).$$

The Hukuhara difference (H-difference) of two fuzzy numbers v and w has been introduced as a fuzzy number $u=v{\ominus }_{H}w$ if and only if $u\oplus w=v$. The H-difference is unique, but it does not always exist. If $v{\ominus }_{H}w$ exists, then its parametric form is

$$\left(v{\ominus }_{H}w\right)\left(r\right)=(\underline{v}\left(r\right)-\underline{w}\left(r\right),\overline{v}\left(r\right)-\overline{w}\left(r\right)).$$

Definition 4 

([26]). The generalized Hukuhara difference (gH-difference) of two fuzzy numbers v and w is the fuzzy number u if it exists and

$$v{\ominus }_{gH}w=uifandonlyif$$

$$\left(i\right)v=w\oplus uor\left(ii\right)w=v\oplus (-1)\odot u.$$

Hence, its r-levels are

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

Sufficient conditions for the existence of the $gH$-difference are obtained in [26].

For an interval $[c,d]$, we define the norm

$$\parallel [c,d]\parallel =max\left\{\right|c|,|d\left|\right\}.$$

Then, the Hausdorff distance between two fuzzy numbers v and w is given as

$$D(v,w)=\underset{r\in [0,1]}{sup}{\{\parallel \left[v\right]}^r{\ominus }_{gH}{\left[w\right]}^r\parallel \}.$$

The metric D is well defined, since the $gH$-difference of the intervals ${\left[v\right]}^r{\ominus }_{gH}{\left[w\right]}^r$ always exists. Hence $(E^1,D)$ is a complete metric space.

The following properties for the $gH$-difference are given in [27].

Proposition 1. 

Suppose that v and w are fuzzy numbers, then

(i) 

if $v{\ominus }_{H}w$ exists, then $v{\ominus }_{gH}w=v{\ominus }_{H}w$ or $v{\ominus }_{gH}w=-\left(w{\ominus }_{H}v\right)$;

(ii) 

if $v{\ominus }_{gH}w$ exists, then it is unique;

(iii) 

if $v{\ominus }_{gH}w$ exists in the sense $\left(ii\right)$, then $w{\ominus }_{gH}v$ exists in the sense $\left(i\right)$ and vice versa;

(iv) 

$(v\oplus w){\ominus }_{gH}w=v$;

(v) 

$\tilde{0}{\ominus }_{gH}\left(v{\ominus }_{gH}w\right)=w{\ominus }_{gH}v$;

(vi) 

if $v{\ominus }_{gH}w=w{\ominus }_{gH}v=u$ if and only if $u=-u$.

2.1. Fuzzy Function of One-Variable

In the following part of the article, we introduce some definitions and some properties of the fuzzy-valued function $\phi :\mathbb{R}\to E^1$. The endpoint functions $\underline{\phi }(.,r),\overline{\phi }(.,r):\mathbb{R}\to \mathbb{R}$ are called the upper and lower functions of the fuzzy-valued function $\phi$ for each $r\in [0,1]$. Then, the parametric form of the function $\phi$ is

$$\phi (t,r)=\left(\underline{\phi }(t,r),\overline{\phi }(t,r)\right).$$

A fuzzy-valued function is a function that produces an image of a crisp domain in a fuzzy set.

Example 1. 

Consider two crisp sets, $A=\{1,2,3\}$ and $B=\{3,4,6,8,10,15,17\}$.

A fuzzy-valued function g maps the elements in A to the power set $\tilde{P}\left(B\right)$ in the following manner.

$$g\left(1\right)=B_1,g\left(2\right)=B_2,g\left(3\right)=B_3,$$

where $\tilde{P}\left(B\right)=\{B_1,B_2,B_3\}$, $B_1=\{(2,0.3),(5,0.7),(7,0.3)\}$, $B_2=\{(3,0.3),(6,0.7),(8,0.3)\}$, $B_3=\{(7,0.3),(9,0.7),(15,0.3)\}$.

The function g maps element $1\in A$ to element $2\in B_1$ with degree $0.3$, to element $5\in B_1$ with $0.7$, and to element $7\in B_1$ with $0.3$. Now, we apply the r-cut operation to the fuzzy-valued function.

$$g\left(1\right)=\left\{\begin{array}{cc}\{2,9\}\hfill & forr=0.3\hfill \\ \left\{5\right\}\hfill & forr=0.7\hfill \end{array}\right.$$

$$g\left(2\right)=\left\{\begin{array}{cc}\{3,8\}\hfill & forr=0.3\hfill \\ \left\{6\right\}\hfill & forr=0.7\hfill \end{array}\right.$$

$$g\left(3\right)=\left\{\begin{array}{cc}\{7,15\}\hfill & forr=0.3\hfill \\ \left\{9\right\}\hfill & forr=0.7\hfill \end{array}\right.$$

Lemma 1 

([18]). Let $\phi :\mathbb{R}\to E^1$ be a fuzzy-valued function and real numbers ${\gamma }_1$ and ${\gamma }_2$ such that ${\gamma }_1,{\gamma }_2\ge 0$ or ${\gamma }_1,{\gamma }_2\le 0$, then

$${\gamma }_1\odot \phi \left(t\right){\ominus }_{gH}{\gamma }_2\odot \phi \left(t\right)=({\gamma }_1-{\gamma }_2)\odot \phi \left(t\right).$$

Let $\phi :[0,\infty )\to E^1$ be a fuzzy-valued function with parametric form $\phi \left(t\right)=\left(\underline{\phi }(t,r),\overline{\phi }(t,r)\right)$. Suppose that, for all fixed $r\in [0,1]$, the crisp functions $\underline{\phi }(t,r)$ and $\overline{\phi }(t,r)$ are integrable on $[0,\xi ]$, for every $0\le \xi <\infty$, and that there exist two positive constants $\underline{L}\left(\xi \right)$ and $\overline{L}\left(\xi \right)$ such that

$$\underset{0}{\overset{\xi }{\int }}|\underline{\phi }(t,r)|dt\le \underline{L}\left(\xi \right),\underset{0}{\overset{\xi }{\int }}\left|\overline{\phi }(t,r)\right|dt\le \overline{L}\left(\xi \right)$$

for every $0\le \xi <\infty$. Then $\phi \left(t\right)$ is fuzzy Riemann integrable on $[0,\infty )$, its improper fuzzy integral

$$\underset{0}{\overset{\infty }{\int }}\phi \left(t\right)dt\in E^1$$

and

$$\underset{0}{\overset{\infty }{\int }}\phi \left(t\right)dt=\left(\underset{0}{\overset{\infty }{\int }}\underline{\phi }(t,r)dt,\underset{0}{\overset{\infty }{\int }}\overline{\phi }(t,r)dt\right).$$

Lemma 2 

([18]). Let ${\gamma }_1,{\gamma }_2\ge 0$ or ${\gamma }_1,{\gamma }_2\le 0$. If fuzzy-valued functions ${\phi }_1,{\phi }_2:[0,\infty )\to E^1$ are improper fuzzy Rieman integrable on $[0,\infty )$, then

(i) 

$\underset{0}{\overset{\infty }{\int }}\left[{\gamma }_1\odot {\phi }_1\left(t\right){\ominus }_{gH}{\gamma }_2\odot {\phi }_2\left(t\right)\right]dt={\gamma }_1\odot \underset{0}{\overset{\infty }{\int }}{\phi }_1\left(t\right)dt{\ominus }_{gH}{\gamma }_2\odot \underset{0}{\overset{\infty }{\int }}{\phi }_2\left(t\right)dt$;

(ii) 

$\underset{0}{\overset{\infty }{\int }}\left[{\gamma }_1\odot {\phi }_1\left(t\right)\oplus {\gamma }_2\odot {\phi }_2\left(t\right)\right]dt={\gamma }_1\odot \underset{0}{\overset{\infty }{\int }}{\phi }_1\left(t\right)dt\oplus {\gamma }_2\odot \underset{0}{\overset{\infty }{\int }}{\phi }_2\left(t\right)dt$.

