Application of Double Sumudu-Generalized Laplace Decomposition Method for Solving 2+1-Pseudoparabolic Equation

Abstract

The main purpose of this research paper is to discuss the solution of the singular two-dimensional pseudoparabolic equation by employing the double Sumudu-generalized Laplace transform decomposition method (DSGLTDM). We establish two theorems related to the partial derivatives. Furthermore, to investigate the relevance of the proposed method to solving singular two-dimensional pseudo parabolic equations, three examples are provided.

1. Introduction

Partial differential equations have several implementations in mathematical physics. One of these implementations is the pseudoparabolic (also known as Sobolev-type differential equation) which is considered an important problem occurring in a kind of physical situation like the flow of fluid out of fissured rocks, thermodynamics, and wave propagation. The parabolic equation happens in many areas of applied mathematics physic, for example, the heat diffusion equation and fluid mechanics; for more details, see [1,2,3,4]. The Adomian decomposition and the series expansion methods are used in solutions of fractional diffusion equation problems in [5,6]. The existence, uniqueness, and continuous dependence of powerful solutions of one-dimensional pseudoparabolic equation were studied in [7]. Presently, several researchers have proposed a precise solution to a one-dimensional connected parabolic equation; see [8,9]. The author in [10] suggested the modification of the double Laplace decomposition method to find the analytical approximation solution of a coupled system of pseudoparabolic equations with initial conditions. Newly, in [11], the authors applied the three-dimensional Laplace Adomian decomposition method to solve singular pseudoparabolic equations. The convergence of the Adomian method was studied by several researchers (we refer the readers to see [12,13,14,15]). The author in [16] proposed Sumudu transform, later used by Belgacem et al. in [17] to generalize the existence of Sumudu differentiation, integration, and convolution theorems. It was also applied by the same authors for solving an integral production depreciation problem in [18]. In [19], the researcher employed the Sumudu transform to obtain the solution of Abel’s integral equation, an integrodifferential equation, a dynamic system with delayed time signals, and a differential dynamic system. Moreover, the author in [20] expanded the single Sumudu transform to a double Sumudu transform with a focus on solutions to partial differential equations. Ahmeda et al. [21] discussed the convergence of the double Sumudu transformation and used it to obtain the solution of the Volterra integro-partial differential equation. The generalized Laplace transforms were used to study the solution of partial differential equations (PDEs) and also, the suggested ideas gave an easy solution to engineering problems by freely selecting integer $\alpha$ in the definition [22]. Recently, the researchers in [23] studied the solution of fractional third-order dispersive partial differential equations and symmetric KdV by utilizing Sumudu-generalized Laplace transform decomposition.

The main goal of this work is to generate a new method by combining the decomposition method and double Sumudu-generalized Laplace transform, which is called the double Sumudu-generalized Laplace transform decomposition method, to solve 2+1-singular pseudoparabolic equations. Here, we list some definitions that are applied in this work.

2. Properties of Double Sumudu-Generalized Laplace Transform

The definitions and existence conditions of the double Sumudu-generalized Laplace transform are presented. Here, we work with the double Sumudu-generalized Laplace transform, which is defined by

$$S_x{S}_y{G}_t\left(f\left(x,y,t\right)\right)=F\left({\zeta }_1,{\zeta }_2,s\right)=\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{x}{{\zeta }_1}+\frac{y}{{\zeta }_2}+\frac{t}{s}\right)}f\left(x,y,t\right)dtdydx$$

(1)

Existence Condition for the Double Sumudu-Generalized Laplace Transform

In the following, the conditions for the existence of the double Sumudu-generalized Laplace transform are offered. If $f\left(x,y,t\right)$ is an exponential order $a_1,a_2$ and b as $x\to \infty$, $y\to \infty ,t\to \infty ,$ and if $\exists R>0$ similarly applies for all $x>X$, $y>Y$ and $t>T$

$$\left|f\left(x,y,t\right)\right|\le R{e}^{a_{1}x+a_{2}y+bt},$$

(2)

for some $X,Y$ and T, then we write

$$f\left(x,y,t\right)=O\left(e^{a_{1}x+a_{2}y+bt}\right)\mathrm{as}y\to \infty ,y\to \infty ,t\to \infty ,$$

and similarly,

$$\underset{\begin{array}{c}x\to \infty \\ y\to \infty \\ t\to \infty \end{array}}{lim}{e}^{-\frac{1}{\mu }x-\frac{1}{\eta }y-\frac{1}{\epsilon }t}\left|f\left(x,y,t\right)\right|=R\underset{\begin{array}{c}x\to \infty \\ y\to \infty \\ t\to \infty \end{array}}{lim}{e}^{-\left(\frac{1}{{\lambda }_1}-a_1\right)x-\left(\frac{1}{{\lambda }_2}-a_2\right)y-\left(\frac{1}{\eta }-c\right)t}=0,$$

(3)

whenever $\frac{1}{{\lambda }_1}>a,$$\frac{1}{\eta }>c$ and $\frac{1}{{\lambda }_2}>b.$ The function $f\left(x,y,t\right)$ does not develop swifter than $K\left(x,y,t\right)$ as $x\to \infty$, $y\to \infty ,t\to \infty$.

Theorem 1.