Based on the gH-difference, we obtain the following definition for the gH-derivative.

Definition 5 

([26]). 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 fuzzy-valued function $\phi :(c,d)\to E^1$ at the point $t_0$ is fuzzy number ${\phi }_{gH}^{\prime }\left(t_0\right)$ defined as

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

(1)

if limit exists.

Definition 6 

([26]). The fuzzy-valued function $\phi ;(c,d)\to E^1$ is called

(i) 

gH-differentiable

at the point $t_0\in (c,d)$ if

$${\phi }_{i,gH}^{\prime }(t_0,r)=\left({\underline{\phi }}^{\prime }(t_0,r),{\overline{\phi }}^{\prime }(t_0,r)\right)forall0\le r\le 1,$$

(2)

(ii) 

gH-differentiable at the point $t_0\in (c,d)$ if

$${\phi }_{ii,gH}^{\prime }(t_0,r)=\left({\overline{\phi }}^{\prime }(t_0,r),{\underline{\phi }}^{\prime }(t_0,r)\right)forall0\le r\le 1$$

(3)

provided that the functions $\underline{\phi }(.,r)$ and $\overline{\phi }(.,r)$ are differentiable at the point $t_0\in (c,d)$.

Now, we will give some basic properties for the gH-differentiable function.

Theorem 1 

([28]). Let the fuzzy-valued functions $\phi ,\psi :(c,d)\to E^1$ be gH-differentiable, then $\phi \left(t\right){\ominus }_{gH}\psi \left(t\right)$ is gH-differentiable, and

$${\left(\phi \left(t\right){\ominus }_{gH}\psi \left(t\right)\right)}_{gH}^{\prime }={\phi }_{gH}^{\prime }\left(t\right){\ominus }_{gH}{\psi }_{gH}^{\prime }\left(t\right).$$

Theorem 2 

([28]). Let the fuzzy-valued function $\phi :(c,d)\to E^1$ be $gH$-differentiable and the functions $\psi :(c,d)\to [0,\infty )$ be differentiable. Then

$${\left(\psi \left(t\right)\odot \phi \left(t\right)\right)}_{gH}^{\prime }=\psi \left(t\right)\odot {\phi }_{gH}^{\prime }\left(t\right)\oplus {\psi }^{\prime }\left(t\right)\odot \phi \left(t\right).$$

Theorem 3 

([26]). Let $\phi :[c,d]\to E^1$ be $gH$-differentiable in the interval $(c,d)$. Then

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

Theorem 4 

([29]). Let $\phi :[c,d]\to E^1$ be $gH$-differentiable in the interval $(c,d)$ and $\psi :[c,d]\to \mathbb{R}$ be differentiable functions in the interval $(c,d)$. Then

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

2.2. Fuzzy Function of Two-Variable

Let $u(x,t,r)=\left(\underline{u}(x,t,r),\overline{u}(x,t,r)\right)$ be the parametric form of the fuzzy function $u:D\subset \mathbb{R}\times \mathbb{R}\to E^1$ for all $0\le r\le 1$.

Definition 7 

([29]). We say that the fuzzy-valued function $u:D\to E^1$ is continuous at the point $(x_0,t_0)\in D$ if for each $\epsilon >0$ there is $\delta >0$ such that $D(u(x,t),u(x_0,t_0))<\epsilon$ whenever $|x-x_0|+|t-t_0|<\delta .$

Definition 8 

([28]). Let the point $(x_0,t_0)\in D$ and real constants h, k be such that $(x_0+h,t_0)\in D$, $(x_0,t_0+k)\in D$. Then, the first generalized Hukuhara partial derivative ($gH$-p-derivative) of fuzzy function $u:D\to E^1$ at $(x_0,t_0)$ with respect to x and t are fuzzy numbers $u_{x,gH}^{\prime }(x_0,t_0)$ and $u_{t,gH}^{\prime }(x_0,t_0)$ defined by

$$u_{x,gH}^{\prime }(x_0,t_0)=\underset{h\to 0}{lim}{\displaystyle \frac{u(x_0+h,t_0){\ominus }_{gH}u(x_0,t_0)}{h}},$$

(4)

$$u_{t,gH}^{\prime }(x_0,t_0)=\underset{k\to 0}{lim}{\displaystyle \frac{u(x_0,t_0+k){\ominus }_{gH}u(x_0,t_0)}{k}}.$$

(5)

provided that limits exist in Equations $\left(4\right)$ and $\left(5\right)$.

Definition 9 

([28]). Let $u:D\to E^1$ be a fuzzy-valued function, $(x_0,t_0)\in D$ and $\underline{u}(x,t,r)$, $\overline{u}(x,t,r)$ both be partial differentiable

at the point $(x_0,t_0)$ with respect to t. Then, we say that $u(x,t)$ is

(i) 

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

$$u_{t,\left(i\right)-gH}^{\prime }(x_0,t_0,r)=\left({\underline{u}}_t^{\prime }(x_0,t_0,r),{\overline{u}}_t^{\prime }(x_0,t_0,r)\right),r\in [0,1],$$

(6)

(ii) 

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

$$u_{t,\left(ii\right)-gH}^{\prime }(x_0,t_0,r)=\left({\overline{u}}_t^{\prime }(x_0,t_0,r),{\underline{u}}_t^{\prime }(x_0,t_0,r)\right),r\in [0,1].$$

(7)

Theorem 5 

([28]). Let the fuzzy-valued function $u:D\to E^1$ be $gH$-p-differentiable with respect to t and $c>0$. Then, ${\left(c\odot u(x,t)\right)}_{t,gH}^{\prime }$ exists and

$${\left(c\odot u(x,t)\right)}_{t,gH}^{\prime }=c\odot u_{t,gH}^{\prime }(x,t).$$

Theorem 6 

([29]). Let $u:[0,\infty )\times [0,\infty )\to E^1$ be a fuzzy-valued function. Suppose that $\underset{0}{\overset{\infty }{\int }}u(x,\eta )d\eta$ is convergent for each $x\in [0,\infty )$ and $\underset{0}{\overset{\infty }{\int }}u(\xi ,t)d\xi$ is convergent for each $t\in [0,\infty )$. Then

$$\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}u(\xi ,\eta )d\eta d\xi =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}u(\xi ,\eta )d\xi d\eta .$$

3. Basic Definitions and Theorems for Yang and General Fuzzy Transforms

In this section, we give the definitions and some fundamental properties of the fuzzy Yang and fuzzy General transforms.

3.1. Fuzzy Yang Transform

The definition of fuzzy Yang transform (FYT) is introduced in [17].

Definition 10. 

Let $\phi :[0,\infty )\to E^1$ and the function $e^{-{\displaystyle \frac{x}{\alpha }}}\odot \phi \left(x\right)$ be fuzzy improper integrable on $[0,\infty )$ for $\alpha >0$. Then, the FYT of $\phi \left(x\right)$ is defined as

$$\mathrm{\Phi }\left(\alpha \right)=Y_x\left[\phi \left(x\right)\right]=\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot \phi \left(x\right)dx,$$

(8)

where x and α are transform variables.

Definition 11. 

The fuzzy inverse Yang transform is given by

$$Y_{\alpha }^{-1}\left[\mathrm{\Phi }\left(\alpha \right)\right]={\displaystyle \frac{1}{2\pi i}}\underset{a-i\infty }{\overset{a+i\infty }{\int }}{e}^{{\displaystyle \frac{x}{\alpha }}}\odot \mathrm{\Phi }\left(\alpha \right)d\alpha ,$$

(9)

where the function $\mathrm{\Phi }\left(\alpha \right)$ must be analytic for all α such that $Re\alpha >a$.

Definition 12. 

We say that the fuzzy-valued function $\phi :[0,\infty )\to E^1$ is of exponential order $c>0$ if there exists a constant $K>0$, such that for all $x>X$

$$D(\phi \left(x\right),\tilde{0})\le K{e}^{cx}.$$

Theorem 7. 

Let the fuzzy function $\phi :[0,\infty )\to E^1$ be continuous in every interval $0<x<X<\infty$ of exponential order $c>0$. Then, the FYT of $\phi \left(x\right)$ exists for all α provided that $Re\left({\displaystyle \frac{1}{\alpha }}\right)>c$.

Proof. 

Using Definition 10 and the properties of fuzzy improper integral, we obtain

$$\begin{array}{cc}D(\mathrm{\Phi }\left(\alpha \right),\tilde{0})\hfill & =D(\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot \phi \left(x\right)dx,\tilde{0})\le \underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}D(\phi \left(x\right),\tilde{0})dx\le \hfill \\ & \le K\underset{0}{\overset{\infty }{\int }}{e}^{-({\displaystyle \frac{1}{\alpha }}-c)x}dx={\displaystyle \frac{K\alpha }{1-c\alpha }},forRe\left({\displaystyle \frac{1}{\alpha }}\right)>c,\hfill \end{array}$$

where $Y_x\left[\varphi \left(x\right)\right]=\mathrm{\Phi }\left(\alpha \right)$. □

In [30], the classical Yang transform is applied on some special functions.

(i)

$Y_x\left[1\right]=\alpha$;

(ii)

$Y_x\left[x^m\right]=(m!){\alpha }^{m+1},$ where m is positive integer;

(iii)

$Y_x\left[e^{cx}\right]={\displaystyle \frac{\alpha }{1-c\alpha }},$ where $c\in \mathbb{R}$;

(iv)

$Y_x[sincx]={\displaystyle \frac{\alpha }{1+c^2{\alpha }^2}},$ where $c\in \mathbb{R}$;

(v)

$Y_x[coscx]={\displaystyle \frac{c{\alpha }^2}{1+c^2{\alpha }^2}},$ where $c\in \mathbb{R}$.

Using Definition 10, we obtain the following useful properties of the FYT.

Theorem 8. 

Let for fuzzy-valued functions ${\phi }_1,{\phi }_2:[0,\infty )\to E^1$ exist a FYT. Then, the FYT of functions ${\gamma }_1\odot {\phi }_1\left(x\right)\oplus {\gamma }_2\odot {\phi }_2\left(x\right)$ and ${\gamma }_1\odot {\phi }_1\left(x\right){\ominus }_{gH}{\gamma }_2\odot {\phi }_2\left(x\right)$ exist and

(i) 

${\gamma }_1\odot Y_x\left[{\phi }_1\left(x\right)\right]\oplus {\gamma }_2\odot Y_x\left[{\phi }_2\left(x\right)\right]=Y_x[{\gamma }_1\odot {\phi }_1\left(x\right)\oplus {\gamma }_2\odot {\phi }_2\left(x\right)]$;

(ii) 

${\gamma }_1\odot Y_x\left[{\phi }_1\left(x\right)\right]{\ominus }_{gH}{\gamma }_2\odot Y_x\left[{\phi }_2\left(x\right)\right]=Y_x[{\gamma }_1\odot {\phi }_1\left(x\right){\ominus }_{gH}{\gamma }_2\odot {\phi }_2\left(x\right)],$

where the real numbers ${\gamma }_1$ and ${\gamma }_2$ such that ${\gamma }_1,{\gamma }_2\ge 0$ or ${\gamma }_1,{\gamma }_2\le 0$.

Proof. 

We prove case (ii). Using Lemma 2, we obtain

$$\begin{array}{cc}{\gamma }_1\odot Y_x\left[{\phi }_1\left(x\right)\right]{\ominus }_{gH}{\gamma }_2\odot Y_x\left[{\phi }_2\left(x\right)\right]\hfill & =\underset{0}{\overset{\infty }{\int }}{\gamma }_1{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot {\phi }_1\left(x\right)dx{\ominus }_{gH}\underset{0}{\overset{\infty }{\int }}{\gamma }_2{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot {\phi }_2\left(x\right)dx=\hfill \\ & =\underset{0}{\overset{\infty }{\int }}({\gamma }_1{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot {\phi }_1\left(x\right){\ominus }_{gH}{\gamma }_2{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot {\phi }_2\left(x\right))dx=\hfill \\ & =\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}({\gamma }_1\odot {\phi }_1\left(x\right){\ominus }_{gH}{\gamma }_2\odot {\phi }_2\left(x\right))dx=\hfill \\ & =Y_x[{\gamma }_1\odot {\phi }_1\left(x\right){\ominus }_{gH}{\gamma }_2\odot {\phi }_2\left(x\right)].\hfill \end{array}$$

Analogously, we prove case (i). □

Theorem 9. 

Let the fuzzy-valued function $\phi :[0,\infty )\to E^1$ be continuous of exponential order $c>0$ and ${\phi }_{gH}^{\prime }\left(x\right)$ be continuous in every interval $0\le x\le X$. Then

(i) 

$Y_x\left[{\phi }_{gH}^{\prime }\left(x\right)\right]=(-1)\odot \phi \left(0\right){\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}\odot \mathrm{\Phi }\left(\alpha \right)$;

(ii) 

$Y_x\left[{\phi }_{gH}^{\prime \prime }\left(x\right)\right]=(-1)\odot {\phi }_{gH}^{\prime }\left(0\right){\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot \phi \left(0\right){\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot \mathrm{\Phi }\left(\alpha \right)\right)$,

where $Y_x\left[\phi \left(x\right)\right]=\mathrm{\Phi }\left(\alpha \right)$ and $Re\left({\displaystyle \frac{1}{\alpha }}\right)>c$.

Proof. 

Using the definition of the improper fuzzy Riemann integral and Theorem 4, we obtain

$$\begin{array}{cc}{Y}_x\left[{\phi }_{gH}^{\prime }\left(x\right)\right]\hfill & =\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot {\phi }_{gH}^{\prime }\left(x\right)dx=\underset{\eta \to \infty }{lim}\underset{0}{\overset{\eta }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot {\phi }_{gH}^{\prime }\left(x\right)dx=\hfill \\ & =\underset{\eta \to \infty }{lim}\left(e^{-{\displaystyle \frac{x}{\alpha }}}{\odot \phi \left(x\right)|}_0^{\eta }\right){\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}\underset{\eta \to \infty }{lim}\underset{0}{\overset{\eta }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot \phi \left(x\right)dx\hfill \\ & =\underset{\eta \to \infty }{lim}{e}^{-{\displaystyle \frac{\eta }{\alpha }}}\odot \phi \left(\eta \right){\ominus }_{gH}\phi \left(0\right){\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}\underset{\eta \to \infty }{lim}\underset{0}{\overset{\eta }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot \phi \left(x\right)dx.\hfill \end{array}$$

The fuzzy-valued function $\phi$ is of exponential order $c>0$. Then there exist $K>0$ and $X>0$, such that

$$D(e^{-{\displaystyle \frac{\eta }{\alpha }}}\odot \phi \left(\eta \right),\tilde{0})\le K{e}^{c\eta }$$

for all $\eta >X$. Hence

$$\underset{\eta \to \infty }{lim}D(e^{-{\displaystyle \frac{\eta }{\alpha }}}\odot \phi \left(\eta \right),\tilde{0})\le \underset{\eta \to \infty }{lim}K{e}^{-({\displaystyle \frac{1}{\alpha }}-c)\eta }=0$$

for $Re\left({\displaystyle \frac{1}{\alpha }}\right)>c$.

By the above equation and Proposition 1, we obtain

$$Y_x\left[{\phi }_{gH}^{\prime }\left(x\right)\right]=(-1)\odot \phi \left(0\right){\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}\odot Y_x\left[\phi \left(x\right)\right].$$

(10)

We will prove Equation (ii). Using Definition 10 and (10), we have

$$\begin{array}{cc}{Y}_x\left[{\phi }_{gH}^{\prime \prime }\left(x\right)\right]\hfill & =(-1)\odot {\phi }_{gH}^{\prime }\left(0\right){\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}\odot Y_x\left[{\phi }_{gH}^{\prime }\left(x\right)\right]=\hfill \\ & =(-1)\odot {\phi }_{gH}^{\prime }\left(0\right){\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}\left((-1)\odot \phi \left(0\right){\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}\odot Y_x\left[\phi \left(x\right)\right]\right)=\hfill \\ & =(-1)\odot {\phi }_{gH}^{\prime }\left(0\right){\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot \phi \left(0\right){\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot \mathrm{\Phi }\left(\alpha \right)\right).\hfill \end{array}$$

□

Corollary 1. 