The function $f\left(x,y,t\right)$ is defined on $(0,X),\left(0,Y\right)$ and $(0,T)$ and is of exponential order $\left(x,y,t\right)$, then the double Sumudu-generalized Laplace transform of $f\left(x,y,t\right)$ exists for all $\Re (\frac{1}{{\zeta }_1})>\frac{1}{{\lambda }_1}$, $\Re (\frac{1}{{\zeta }_2})>\frac{1}{{\lambda }_2}$ $\Re \frac{1}{s}>\frac{1}{\eta }$

Proof. 

By putting Equation (1) into Equation (2), we obtain

$$\begin{array}{ccc}\hfill \left|F\left({\zeta }_1,{\zeta }_2,s\right)\right|& =& \left|\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{x}{{\zeta }_1}+\frac{y}{{\zeta }_2}+\frac{t}{s}\right)}f\left(x,y,t\right)dxdydt\right|\hfill \\ & \le & R\left|\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{\left(\frac{1}{u_1}-a\right)x-\left(\frac{1}{u_2}-b\right)y-\left(\frac{1}{v}-c\right)t}dxdydt\right|\hfill \\ & =& \frac{R{s}^{\alpha +1}}{\left(1-a{\zeta }_1\right)\left(1-c{\zeta }_2\right)\left(1-bs\right)}.\hfill \end{array}$$

(4)

From using the condition $\Re (\frac{1}{{\zeta }_1})>\frac{1}{{\lambda }_1}$, $\Re (\frac{1}{{\zeta }_2})>\frac{1}{{\lambda }_2}$ $\Re \frac{1}{s}>\frac{1}{\eta }$, we yield

$$\underset{\begin{array}{c}x\to \infty \\ y\to \infty \\ t\to \infty \end{array}}{lim}\left|F(u_1,u_2,v)\right|=0\mathrm{or}\underset{{}_{{}_{\begin{array}{c}x\to \infty \\ y\to \infty \\ t\to \infty \end{array}}}}{lim}F(u_1,u_2,v)=0.$$

□

Theorem 2.

If the double Sumudu-generalized Laplace transform of the function $f\left(x,y,t\right)$ is presented by $S_x{S}_y{G}_t\left(f\left(x,y,t\right)\right)=F\left({\zeta }_1,{\zeta }_2,s\right)$, then the double Sumudu-generalized Laplace transform of the function

$$xyf\left(x,y,t\right),$$

is determined by

$$S_x{S}_y{G}_t\left[xyf\left(x,y,t\right)\right]={\zeta }_1{\zeta }_2\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right).$$

(5)

Proof. 

By applying a partial derivative according to ${\zeta }_1$ for Equation (1), we obtain

$$\begin{array}{ccc}\hfill \frac{\partial F({\zeta }_1,{\zeta }_2,s)}{\partial {\zeta }_1}& =& \frac{\partial }{\partial {\zeta }_1}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_1}x+\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}f\left(x,y,t\right)dxdydt,\hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{\partial }{\partial {\zeta }_1}\frac{1}{{\zeta }_1}{e}^{-\frac{1}{{\zeta }_1}x}f\left(x,y,t\right)dx\right)dydt,\hfill \end{array}$$

(6)

and by computing the partial derivative within the brackets, we obtain

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }\frac{\partial }{\partial {\zeta }_1}\frac{1}{{\zeta }_1}{e}^{-\frac{1}{{\zeta }_1}x}f\left(x,y,t\right)dx& =& {\int }_0^{\infty }\left(\frac{1}{{\zeta }_1^3}x-\frac{1}{{\zeta }_1^2}\right)e^{-\frac{1}{{\zeta }_1}x}f\left(x,y,t\right)dx\hfill \\ & =& {\int }_0^{\infty }\frac{1}{{\zeta }_1^3}x{e}^{-\frac{1}{{\zeta }_1}x}f\left(x,y,t\right)dx\hfill \\ & & -{\int }_0^{\infty }\frac{1}{{\zeta }_1^2}{e}^{-\frac{1}{{\zeta }_1}x}f\left(x,y,t\right)dx,\hfill \end{array}$$

(7)

then substituting Equation (7) into Equation (6), we obtain

$$\begin{array}{ccc}\hfill \frac{\partial F({\zeta }_1,{\zeta }_2,s)}{\partial {\zeta }_1}& =& {\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{{\zeta }_1^3}x{e}^{-\frac{1}{{\zeta }_1}x}f\left(x,y,t\right)dx\right)dydt\hfill \\ & & -{\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{{\zeta }_1^2}{e}^{-\frac{1}{{\zeta }_1}x}f\left(x,y,t\right)dx\right)dydt,\hfill \end{array}$$