Let $u(x,t)$ be a fuzzy-valued function. Then, the FYT for partial derivatives of $u(x,t)$ is as follows

(i) 

$Y_x\left[u_{x,gH}^{\prime }(x,t)\right]=(-1)\odot u(0,t){\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}\odot U(\alpha ,t)$;

(ii) 

$Y_x\left[u_{xx,gH}^{\prime \prime }(x,t)\right]=(-1)\odot u_{x,gH}^{\prime }(0,t){\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot u(0,t){\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot U(\alpha ,t)\right)$,

where $Y_x\left[u(x,t)\right]=U(\alpha ,t)$.

3.2. Fuzzy General Transform

The fuzzy General transform is introduced in [16].

Definition 13. 

Let $\psi :[0,\infty )\to E^1$ and the function $e^{-\tau \left(\beta \right)t}\odot \psi \left(t\right)$ be improper fuzzy Riemann integrable on $[0,\infty )$ for some $\sigma \left(\beta \right)$. Then, the fuzzy General transform (FGT) of a function $\psi \left(t\right)$ is defined as

$$\mathrm{\Psi }\left(\beta \right)=J_t\left[\psi \left(t\right)\right]=\sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}{e}^{-\tau \left(\beta \right)t}\odot \psi \left(t\right)dt,$$

(11)

where $\sigma \left(\beta \right)\ne 0$ and $\tau \left(\beta \right)$ are positive real functions.

If $\sigma \left(\beta \right)=1$ and $\tau \left(\beta \right)={\displaystyle \frac{1}{\alpha }}$, then FGT (11) gives the FYT (8).

Definition 14. 

The fuzzy inverse General transform is given by

$$J_{\beta }^{-1}\left[\mathrm{\Psi }\left(\beta \right)\right]={\displaystyle \frac{1}{2\pi i}}\underset{a-i\infty }{\overset{a+i\infty }{\int }}{\displaystyle \frac{1}{\sigma \left(\beta \right)}}{e}^{\tau \left(\beta \right)t}{\tau }^{\prime }\left(\beta \right)\odot \mathrm{\Psi }\left(\beta \right)d\beta ,$$

(12)

where $\mathrm{\Psi }\left(\beta \right)$ must be analytic for all β such that $Re\beta >a$.

Theorem 10. 

Let the fuzzy-valued function $\psi :[0,\infty )\to E^1$ be a continuous function in every finite interval $(0,T)$ of exponential order $d>0$. Then, the FGT of $\psi \left(t\right)$ exists for all β, such that $Re\left(\tau \right(\beta \left)\right)>d$.

Proof. 

From Definition 12, it follows that there exist $Q>0$ and $T>0$, such that

$$D(\psi \left(t\right),\tilde{0})\le Q{e}^{dt}forallt>T.$$

Hence

$$\begin{array}{cc}D(\mathrm{\Psi }\left(\beta \right),\tilde{0})\hfill & =D(\sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}{e}^{-\tau \left(\beta \right)t}\odot \psi \left(t\right)dt,\tilde{0})\le \hfill \\ & \le \sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}{e}^{-\tau \left(\beta \right)t}D(\psi \left(t\right),\tilde{0})dt\le \hfill \\ & \le Q\sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}{e}^{-\left(\tau \right(\beta )-d)t}dt={\displaystyle \frac{Q\sigma \left(\beta \right)}{\tau \left(\beta \right)-d}},\hfill \end{array}$$

where $J_t\left[\psi \left(t\right)\right]=\mathrm{\Psi }\left(\beta \right)$ and $Re\left(\tau \right(\beta \left)\right)>d.$ □

The classical General transform for some special functions is given in [31].

(i)

$J_t\left[1\right]={\displaystyle \frac{\sigma \left(\beta \right)}{\tau \left(\beta \right)}}$;

(ii)

$J_t\left[t^n\right]={\displaystyle \frac{(n!)\sigma \left(\beta \right)}{q^{n+1}\left(\beta \right)}},$ where n is a positive integer;

(iii)

$J_t\left[e^{dt}\right]={\displaystyle \frac{\sigma \left(\beta \right)}{\tau \left(\beta \right)-d}},$ where $d\in \mathbb{R}$;

(iv)

$J_t[sindt]={\displaystyle \frac{d\sigma \left(\beta \right)}{d^2+{\tau }^2\left(\beta \right)}},$ where $d\in \mathbb{R}$;

(v)

$J_t[cosdt]={\displaystyle \frac{\sigma \left(\beta \right)\tau \left(\beta \right)}{d^2+{\tau }^2\left(\beta \right)}},$ where $d\in \mathbb{R}$.

Using Definition 13, we obtain the basic properties of the FGT.

Theorem 11. 

Let for fuzzy-valued functions ${\psi }_1,{\psi }_2:{\mathbb{R}}_{+}\to E^1$ exist an FGT. Then, the FGT of functions ${\gamma }_1\odot {\psi }_1\left(t\right)\oplus {\gamma }_2\odot {\psi }_2\left(t\right)$ and ${\gamma }_1\odot {\psi }_1\left(t\right){\ominus }_{gH}{\gamma }_2\odot {\psi }_2\left(t\right)$ exists and

(i) 

${\gamma }_1\odot G_t\left[{\psi }_1\left(x\right)\right]{\ominus }_{gH}{\gamma }_2\odot G_t\left[{\psi }_2\left(x\right)\right]=G_t[a_1\odot {\psi }_1\left(x\right){\ominus }_{gH}{\gamma }_2\odot {\psi }_2\left(x\right)]$;

(ii) 

${\gamma }_1\odot G_t\left[{\psi }_1\left(x\right)\right]\oplus {\gamma }_2\odot G_t\left[{\psi }_2\left(x\right)\right]=G_t[{\gamma }_1\odot {\psi }_1\left(x\right)\oplus {\gamma }_2\odot {\psi }_2\left(x\right)]$,

where ${\gamma }_1,{\gamma }_2\ge 0$ or ${\gamma }_1,{\gamma }_2\le 0$.

Proof. 

Analogously in Theorem 8. □

Theorem 12. 

Let the fuzzy-valued function $\psi :[0,\infty )\to E^1$ be continuous of exponential order $d>0$ and ${\psi }_{gH}^{\prime }\left(t\right)$ be continuous in every interval $0\le t\le T$. Then

(i) 

$J_t\left[{\psi }_{gH}^{\prime }\left(t\right)\right]=(-1)\sigma \left(\beta \right)\odot \psi \left(0\right){\ominus }_{gH}(-1)\tau \left(\beta \right)\odot \mathrm{\Psi }\left(\beta \right)$;

(ii) 

$J_t\left[{\psi }_{gH}^{\prime \prime }\left(t\right)\right]=(-1)\sigma \left(\beta \right)\odot {\psi }_{gH}^{\prime }\left(0\right){\ominus }_{gH}\left(\sigma \left(\beta \right)\tau \left(\beta \right)\odot \psi \left(0\right){\ominus }_{gH}{\tau }^2\left(\beta \right)\odot \mathrm{\Psi }\left(\beta \right)\right)$,

where $J_t\left[\psi \left(t\right)\right]=\mathrm{\Psi }\left(\beta \right)$ and $Re\left(\tau \right(\beta \left)\right)>d$ we have.

Proof. 

Analogously in Theorem 9. □

Corollary 2. 

Let $u(x,t)$ be a fuzzy-valued function. Then, the FYT for partial derivatives of $u(x,t)$ is as follows:

(i) 

$J_t\left[u_{t,gH}^{\prime }(x,t)\right]=(-1)\sigma \left(\beta \right)\odot u(x,0){\ominus }_{gH}(-1)\tau \left(\beta \right)\odot U(x,\beta )$;

(ii) 

$J_t\left[u_{tt,gH}^{\prime \prime }(x,t)\right]=(-1)\sigma \left(\beta \right)\odot u_{t,gH}^{\prime }(x,0){\ominus }_{gH}\left(\sigma \left(\beta \right)\tau \left(\beta \right)\odot u(x,0){\ominus }_{gH}{\tau }^2\left(\beta \right)\odot U(x,\beta )\right)$,

where $J_t\left[u(x,t)\right]=U(x,\beta )$.

4. Double Fuzzy Yang-General Transform

In this part, we introduce DFY-GT, which combines the Yang and General first-order fuzzy transforms. The definition and some of the fundamental properties of this integral transform are presented.

Definition 15. 

Let $u:[0,\infty )\times [0,\infty )\to E^1$, the function $e^{-{\displaystyle \frac{x}{\alpha }}-\tau \left(\beta \right)t}\odot u(x,t)$ be improper fuzzy integrable on $[0,\infty )\times [0,\infty )$ for some $\tau \left(\beta \right)>0$ and $\alpha >0$. Then, the fuzzy Yang–General transform of a function $u(x,t)$ is defined as

$$U(\alpha ,\beta )=Y_x{J}_t\left[u(x,t)\right]=\tau \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot u(x,t)dxdt,$$

(13)

where $\sigma \left(\beta \right)$ is a real positive function.

Definition 16. 

Double fuzzy inverse Yang–General transform is denoted by $Y_{\alpha }^{-1}{G}_{\beta }^{-1}$ and

$$Y_{\alpha }^{-1}{J}_{\beta }^{-1}\left[U(\alpha ,\beta )\right]=Y_{\alpha }^{-1}\left[U(\alpha ,t)\right]=u(x,t).$$

(14)

Definition 17. 

A fuzzy function $u:[0,\infty )\times [0,\infty )\to E^1$ is called exponentially ordered $c>0$, $d>0$, if there exist positive real constants $K,X,T$ such that for all $x>X$, $t>T$

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

Theorem 13. 

Let the fuzzy function $u:[0,\infty )\times [0,\infty )\to E^1$ be a continuous in $(0,X)\times (0,T)$ of exponential order $c>0$, $d>0$. Then, the DFY-GT of $u(x,t)$ exists for all α and $\tau \left(\beta \right)$ with $Re\left({\displaystyle \frac{1}{\alpha }}\right)>c$, $Re\left(\tau \right(\beta \left)\right)>d$.

Proof. 

Let $Y_x{J}_t\left[u(x,t)\right]=U(\alpha ,\beta ))$. Then

$$\begin{array}{cc}D(U(\alpha ,\beta ),\tilde{0})\hfill & =D(\sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\tau \left(\beta \right)t}\odot u(x,t)dxdt,\tilde{0})\le \hfill \\ & \le \sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\tau \left(\beta \right)t}D(u(x,t),\tilde{0})dxdt\le \hfill \\ & \le K\sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-({\displaystyle \frac{1}{\alpha }}-c)x}{e}^{-\left(\tau \right(\beta )-d)t}dxdt=\hfill \\ & ={\displaystyle \frac{K\alpha \sigma \left(\beta \right)}{(1-c\alpha )\left(\tau \right(\beta )-d)}}.\hfill \end{array}$$

The improper double fuzzy integral converges for all $Re\left({\displaystyle \frac{1}{\alpha }}\right)>c$, $Re\left(\tau \right(\beta \left)\right)>d$. Thus, $Y_x{J}_t\left[u(x,t)\right]$ exists. □

Lemma 3. 

Let $\phi ,\psi :(0,\infty )\to E^1$ and the fuzzy function $u(x,t)=\phi \left(x\right)\psi \left(t\right)$. Then,

$$Y_x{J}_t\left[u(x,t)\right]=Y_x\left[\phi \left(x\right)\right]J_t\left[\psi \left(t\right)\right].$$

Proof. 

Using Definition 15, we find

$$\begin{array}{cc}{Y}_x{J}_t\left[u(x,t)\right]\hfill & =Y_x{J}_t\left[\phi \left(x\right)\psi \left(t\right)\right]=\tau \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot \left(\phi \left(x\right)\psi \left(t\right)\right)dxdt=\hfill \\ & =\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}\odot \phi \left(x\right)dx\tau \left(\beta \right)\underset{0}{\overset{\infty }{\int }}{e}^{-\sigma \left(\beta \right)t}\odot \psi \left(t\right)dt=\hfill \\ & =Y_x\left[\phi \left(x\right)\right]J_t\left[\psi \left(t\right)\right].\hfill \end{array}$$

□

Lemma 4. 

Let $u(x,t)=a\in E^1$. Then, we have

$$Y_x{J}_t\left[u(x,t)\right]={\displaystyle \frac{\alpha \tau \left(\beta \right)}{\sigma \left(\beta \right)}}\odot a$$

for $x>0$ and $t>0$.

Proof. 

Using Definition 15, we have

$$\begin{array}{cc}{Y}_x{J}_t\left[u(x,t)\right]\hfill & =\tau \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot adxdt=\hfill \\ & =\left(\tau \left(\beta \right)\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}}dx\underset{0}{\overset{\infty }{\int }}{e}^{-\sigma \left(\beta \right)t}dt\right)\odot a=Y_x\left[1\right]J_t\left[1\right]\odot a={\displaystyle \frac{\alpha \tau \left(\beta \right)}{\sigma \left(\beta \right)}}\odot a.\hfill \end{array}$$

□

By using Yang and General transform on some special functions, we obtain

(i)

$Y_x{J}_t\left[1\right]={\displaystyle \frac{\alpha \tau \left(\beta \right)}{\sigma \left(\beta \right)}};$

(ii)

$Y_x{J}_t\left[x^m{t}^n\right]={\displaystyle \frac{m!n!{\alpha }^{m+1}\tau \left(\beta \right)}{{\sigma }^{n+1}\left(\beta \right)}},$ where $m,n$ are positive integers;

(iii)

$Y_x{J}_t\left[e^{cx+dt}\right]={\displaystyle \frac{\alpha \tau \left(\beta \right)}{(1-c\alpha )\left(\sigma \right(\beta )-d)}},$ where $c,d\in \mathbb{R}$;

(iv)

$Y_x{J}_t\left[e^{i(cx+dt)}\right]={\displaystyle \frac{\alpha \tau \left(\beta \right)}{(1-ic\alpha )\left(\sigma \right(\beta )-id)}}=\alpha \tau \left(\beta \right){\displaystyle \frac{(1+ic\alpha )\sigma \left(\beta \right)+id)}{(1+c^2{\alpha }^2)({\sigma }^2\left(\beta \right)+d^2)}}=$

$=\alpha \tau \left(\beta \right){\displaystyle \frac{\sigma \left(\beta \right)-cd\alpha +i\left(c\alpha \sigma \right(\beta )+d)}{(1+c^2{\alpha }^2)({\sigma }^2\left(\beta \right)+d^2)}},$ where $c,d\in \mathbb{R}.$

Consequently,

$$Y_x{J}_t[cos(cx+dt)]=\alpha \tau \left(\beta \right){\displaystyle \frac{\sigma \left(\beta \right)-cd\alpha }{(1+c^2{\alpha }^2)({\sigma }^2\left(\beta \right)+d^2)}},$$

$$Y_x{J}_t[sin(cx+dt)]=\alpha \tau \left(\beta \right){\displaystyle \frac{c\alpha \sigma \left(\beta \right)+d}{(1+c^2{\alpha }^2)({\sigma }^2\left(\beta \right)+d^2)}}.$$

Now, we present some properties of DFY-GT.

Using Theorems 8 and 11, we obtain that double fuzzy Yang–General transform is a linear transformation.

Theorem 14. 