(8)

and by taking the derivative according to ${\zeta }_2$ for Equation (8), we achieve

$$\begin{array}{ccc}\hfill \frac{{\partial }^{2}F({\zeta }_1,{\zeta }_2,s)}{\partial {\zeta }_1\partial {\zeta }_2}& =& \frac{s^{\alpha }}{{\zeta }_1^3}{\int }_0^{\infty }{\int }_0^{\infty }x{e}^{-\left(\frac{1}{{\zeta }_1}x+\frac{1}{s}t\right)}\left({\int }_0^{\infty }{e}^{-\frac{1}{{\zeta }_2}y}\left(\frac{1}{{\zeta }_2^3}y-\frac{1}{{\zeta }_2^2}\right)f\left(x,y,t\right)\right)dxdydt\hfill \\ & & -\frac{s^{\alpha }}{{\zeta }_1^2}{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{{\zeta }_1}x+\frac{1}{s}t\right)}\left({\int }_0^{\infty }{e}^{-\frac{1}{{\zeta }_2}y}\left(\frac{1}{{\zeta }_2^3}y-\frac{1}{{\zeta }_2^2}\right)f\left(x,y,t\right)\right)dxdydt.\hfill \end{array}$$

(9)

Then, Equation (9), becomes

$$\begin{array}{ccc}\hfill \frac{{\partial }^{2}F({\zeta }_1,{\zeta }_2,s)}{\partial {\zeta }_1\partial {\zeta }_2}& =& \frac{1}{{\zeta }_1^2{\zeta }_2^2}{S}_x{S}_y{G}_t\left[xyf\left(x,y,t\right)\right]-\frac{1}{{\zeta }_1^2{\zeta }_2}{S}_x{S}_y{G}_t\left[xf\left(x,y,t\right)\right]\hfill \\ & & -\frac{1}{{\zeta }_1{\zeta }_2^2}{S}_x{S}_y{G}_t\left[yf\left(x,y,t\right)\right]+\frac{1}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[f\left(x,y,t\right)\right],\hfill \end{array}$$

(10)

by arranging the above equation, we achieve

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[xy\left(x,y,t\right)\right]& =& {\zeta }_1^2{\zeta }_2^2\frac{{\partial }^{2}F({\zeta }_1,{\zeta }_2,s)}{\partial {\zeta }_1\partial {\zeta }_2}+{\zeta }_1^2{\zeta }_2\frac{\partial F({\zeta }_1,{\zeta }_2,s)}{\partial {\zeta }_1}\hfill \\ & & +{\zeta }_1{\zeta }_2^2\frac{\partial F({\zeta }_1,{\zeta }_2,s)}{\partial {\zeta }_2}+{\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s),\hfill \end{array}$$

and thus, through simplifying, we obtain

$$S_x{S}_y{G}_t\left[xyf\left(x,y,t\right)\right]={\zeta }_1{\zeta }_2\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right).$$

The proof is finished. □

The following theorem offers the double Sumudu-generalized Laplace transform of the partial derivatives $xy{\psi }_t$.

Theorem 3.

The double Sumudu-generalized Laplace transform of the partial derivatives $xy{\psi }_t$ is presented by

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[xy{\psi }_t\right]& =& \frac{{\zeta }_1{\zeta }_2}{s}\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2\Psi ({\zeta }_1,{\zeta }_2,s)\right)\hfill \\ & & -{\zeta }_1{\zeta }_2{s}^{\alpha }\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2\Psi ({\zeta }_1,{\zeta }_2,0)\right),\hfill \end{array}$$

(11)

Proof. 

By taking the partial derivative according to ${\zeta }_1$ for Equation (22), we have

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\zeta }_1}\left(S_x{S}_y{G}_t\left[{\psi }_t\right]\right)& =& \frac{\partial }{\partial {\zeta }_1}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_1}x+\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}{\psi }_{t}dxdydt,\hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{\partial }{\partial {\zeta }_1}\frac{1}{{\zeta }_1}{e}^{-\frac{1}{{\zeta }_1}x}{\psi }_{t}dx\right)dydt,\hfill \end{array}$$

(12)

we calculate the partial derivative inside brackets as shown below:

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }\frac{\partial }{\partial {\zeta }_1}\frac{1}{{\zeta }_1}{e}^{-\frac{1}{{\zeta }_1}x}\frac{\partial \psi }{\partial t}dx& =& {\int }_0^{\infty }\left(\frac{1}{{\zeta }_1^3}x-\frac{1}{{\zeta }_1^2}\right)e^{-\frac{1}{{\zeta }_1}x}{\psi }_{t}dx\hfill \\ & =& {\int }_0^{\infty }\frac{1}{{\zeta }_1^3}x{e}^{-\frac{1}{{\zeta }_1}x}{\psi }_{t}dx\hfill \\ & & -{\int }_0^{\infty }\frac{1}{{\zeta }_1^2}{e}^{-\frac{1}{{\zeta }_1}x}{\psi }_{t}dx,\hfill \end{array}$$

(13)

substituting Equation (13) into Equation (12), we obtain

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\zeta }_1}\left(S_x{S}_y{G}_t\left[{\psi }_t\right]\right)& =& {\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{{\zeta }_1^3}x{e}^{-\frac{1}{{\zeta }_1}x}{\psi }_{t}dx\right)dydt\hfill \\ & & -{\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{{\zeta }_1^2}{e}^{-\frac{1}{{\zeta }_1}x}{\psi }_{t}dx\right)dydt,\hfill \end{array}$$