Let $u_1,u_2:[0,\infty )\times [0,\infty )\to E^1$ be fuzzy-valued functions. Then,

(i) 

${\gamma }_1\odot Y_x{J}_t\left[u_1(x,t)\right]{\ominus }_{gH}{\gamma }_2\odot Y_x{J}_t\left[u_2(x,t)\right]=Y_x{J}_t[{\gamma }_1\odot u_1(x,t){\ominus }_{gH}{\gamma }_2\odot u_2(x,t)]$;

(ii) 

${\gamma }_1\odot Y_x{J}_t\left[u_1(x,t)\right]\oplus {\gamma }_2\odot Y_x{J}_t\left[u_2(x,t)\right]=Y_x{J}_t[{\gamma }_1\odot u_1(x,t)\oplus {\gamma }_2\odot u_2(x,t)]$,

where ${\gamma }_1,{\gamma }_2>0$ or ${\gamma }_1,{\gamma }_2<0$.

From Theorem 14, it follows that the double fuzzy inverse Yang–General transform is also a linear transformation.

Theorem 15. 

Let $u:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to E^1$ be a periodic function of periods ξ and η such that

$$u(x+\xi ,t+\eta )=u(x,t)$$

and $Y_x{J}_t\left[u(x,t)\right]$ exists. Then,

$$Y_x{J}_t\left[u(x,t)\right]=U(\alpha ,\beta )={\displaystyle \frac{\tau \left(\beta \right)}{1-\tau \left(\beta \right)e^{-\frac{\xi }{\alpha }-\sigma \left(\beta \right)\eta }}}\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\eta }{\int }}{e}^{-{\displaystyle \frac{\xi }{\alpha }}-\sigma \beta )\eta }\odot u(x,t)dxdt.$$

Proof. 

Using the Definition 15 and properties of improper fuzzy integral, we find

$$\begin{array}{cc}{Y}_x{J}_t\left[u(x,t)\right]\hfill & =\tau \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot u(x,t)dxdt=\hfill \\ & =\tau \left(\beta \right)\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\eta }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot u(x,t)dxdt\oplus \tau \left(\beta \right)\underset{\xi }{\overset{\infty }{\int }}\underset{\eta }{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot u(x,t)dxdt.\hfill \end{array}$$

Putting $x=\xi +\rho$ and $t=\eta +\delta$ on second integral and using the periodicity of the function $u(x,t)$, we obtain

$$\begin{array}{c}\underset{\xi }{\overset{\infty }{\int }}\underset{\eta }{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot u(x,t)dxdt=\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{\xi +\rho }{\alpha }}-\sigma \left(\beta \right)(\eta +\tau )}\odot u(\xi +\rho ,\eta +\tau )d\rho d\delta =\hfill \\ =e^{-{\displaystyle \frac{\xi }{\alpha }}-\sigma \left(\beta \right)\eta }\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{\rho }{\alpha }}-\sigma \left(\beta \right)\delta }\odot u(\rho ,\delta )d\rho d\delta =e^{-{\displaystyle \frac{\xi }{\alpha }}-\sigma \left(\beta \right)\eta }{Y}_x{J}_t\left[u(x,t)\right].\hfill \end{array}$$

By substituting into the above equation, we obtain

$$\begin{array}{cc}{Y}_x{J}_t\left[u(x,t)\right]\hfill & =\tau \left(\beta \right)\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\eta }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot u(x,t)dxdt\oplus \hfill \\ & \oplus \tau \left(\beta \right)e^{-{\displaystyle \frac{\xi }{\alpha }}-\sigma \left(\beta \right)\eta }{Y}_x{J}_t\left[u(x,t)\right].\hfill \end{array}$$

This equation can be simplified into

$$Y_x{J}_t\left[u(x,t)\right]={\displaystyle \frac{\tau \left(\beta \right)}{1-\tau \left(\beta \right)e^{-\frac{\xi }{\alpha }-\sigma \left(\beta \right)\eta }}}\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\eta }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-\sigma \left(\beta \right)t}\odot u(x,t)dxdt.$$

□

Theorem 16. 

Let $Y_x{J}_t\left[u(x,t)\right]$ exists. Then

$$Y_x{J}_t\left[u(x-\rho ,t-\delta )H(x-\rho ,t-\delta )\right]=e^{-{\displaystyle \frac{\rho }{\alpha }}-q\left(\beta \right)\delta }\odot Y_x{J}_t\left[u(x,t)\right],$$

(15)

where

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

is the Heaviside function.

Proof. 

By Definition 15, we obtain

$$\begin{array}{c}{Y}_x{J}_t\left[u(x-\rho ,t-\delta )H(x-\rho ,t-\delta )\right]=\hfill \\ =\sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-q\left(\beta \right)t}\odot u(x-\rho ,t-\delta )H(x-\rho ,t-\delta )dxdt=\hfill \\ =\sigma \left(\beta \right)\underset{\rho }{\overset{\infty }{\int }}\underset{\delta }{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{x}{\alpha }}-q\left(\beta \right)t}\odot u(x-\rho ,t-\delta )dxdt.\hfill \end{array}$$

Substituting $x-\rho ={\rho }_1$ and $t-\delta ={\delta }_1$, we obtain

$$\begin{array}{c}{Y}_x{J}_t\left[u(x-\rho ,t-\eta )H(x-\rho ,t-\eta )\right]=\hfill \\ =\sigma \left(\beta \right)\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{\rho +{\rho }_1}{\alpha }}-q\left(\beta \right)(\delta +{\delta }_1)}\odot u({\rho }_1,{\delta }_1)d{\rho }_{1}d{\delta }_1=\hfill \\ =\sigma \left(\beta \right)e^{-{\displaystyle \frac{\rho }{\alpha }}-q\left(\beta \right)\delta }\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-{\displaystyle \frac{{\rho }_1}{\alpha }}-\tau \left(\beta \right){\delta }_1}\odot u({\rho }_1,{\delta }_1)d{\rho }_{1}d{\delta }_1=\hfill \\ =e^{-{\displaystyle \frac{\rho }{\alpha }}-q\left(\beta \right)\delta }\odot Y_x{J}_t\left[u(x,t)\right].\hfill \end{array}$$

□

Theorem 17. 

Let $Y_x{J}_t\left[u(x,t)\right]=U(\alpha ,\beta )$. Then

(i) 

$Y_x{J}_t\left[u_{x,gH}^{\prime }(x,t)\right]=(-1)\odot J_t\left[u(0,t)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot U(\alpha ,\beta )$;

(ii) 

$Y_x{J}_t\left[u_{t,gH}^{\prime }(x,t)\right]=(-1)\sigma \left(\beta \right)\odot Y_x\left[u(x,0)\right]{\ominus }_{gH}(-1)\tau \left(\beta \right)\odot U(\alpha ,\beta )$;

(iii) 

$Y_x{J}_t\left[u_{xx}^{\prime \prime }(x,t)\right]=(-1)\odot J_t\left[u_x^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot J_t\left[u(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot U(\alpha ,\beta )\right)$;

(iv) 

$Y_x{J}_t\left[u_{tt,gH}^{\prime \prime }(x,t)\right]=$

$=(-1)\sigma \left(\beta \right)\odot Y_x\left[u_{t,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\left(\sigma \left(\beta \right)\tau \left(\beta \right)\odot Y_x\left[u(x,0)\right]{\ominus }_{gH}{\tau }^2\left(\beta \right)\odot U(\alpha ,\beta )\right)$;

(v) 

$Y_x{J}_t\left[u_{xt,gH}^{\prime \prime }(x,t)\right]=$

$=(-1)\sigma \left(\beta \right)\odot Y_x\left[u_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\tau \left(\beta \right)\odot \left(J_t\left[u(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}U(\alpha ,\beta )\right)$;

(vi) 

$Y_x{J}_t\left[u_{tx,gH}^{\prime \prime }(x,t)\right]=$

$=(-1)\odot J_t\left[u(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{\alpha }}\left(\sigma \left(\beta \right)\odot Y_x\left[u_{t,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\tau \left(\beta \right)\odot U(\alpha ,\beta )\right).$

Proof. 

Using Definition 15 and Corollary 1, we obtain

$$\begin{array}{c}{Y}_x{J}_t\left[u_{x,gH}^{\prime }(x,t)\right]=J_t\left[Y_x\left[u_{x,gH}^{\prime }(x,t)\right]\right]=J_t\left[(-1)\odot u(0,t){\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot Y_x\left[u(x,t)\right]\right]=\hfill \\ =(-1)\odot J_t\left[u(0,t)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot J_t\left[Y_x\left[u(x,t)\right]\right]=\hfill \\ =(-1)\odot J_t\left[u(0,t)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot U(\alpha ,\beta ).\hfill \end{array}$$

Similarly, we prove the case $\left(ii\right)$. Indeed,

$$\begin{array}{c}{Y}_x{J}_t\left[u_{t,gH}^{\prime }(x,t)\right]=Y_x\left[J_t\left[u_{t,gH}^{\prime }(x,t)\right]\right]=\hfill \\ =Y_x\left[(-1)\sigma \left(\beta \right)\odot u(x,0){\ominus }_{gH}(-1)\tau \left(\beta \right)\odot J_t\left[u(x,t)\right]\right]=\hfill \\ =(-1)\sigma \left(\beta \right)\odot Y_x\left[u(x,0)\right]{\ominus }_{gH}(-1)\tau \left(\beta \right)\odot Y_x\left[J_t\left[u(x,t)\right]\right]=\hfill \\ =(-1)\sigma \left(\beta \right)\odot Y_x\left[u(x,0)\right]{\ominus }_{gH}(-1)\tau \left(\beta \right)\odot U(\alpha ,\beta ).\hfill \end{array}$$

Now, we will prove case (iii). Applying Definition 15 and case (ii) of Corollary 1, we have

$$\begin{array}{c}{Y}_x{J}_t\left[u_{xx,gH}^{\prime \prime }(x,t)\right]=J_t\left[Y_x\left[u_{xx,gH}^{\prime \prime }(x,t)\right]\right]=\hfill \\ =J_t\left[(-1)\odot u_{x,gH}^{\prime }(0,t){\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot u(0,t){\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot Y_x\left[u(x,t)\right]\right)\right]=\hfill \\ =(-1)\odot J_t\left[u_{x,gH}^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot J_t\left[u(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot J_t\left[Y_x\left[u(x,t)\right]\right]\right)=\hfill \\ =(-1)\odot J_t\left[u_{x,gH}^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot J_t\left[u(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot U(\alpha ,\beta )\right).\hfill \end{array}$$

By Definition 15 and case (ii) of Corollary 2, we obtain case (iv). Really,

$$\begin{array}{c}{Y}_x{J}_t\left[u_{tt,gH}^{\prime \prime }(x,t)\right]=Y_x\left[J_t\left[u_{tt,gH}^{\prime \prime }(x,t)\right]\right]=\hfill \\ =Y_x\left[(-1)\sigma \left(\beta \right)\odot u_{t,gH}^{\prime }(x,0){\ominus }_{gH}\left(\sigma \left(\beta \right)\tau \left(\beta \right)\odot u(x,0){\ominus }_{gH}{\tau }^2\left(\beta \right)\odot J_t\left[u(x,t)\right]\right)\right]=\hfill \\ =(-1)\sigma \left(\beta \right)\odot Y_x\left[u_{t,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\left(\sigma \left(\beta \right)\tau \left(\beta \right)\odot Y_x\left[u(x,0)\right]{\ominus }_{gH}{\tau }^2\left(\beta \right)\odot Y_x\left[J_t\left[u(x,t)\right]\right]\right)=\hfill \\ =(-1)\sigma \left(\beta \right)\odot Y_x\left[u_{t,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\left(\sigma \left(\beta \right)\tau \left(\beta \right)\odot Y_x\left[u(x,0)\right]{\ominus }_{gH}{\tau }^2\left(\beta \right)\odot U(\alpha ,\beta )\right).\hfill \end{array}$$

We will prove the case (v).

$$\begin{array}{c}{Y}_x{J}_t\left[u_{xt,gH}^{\prime \prime }(x,t)\right]=Y_x\left[J_t\left[u_{xt,gH}^{\prime \prime }(x,t)\right]\right]=\hfill \\ =Y_x\left[(-1)\sigma \left(\beta \right)\odot u_{x,gH}^{\prime }(x,0){\ominus }_{gH}(-1)\tau \left(\beta \right)\odot J_t\left[u_{x,gH}^{\prime }(x,t)\right]\right]=\hfill \\ =(-1)\sigma \left(\beta \right)\odot Y_x\left[u_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}(-1)\tau \left(\beta \right)\odot J_t\left[Y_x\left[u_{x,gH}^{\prime }(x,t)\right]\right]=\hfill \\ =(-1)\sigma \left(\beta \right)\odot Y_x\left[u_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}(-1)\tau \left(\beta \right)\odot J_t\left[Y_x\left[u_{x,gH}^{\prime }(x,t)\right]\right]=\hfill \\ =(-1)\sigma \left(\beta \right)\odot Y_x\left[u_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}(-1)\tau \left(\beta \right)\odot J_t\left[(-1)u(0,t){\ominus }_{gH}{\displaystyle \frac{1}{\alpha }}{Y}_x\left[u(x,t)\right]\right]=\hfill \\ =(-1)\sigma \left(\beta \right)\odot Y_x\left[u_{x,gH}^{\prime }(x,0)\right]{\ominus }_{gH}\tau \left(\beta \right)\odot \left(J_t\left[u(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{(-1)}{\alpha }}U(\alpha ,\beta )\right).\hfill \end{array}$$

The proof of case (vi) is analogous to case (v). □

5. Applications of Double Fuzzy Yang–General Transform

In this section, we illustrate the application of DFY-GT and obtain the solution framework for the linear fuzzy advection–diffusion equation given by

$$u_{t,gH}^{\prime }(x,t)={\kappa }_1\odot u_{xx,gH}^{″}(x,t){\ominus }_{gH}{\kappa }_2\odot u_{x,gH}^{\prime }(x,t),x\ge 0t\ge 0,$$

(16)

with the initial condition

$$u(x,0)=f_0\left(x\right),$$

(17)

and the boundary conditions

$$u(0,t)=g_0\left(t\right),u_x^{\prime }(0,t)=g_1\left(t\right).$$

(18)

The constant ${\kappa }_1>0$ is the diffusion coefficient and the constant ${\kappa }_2$ is the advection velocity.

Applying DFY-GT on both sides of Equation (16), we obtain

$$Y_x{J}_t\left[u_{t,gH}^{\prime }(x,t)\right]={\kappa }_1\odot Y_x{J}_t\left[u_{xx,gH}^{\prime \prime }(x,t)\right]{\ominus }_{gH}{\kappa }_2\odot Y_x{J}_t\left[u_{x,gH}^{\prime }(x,t)\right].$$

Let $U(\alpha ,\beta )=Y_x{J}_t\left[u(x,t)\right]$. Using the differentiation property of the DFY-GT (Theorem 17), we obtain

$$\begin{array}{c}(-1)\sigma \left(\beta \right)\odot Y_x\left[u(x,0)\right]{\ominus }_{gH}(-1)\tau \left(\beta \right)\odot U(\alpha ,\beta )=\hfill \\ ={\kappa }_1\left[(-1)\odot J_t\left[u_{x,gH}^{\prime }(0,t)\right]{\ominus }_{gH}\left({\displaystyle \frac{1}{\alpha }}\odot J_t\left[u(0,t)\right]{\ominus }_{gH}{\displaystyle \frac{1}{{\alpha }^2}}\odot U(\alpha ,\beta )\right)\right]{\ominus }_{gH}\hfill \\ {\ominus }_{gH}{\kappa }_2\left[(-1)\odot J_t\left[u(0,t)\right]{\ominus }_{gH}(-1){\displaystyle \frac{1}{\alpha }}\odot U(\alpha ,\beta )\right].\hfill \end{array}$$

We apply the fuzzy Yang transform to Equation (17) and the fuzzy General transform to Equation (18).