(14)

by taking the partial derivative according to ${\zeta }_2$ for Equation (14),

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left(S_x{S}_y{G}_t\left[{\psi }_t\right]\right)& =& \frac{\partial }{\partial {\zeta }_2}\left({\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{{\zeta }_1^3}x{e}^{-\frac{1}{{\zeta }_1}x}{\psi }_{t}dx\right)dydt\right)\hfill \\ & & -\frac{\partial }{\partial {\zeta }_2}\left({\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{{\zeta }_2}{e}^{-\left(\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{{\zeta }_1^2}{e}^{-\frac{1}{{\zeta }_1}x}{\psi }_{t}dx\right)dydt\right),\hfill \\ & & \hfill \end{array}$$

(15)

and therefore, Equation (15) becomes

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left(S_x{S}_y{G}_t\left[{\psi }_t\right]\right)& =& \frac{1}{{\zeta }_1^2{\zeta }_2^2}\left(\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{{\zeta }_1}x+\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}xy{\psi }_{t}dxdydt\right)\hfill \\ & & +\frac{1}{{\zeta }_1{\zeta }_2}\left(\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{{\zeta }_1}x+\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}{\psi }_{t}dxdydt\right)\hfill \\ & & -\frac{1}{{\zeta }_1{\zeta }_2^2}\left(\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{{\zeta }_1}x+\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}y{\psi }_{t}dxdydt\right)\hfill \\ & & -\frac{1}{{\zeta }_1^2{\zeta }_2}\left(\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{{\zeta }_1}x+\frac{1}{{\zeta }_2}y+\frac{1}{s}t\right)}x{\psi }_{t}dxdydt\right),\hfill \end{array}$$

(16)

and hence,

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial u_1\partial u_2}\left(S_x{S}_y{G}_t\left[{\psi }_t\right]\right)& =& \frac{1}{{\zeta }_1^2{\zeta }_2^2}{S}_x{S}_y{G}_t\left[xy{\psi }_t\right]+\frac{1}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[{\psi }_t\right]\hfill \\ & & -\frac{1}{{\zeta }_1{\zeta }_2^2}{S}_x{S}_y{G}_t\left[y{\psi }_t\right]-\frac{1}{{\zeta }_1^2{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\psi }_t\right],\hfill \end{array}$$

(17)

by reordering Equation (17), we prove Equation (11):

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[xy{\psi }_t\right]& =& \frac{{\zeta }_1{\zeta }_2}{s}\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2\Psi ({\zeta }_1,{\zeta }_2,s)\right)\hfill \\ & & -{\zeta }_1{\zeta }_2{s}^{\alpha }\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2\Psi ({\zeta }_1,{\zeta }_2,0)\right),\hfill \end{array}$$

□

Double Sumudu transform of function $\psi (x,t)$ is given by $S_x{S}_t\left[\psi \left(x,t\right)\right]=\psi ({\zeta }_1,s)$. Moreover, the Sumudu-generalized Laplace transform of ${\psi }_x,$${\psi }_{xx}$, ${\psi }_{tt}$ and ${\psi }_t$ is determined by

$$\begin{array}{ccc}\hfill S_x{G}_t\left[{\psi }_x\right]& =& \frac{\psi ({\zeta }_1,s)-\psi \left(0,s\right)}{{\zeta }_1},\hfill \\ \hfill S_x{G}_t\left({\psi }_{xx}\right)& =& \frac{\psi ({\zeta }_1,s)}{{\zeta }_1^2}-\frac{\psi (0,s)}{{\zeta }_1^2}-\frac{{\psi }_x(0,s)}{{\zeta }_1}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill S_x{G}_t\left[{\psi }_t\right]& =& \frac{\psi ({\zeta }_1,s)}{s}-s^{\alpha }\psi \left({\zeta }_1,0\right),\hfill \\ \hfill S_x{G}_t\left({\psi }_{tt}\right)& =& \frac{\psi ({\zeta }_1,s)}{s^2}-s^{\alpha -1}\psi \left({\zeta }_1,0\right)-s^{\alpha }{\psi }_t\left({\zeta }_1,0\right).\hfill \end{array}$$

Double Sumudu-generalized Laplace transform of the function $\psi (x,y,t)$ is presented by $S_x{S}_y{G}_t\left[\psi \left(x,y,t\right)\right]=\psi ({\zeta }_1,{\zeta }_2,s)$; therefore, the double Sumudu -generalized Laplace transform of ${\psi }_x,$${\psi }_{xx},$${\psi }_t$ and ${\psi }_{tt}$ is provided by

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[{\psi }_x\right]& =& \frac{\psi ({\zeta }_1,{\zeta }_2,s)-\psi \left(0,{\zeta }_2,s\right)}{{\zeta }_1},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left({\psi }_{xx}\right)& =& \frac{\psi ({\zeta }_1,{\zeta }_2,s)}{{\zeta }_1^2}-\frac{\psi \left(0,{\zeta }_2,s\right)}{{\zeta }_1^2}-\frac{{\psi }_x(0,{\zeta }_2,s)}{{\zeta }_1}\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[{\psi }_y\right]& =& \frac{\psi ({\zeta }_1,{\zeta }_2,s)-\psi \left({\zeta }_1,0,s\right)}{{\zeta }_2},\hfill \\ \hfill S_x{S}_y{G}_t\left({\psi }_{yy}\right)& =& \frac{\psi ({\zeta }_1,{\zeta }_2,s)}{{\zeta }_2^2}-\frac{\psi \left({\zeta }_1,0,s\right)}{{\zeta }_2^2}-\frac{{\psi }_y\left({\zeta }_1,0,s\right)}{{\zeta }_2}\hfill \end{array}$$