Operation of fuzzy Yang transform to the initial condition yields

$$F_0\left(\alpha \right)=Y_x\left[f_0\left(x\right)\right].$$

Fuzzy General transform of the boundary conditions is given by

$$G_0\left(\beta \right)=J_t\left[u(0,t)\right]G_1\left(\beta \right)=J_t\left[u_x^{\prime }(0,t)\right].$$

Substituting into the above equation, we obtain

$$\begin{array}{c}(-1)\sigma \left(\beta \right)\odot F_0\left(\alpha \right){\ominus }_{gH}(-1)\tau \left(\beta \right)\odot U(\alpha ,\beta )=\hfill \\ =(-1){\kappa }_1\odot G_1\left(\beta \right){\ominus }_{gH}\left({\displaystyle \frac{{\kappa }_1}{\alpha }}\odot G_0\left(\beta \right){\ominus }_{gH}{\displaystyle \frac{{\kappa }_1}{{\alpha }^2}}\odot U(\alpha ,\beta )\right){\ominus }_{gH}\hfill \\ {\ominus }_{gH}\left[(-1){\kappa }_2\odot G_0\left(\beta \right){\ominus }_{gH}(-1){\displaystyle \frac{{\kappa }_2}{\alpha }}\odot U(\alpha ,\beta )\right].\hfill \end{array}$$

By using Proposition 1, we obtain

$$\begin{array}{c}\left({\displaystyle \frac{{\kappa }_2}{\alpha }}-{\displaystyle \frac{{\kappa }_1}{{\alpha }^2}}+\tau \left(\beta \right)\right)\odot U(\alpha ,\beta )=\hfill \\ =\sigma \left(\beta \right)\odot F_0\left(\alpha \right)\oplus \left({\kappa }_2-{\displaystyle \frac{{\kappa }_1}{\alpha }}\right)\odot G_0\left(\beta \right)\oplus (-{\kappa }_1)\odot G_1\left(\beta \right).\hfill \end{array}$$

Hence

$$\begin{array}{cc}U(\alpha ,\beta )\hfill & ={\displaystyle \frac{{\alpha }^2\sigma \left(\beta \right)}{\alpha {\kappa }_2-{\kappa }_1+{\alpha }^2\tau \left(\beta \right)}}\odot F_0\left(\alpha \right)\oplus {\displaystyle \frac{{\alpha }^2(\alpha {\kappa }_2-{\kappa }_1)}{\alpha {\kappa }_2-{\kappa }_1+{\alpha }^2\tau \left(\beta \right)}}\odot G_0\left(\beta \right)\oplus \hfill \\ \hfill & \oplus {\displaystyle \frac{-{\alpha }^2{\kappa }_1}{\alpha {\kappa }_2-{\kappa }_1+{\alpha }^2\tau \left(\beta \right)}}\odot G_1\left(\beta \right),\hfill \end{array}$$

(19)

Applying the inverse DFY-GT on both sides of Equation (19), we obtain $u(x,t)$.

6. Examples

Example 2. 

Here, we consider a numerical example solved using the presented method to demonstrate the application of DFY-GT.

$$u_{t,gH}^{\prime }(x,t)=u_{xx,gH}^{\prime \prime }(x,t){\ominus }_{gH}{u}_{x,gH}^{\prime }(x,t),x\ge 0,t\ge 0$$

(20)

The initial and boundary conditions are given as

$$u(x,0,r)=(e^x-x)\odot (1,2,3)$$

(21)

$$u(0,t,r)=(1+t)\odot (1,2,3),u_x^{\prime }(0,t,r)=0\odot (1,2,3).$$

(22)

In this case, ${\kappa }_1={\kappa }_2=1$ and fuzzy functions

$$f_0\left(x\right)=(e^x-x)\odot (1,2,3),g_0\left(x\right)=(1+t)\odot (1,2,3),and{g}_1\left(x\right)=0\odot (1,2,3).$$

Then, applying FYT to Equation (21) and FGT to Equation (22), we find

$$F_0\left(\alpha \right)=Y_x\left[f_0\left(x\right)\right]=\left({\displaystyle \frac{\alpha }{1-\alpha }}-{\alpha }^2\right)\odot (1,2,3),$$

$$G_0\left(\beta \right)=J_t\left[g_0\left(t\right)\right]=\left({\displaystyle \frac{\sigma \left(\beta \right)}{\tau \left(\beta \right)}}+{\displaystyle \frac{\sigma \left(\beta \right)}{q^2\left(\beta \right)}}\right)\odot (1,2,3),$$

$$G_1\left(\beta \right)=J_t\left[g_1\left(t\right)\right]=0\odot (1,2,3).$$

By using Equation (19), we obtain

$$U(\alpha ,\beta )={\displaystyle \frac{{\alpha }^2\sigma \left(\beta \right)}{\alpha -1+{\alpha }^2\tau \left(\beta \right)}}\odot F_0\left(\alpha \right)\oplus {\displaystyle \frac{{\alpha }^2(\alpha -1)}{\alpha -1+{\alpha }^2\tau \left(\beta \right)}}\odot G_0\left(\beta \right).$$

We can simplify the above equation as

$$\begin{array}{cc}U(\alpha ,\beta )\hfill & ={\displaystyle \frac{{\alpha }^2\sigma \left(\beta \right)}{\alpha -1+{\alpha }^2\tau \left(\beta \right)}}\left({\displaystyle \frac{\alpha }{1-\alpha }}-{\alpha }^2\right)\odot (1,2,3)\oplus {\displaystyle \frac{{\alpha }^2(\alpha -1)}{\alpha -1+{\alpha }^2\tau \left(\beta \right)}}\left({\displaystyle \frac{\sigma \left(\beta \right)}{\tau \left(\beta \right)}}+{\displaystyle \frac{\sigma \left(\beta \right)}{{\tau }^2\left(\beta \right)}}\right)\odot (1,2,3)=\hfill \\ & =\left({\displaystyle \frac{\alpha \sigma \left(\beta \right)}{(1-\alpha )\tau \left(\beta \right)}}-{\displaystyle \frac{{\alpha }^2\sigma \left(\beta \right)}{\tau \left(\beta \right)}}+{\displaystyle \frac{\alpha \sigma \left(\beta \right)}{{\tau }^2\left(\beta \right)}}\right)\odot (1,2,3).\hfill \end{array}$$

Finally, taking the inverse DFY-GT of the above equation, we find

$$u(x,t,r)=Y_x^{-1}{J}_t^{-1}\left[\left({\displaystyle \frac{\alpha \sigma \left(\beta \right)}{(1-\alpha )\tau \left(\beta \right)}}-{\displaystyle \frac{{\alpha }^2\sigma \left(\beta \right)}{\tau \left(\beta \right)}}+{\displaystyle \frac{\alpha \sigma \left(\beta \right)}{{\tau }^2\left(\beta \right)}}\right)\odot (1,2,3)\right]$$

Thus, the solution for fuzzy advection–diffusion Equations (20)–(22) is

$$u(x,t,r)=(e^x-x+t)\odot (1,2,3).$$

7. Conclusions

In this paper, a new double fuzzy transform called DFY-GT was introduced. Some fundamental properties of this transform are presented. New theorems related to the existence, linearity, periodicity, and gH-partial derivatives were proven. These results were used to obtain a new simple formula for solving the linear fuzzy advection–diffusion equation. Finally, we constructed a numerical example and obtained the exact solution of the equation considered applying the new double fuzzy integral transform.

This work can be extended to the case where the fuzzy linear advection–diffusion equation is two-dimensional. Furthermore, we can consider analytical solutions of fuzzy nonlinear advection–diffusion problems. In addition, future applications of the DFY-GT can be developed to solve fuzzy parabolic integro-differential equations.