(20)

and

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[{\psi }_t\right]& =& \frac{\psi ({\zeta }_1,{\zeta }_2,s)}{s}-s^{\alpha }\psi \left({\zeta }_1,{\zeta }_2,0\right),\hfill \\ \hfill S_x{S}_y{G}_t\left({\psi }_{tt}\right)& =& \frac{\psi ({\zeta }_1,{\zeta }_2,s)}{s^2}-s^{\alpha -1}\psi \left({\zeta }_1,{\zeta }_2,0\right)-s^{\alpha }{\psi }_t\left({\zeta }_1,{\zeta }_2,0\right).\hfill \end{array}$$

3. Double Sumudu-Generalized Laplace Decomposition Method and 2+1-Dimensional Linear Pseudoparabolic Equation

The following is the procedure showing two problems that are concerning to the linear and nonlinear singular 2+1-D pseudoparabolic equation.

The general singular 2+1-D pseudo parabolic equation is considered as follows:

$${\psi }_t=\frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}{\left(y{\psi }_y\right)}_y+\frac{1}{x}{\left(x{\psi }_x\right)}_{xt}+f\left(x,y,t\right),$$

(21)

dependent on the initial condition

$$\psi \left(x,y,0\right)=f_1\left(x,y\right),$$

(22)

where the functions $f_1\left(x,y\right)$ and $f\left(x,y,t\right)$. Firstly, we obtain the product of both sides of Equation (21) by $xy$, and implementing double Sumudu-generalized Laplace, we yield Equation (21), and implementing double Sumudu transform for Equation (22), we obtain

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[xy{\psi }_t\right]& =& S_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y\right]\hfill \\ & & +S_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_{xt}+xyf\left(x,y,t\right)\right],\hfill \end{array}$$

(23)

and by arranging Equation (23), it becomes

$$\begin{array}{ccc}& & \frac{{\zeta }_1{\zeta }_2}{s}\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2\Psi ({\zeta }_1,{\zeta }_2,s)\right)\hfill \\ & =& {\zeta }_1{\zeta }_2{s}^{\alpha }\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2)\right)\hfill \\ & & +S_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y+yy{\left(x{\psi }_x\right)}_{xt}\right]\hfill \\ & & +{\zeta }_1{\zeta }_2\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right),\hfill \end{array}$$

(24)

and therefore, Equation (24) becomes

$$\begin{array}{ccc}& & \frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2\Psi ({\zeta }_1,{\zeta }_2,s)\right)\hfill \\ & =& s^{\alpha +1}\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2)\right)\hfill \\ & & +\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y+yy{\left(x{\psi }_x\right)}_{xt}\right]\hfill \\ & & +s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right).\hfill \end{array}$$

(25)

By taking the integral for Equation (25), from 0 to ${\zeta }_1$ and 0 to ${\zeta }_2$ according to ${\zeta }_1$ and ${\zeta }_2$, we have

$$\begin{array}{ccc}& & \Psi ({\zeta }_1,{\zeta }_2,s)\hfill \\ & =& \frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}{s}^{\alpha +1}\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2)\right)d{\zeta }_{1}d{\zeta }_2\hfill \\ & & +\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\hfill \\ & & +\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\hfill \\ & & +\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right)d{\zeta }_{1}d{\zeta }_2.\hfill \end{array}$$

(26)

The solution is obtained by using the inverse double Sumudu-generalized Laplace for Equation (26):

$$\begin{array}{ccc}\hfill \psi \left(x,y,t\right)& =& f_1\left(x,y\right)\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_y\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right)d{\zeta }_{1}d{\zeta }_2\right].\hfill \end{array}$$

(27)

where $S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}$ indicates the inverse double Sumudu-generalized Laplace. The double Sumudu-generalized Laplace decomposition method (DSGLTDM) defines the solutions $\psi (x,y,t)$ with the help of infinite series as

$$\psi \left(x,y,t\right)=\sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right).$$

(28)

By substituting Equation (28) into Equation (27), we receive

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right)& =& f_1\left(x,y\right)+S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right)d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\psi }_{nx}\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y\sum _{n=0}^{\infty }{\psi }_{ny}\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\psi }_{nx}\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\right].\hfill \end{array}$$

(29)

By matching both sides of Equation (29), we obtain

$$\begin{array}{ccc}\hfill {\psi }_0\left(x,y,t\right)& =& f_1\left(x,y\right)\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right)d{\zeta }_{1}d{\zeta }_2\right].\hfill \end{array}$$

(30)

In general, the remaining terms are given by

$$\begin{array}{ccc}& & {\psi }_{n+1}\left(x,y,t\right)=\hfill \\ & & S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{nx}\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_{ny}\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{nx}\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\right].\hfill \end{array}$$

(31)

where the inverse double Sumudu-generalized Laplace transform is given by $S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}$. Here, we assume that the inverse exists for Equations (30) and (31). In order to demonstrate the advantages and the preciseness of the (DSGLTDM) for solving singular 2+1-D parabolic equations, we employed the method proposed in the example below.

Example 1.

Consider a singular 2+1-D parabolic equation

$${\psi }_t=\frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}{\left(y{\psi }_y\right)}_y+\left(x^2-y^2\right)cost$$

(32)

subject to the initial condition

$$\psi \left(x,0\right)=0,$$

(33)

and by utilizing Equation (27), we have

$$\begin{array}{ccc}\hfill \psi \left(x,y,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_y\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left(2{\zeta }_1^2-{\zeta }_2^2\right)\left[s^{\alpha +2}-s^{\alpha +4}+s^{\alpha +6}-s^{\alpha +8}+\dots \right].\hfill \end{array}$$

(34)

Therefore,

$$\begin{array}{ccc}\hfill \psi \left(x,y,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_y\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +\left(x^2-y^2\right)\left[t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+\dots \right].\hfill \end{array}$$

our desired recursive relation is given by

$${\psi }_0=\left(x^2-y^2\right)\left[t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+\dots \right],$$

and

$$\begin{array}{ccc}\hfill {\psi }_{n+1}\left(x,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{nx}\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_{ny}\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right],\hfill \end{array}$$

for $n=0,1,2,\dots$; hence, at $n=0$,

$$\begin{array}{ccc}\hfill {\psi }_1\left(x,y,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{0x}\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_{0y}\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[4xy\left(t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+\dots \right)\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & -S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[4xy\left(t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+\dots \right)\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ \hfill {\psi }_1\left(x,y,t\right)& =& 0\hfill \end{array}$$

at $n=1$

$${\psi }_2\left(x,y,t\right)=0$$

and at $n=2,$

$${\psi }_2\left(x,y,t\right)=0.$$

By applying Equation (28), we obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right)& =& {\psi }_0+{\psi }_1+{\psi }_2+\dots .\hfill \\ \hfill \psi \left(x,y,t\right)& =& \left(x^2-y^2\right)\left[t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+\dots \right].\hfill \end{array}$$

Thus, the solution of Equation (32) is given by

$$\psi \left(x,y,t\right)=\left(x^2-y^2\right)sint.$$

Example 2.

Consider the singular 2+1-D pseudoparabolic equation given by

$$\begin{array}{ccc}\hfill {\psi }_t& =& \frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}{\left(y{\psi }_y\right)}_y\hfill \\ & & +\frac{1}{x}{\left(x{\psi }_x\right)}_{xt}+\left(x^2-y^2\right)e^t-4e^t,\hfill \end{array}$$

(35)

subject to the initial condition

$$\psi \left(x,y,0\right)=x^2-y^2.$$

(36)

For the purpose of keep on with our method for Equation (35), we obtain

$$\begin{array}{ccc}\hfill {\psi }_0& =& \left(x^2-y^2\right)\left[1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots \right]\hfill \\ & & -4\left[t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots \right],\hfill \end{array}$$

the exponential function formula is defined by

$$e^t=1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots$$

and

$${\psi }_{n+1}\left(x,y,t\right)=S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[{\Delta }_n\right]d{\zeta }_{1}d{\zeta }_2\right],$$

where

$${\Delta }_n=y{\left(x{\psi }_{nx}\right)}_x+x{\left(y{\psi }_{ny}\right)}_y+y{\left(x{\psi }_{nx}\right)}_{xt},$$

at $n=0$, we have

$${\psi }_1\left(x,y,t\right)=S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[{\Delta }_0\right]d{\zeta }_{1}d{\zeta }_2\right],$$

and therefore,

$$\begin{array}{ccc}\hfill {\Delta }_0& =& y{\left(x{\psi }_{0x}\right)}_x+x{\left(y{\psi }_{0y}\right)}_y+y{\left(x{\psi }_{0x}\right)}_{xt}\hfill \\ & =& 4xy\left[1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots \right],\hfill \end{array}$$

hence

$$\begin{array}{ccc}\hfill {\psi }_1\left(x,y,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[4\left(s^{\alpha +2}+s^{\alpha +3}+s^{\alpha +4}+s^{\alpha +5}+\dots \right)\right],\hfill \\ & =& 4\left[t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots \right].\hfill \end{array}$$

In a similar way, at $n=1$, we have

$${\psi }_2\left(x,y,t\right)=0.$$

In the same way, at $n=1$, we obtain

$${\psi }_3\left(x,y,t\right)=0.$$

Therefore, the approximate solution of Equation (35) is presented as follows:

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right)& =& {\psi }_0+{\psi }_1+{\psi }_2+\dots .\hfill \\ \hfill \psi \left(x,y,t\right)& =& \left(x^2-y^2\right)\left[1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots \right]\hfill \\ & & -4\left[t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots \right]\hfill \\ & & +4\left[t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots \right]\hfill \\ \hfill \psi \left(x,y,t\right)& =& \left(x^2-y^2\right)\left[1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots \right].\hfill \end{array}$$

Hence,

$$\psi \left(x,y,t\right)=\left(x^2-y^2\right)e^t.$$

4. Double Sumudu-Generalized Laplace Decomposition Method and 2+1-Dimensional Nonlinear Pseudoparabolic Equation

We must illustrate the double Sumudu-generalized Laplace decomposition method to solve the singular 2+1-dimensional nonlinear pseudoparabolic equation:

Consider the following general form of the singular 2+1-dimensional nonlinear pseudoparabolic equation of the model:

$$\begin{array}{ccc}\hfill {\psi }_t& =& \frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}{\left(y{\psi }_y\right)}_y+\frac{1}{x}{\left(x{\psi }_x\right)}_{xt}\hfill \\ & & -2\mu \left(x\right){\psi }_x{\psi }_{xx}+\nu \left(x\right){\left({\psi }_x\right)}^2+f\left(x,y,t\right)\hfill \end{array}$$

(37)

with initial condition

$$\psi \left(x,y,0\right)=h_1\left(x,y\right),$$

(38)

where the functions $\mu \left(x\right)$ and $\nu \left(x\right)$ are arbitrary. With a view to procure the solution of Equation (37), first, obtain the product of both sides of Equation (37) by $xy$, and implementing double the Sumudu-generalized Laplace transform, we find

$$\begin{array}{ccc}& & S_x{S}_y{G}_t\left[xy{\psi }_t\right]=S_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y+y{\left(x{\psi }_x\right)}_{xt}\right]\hfill \\ & & +S_x{S}_y{G}_t\left[-2xy\mu \left(x\right){\psi }_x{\psi }_{xx}+xy\nu \left(x\right){\left({\psi }_x\right)}^2+xyf\left(x,y,t\right)\right].\hfill \end{array}$$

(39)

Second, applying theorem 1 and 2 into Equation (39), we obtain

$$\begin{array}{ccc}& & \frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2\Psi ({\zeta }_1,{\zeta }_2,s)\right)\hfill \\ & =& s^{\alpha +1}\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2{H}_1({\zeta }_1,{\zeta }_2)\right)\hfill \\ & & +\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y+y{\left(x{\psi }_x\right)}_{xt}\right]\hfill \\ & & -\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[2xy\mu \left(x\right){\psi }_x{\psi }_{xx}-xy\nu \left(x\right){\left({\psi }_x\right)}^2\right]\hfill \\ & & +s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right).\hfill \end{array}$$

(40)

By taking the integral for Equation (40), from 0 to ${\zeta }_1$ and 0 to ${\zeta }_2$ according to ${\zeta }_1$ and ${\zeta }_2$, we have

$$\begin{array}{ccc}& & \Psi ({\zeta }_1,{\zeta }_2,s)\hfill \\ & =& \frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}{s}^{\alpha +1}\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_2{H}_1({\zeta }_1,{\zeta }_2)\right)d{\zeta }_{1}d{\zeta }_2\hfill \\ & & +\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\hfill \\ & & +\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\hfill \\ & & -\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[2xy\mu \left(x\right){\psi }_x{\psi }_{xx}-xy\nu \left(x\right){\left({\psi }_x\right)}^2\right]d{\zeta }_{1}d{\zeta }_2\hfill \\ & & +\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right)d{\zeta }_{1}d{\zeta }_2.\hfill \end{array}$$

(41)

The third step, now using inverse double Sumudu-generalized Laplace for both sides of Equation (41), the solution of Equation (37) can be written as

$$\begin{array}{ccc}\hfill \psi \left(x,y,t\right)& =& h_1\left(x,y\right)\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_y\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & -S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[2xy\mu \left(x\right){\psi }_x{\psi }_{xx}\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[xy\nu \left(x\right){\left({\psi }_x\right)}^2\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right)d{\zeta }_{1}d{\zeta }_2\right].\hfill \end{array}$$

(42)

By substituting Equation (28) into Equation (42), we obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right)& =& h_1\left(x,y\right)\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\psi }_{nx}\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y\sum _{n=0}^{\infty }{\psi }_{ny}\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\psi }_{nx}\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & -S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[2xy\mu \left(x\right)A_n\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[xy\nu \left(x\right)B_n\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right)d{\zeta }_{1}d{\zeta }_2\right].\hfill \end{array}$$

(43)

where $n=0,1,2,\dots$. Hence, from Equation (43) above, we have

$$\begin{array}{ccc}\hfill {\psi }_0\left(x,y,t\right)& =& h_1\left(x,y\right)+t{g}_2\left(x,y\right)\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}s\frac{{\partial }^2}{\partial {\zeta }_1\partial {\zeta }_2}\left({\zeta }_1{\zeta }_{2}F({\zeta }_1,{\zeta }_2,s)\right)d{\zeta }_{1}d{\zeta }_2\right]\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\psi }_{n+1}\left(x,y,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{nx}\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_{ny}\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{nx}\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & -S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[2xy\mu \left(x\right)A_n\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[xy\nu \left(x\right)B_n\right]d{\zeta }_{1}d{\zeta }_2\right],\hfill \end{array}$$

where nonlinear terms $A_n$ and $B_n$ are addressed as

$$A_n=\sum _{n=0}^{\infty }{\psi }_{nx}{\psi }_{nxx},B_n=\sum _{n=0}^{\infty }{\left({\psi }_{nx}\right)}^2,$$

(44)

where the nonlinear terms ${\psi }_x{\psi }_{xx}$ and ${\left({\psi }_x\right)}^2$ are in the following forms:

$$\begin{array}{cc}\hfill A_0& ={\psi }_{0x}{\psi }_{0xx}\hfill \\ \hfill A_1& ={\psi }_{0x}{\psi }_{1xx}+{\psi }_{x1}{\psi }_{0xx},\hfill \\ \hfill A_2& ={\psi }_{0x}{\psi }_{2xx}+{\psi }_{1x}{\psi }_{1xx}+{\psi }_{2x}{\psi }_{0xx},\hfill \\ \hfill A_3& ={\psi }_{0x}{\psi }_{3xx}+{\psi }_{1x}{\psi }_{2xx}+{\psi }_{2x}{\psi }_{1xx}+{\psi }_{3x}{\psi }_{0xx}.\hfill \end{array}$$

(45)

and

$$\begin{array}{ccc}\hfill B_0& =& {\left({\psi }_{0x}\right)}^2\hfill \\ \hfill B_1& =& 2{\psi }_{0x}{\psi }_{1x}\hfill \\ \hfill B_2& =& 2{\psi }_{0x}{\psi }_{2x}+{\left({\psi }_{1x}\right)}^2\hfill \\ \hfill B_3& =& 2{\psi }_{0x}{\psi }_{3x}+2{\psi }_{1x}{\psi }_{2x}.\hfill \end{array}$$

(46)

To ascribe this method to a linear singular 2+1-dimensional nonlinear pseudoparabolic equation, we give the next example.

Example 3.

The singular 2+1-dimensional nonlinear pseudoparabolic equation is offered by

$$\begin{array}{ccc}\hfill {\psi }_t& =& \frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}{\left(y{\psi }_y\right)}_y+\frac{1}{x}{\left(x{\psi }_x\right)}_{xt}\hfill \\ & & -2\mu \left(x\right){\psi }_x{\psi }_{xx}+\nu \left(x\right){\left({\psi }_x\right)}^2\hfill \\ & & +2\left(x^2-y^2\right)e^{2t}-8e^{2t}\hfill \end{array}$$

(47)

with the initial condition

$$\psi \left(x,y,0\right)=x^2-y^2.$$

(48)

In order to flow with our method for Equation (47), we obtain

$$\begin{array}{ccc}\hfill {\psi }_0\left(x,y,t\right)& =& \left(x^2-y^2\right)\left(1+2t+\frac{{\left(2t\right)}^2}{2!}+\frac{{\left(2t\right)}^3}{3!}+\frac{{\left(2t\right)}^4}{4!}+\dots \right)\hfill \\ & & -8\left(t+\frac{2t^2}{2!}+\frac{4t^3}{3!}+\frac{8t^4}{4!}+\dots \right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\psi }_{n+1}\left(x,y,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{nx}\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_{ny}\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{nx}\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & -S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[2xy\mu \left(x\right)A_n\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[xy\nu \left(x\right)B_n\right]d{\zeta }_{1}d{\zeta }_2\right],\hfill \end{array}$$

the first repeat at $n=0$ is denoted by

$$\begin{array}{ccc}\hfill {\psi }_1\left(x,y,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{0x}\right)}_x\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[x{\left(y{\psi }_{0y}\right)}_y\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[y{\left(x{\psi }_{0x}\right)}_{xt}\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & -S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[2xy\mu \left(x\right)A_0\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & & +S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[xy\nu \left(x\right)B_0\right]d{\zeta }_{1}d{\zeta }_2\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\psi }_1\left(x,y,t\right)& =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\zeta }_1{\zeta }_2}{\int }_0^{{\zeta }_1}{\int }_0^{{\zeta }_2}\frac{s}{{\zeta }_1{\zeta }_2}{S}_x{S}_y{G}_t\left[8xy\left[1+\frac{4t^2}{2!}+\frac{12t^3}{3!}+\frac{32t^4}{4!}+\dots \right]\right]d{\zeta }_{1}d{\zeta }_2\right]\hfill \\ & =& S_{{\zeta }_1}^{-1}{S}_{{\zeta }_2}^{-1}{G}_s^{-1}\left[8\left[s^{\alpha +2}+2s^{\alpha +3}+4s^{\alpha +4}+12s^{\alpha +5}+\dots \right]\right]\hfill \\ & =& 8\left(t+t+\frac{2t^2}{2!}+\frac{4t^3}{3!}+\frac{8t^4}{4!}+\dots \right),\hfill \end{array}$$

at $n=1$, we have

$${\psi }_2\left(x,y,t\right)=0$$

and letting $n=2$, we obtain

$${\psi }_3\left(x,y,t\right)=0,$$

and therefore, by using Equation (28), the series solutions are denoted by

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right)& =& {\psi }_0+{\psi }_1+{\psi }_2+\dots .\hfill \\ & =& \left(x^2-y^2\right)\left(1+2t+\frac{{\left(2t\right)}^2}{2!}+\frac{{\left(2t\right)}^3}{3!}+\frac{{\left(2t\right)}^4}{4!}+\dots \right)\hfill \end{array}$$

and therefore, the accurate solutions become

$$\psi \left(x,y,t\right)=\left(x^2-y^2\right)e^{2t}.$$

5. Conclusions

In this study, we established a new hybrid method, which is named the double Sumudu-generalized Laplace transform decomposition method (DSGLTDM). This approach is successfully applied in singular 2+1-D pseudoparabolic equations. The (DSGLTDM) is an analytical process and works by utilizing the initial conditions only. Three examples are offered to examine the accuracy of the method.