Application of the Double Sumudu-Generalized Laplace Transform Decomposition Method to Solve Singular Pseudo-Hyperbolic Equations

Abstract

In this study, the technique established by the double Sumudu transform in combination with a new generalized Laplace transform decomposition method, which is called the double Sumudu-generalized Laplace transform decomposition method, is applied to solve general two-dimensional singular pseudo-hyperbolic equations subject to the initial conditions. The applicability of the proposed method is analyzed through demonstrative examples. The results obtained show that the procedure is easy to carry out and precise when employed for different linear and non-linear partial differential equations.

1. Introduction

Nonlinear partial differential equations are capable of explaining several phenomena, for instance, diffusion, electrodynamics, fluid dynamics, elasticity, etc. The majority of problems in physics and engineering can be represented by nonlinear differential equations. Pseudo-hyperbolic equations are a mathematical model that appear mostly in the theory of longitudinal and sideways vibrations of pseudo hyperbolic elastic bars (beams). The pseudo-hyperbolic equation is a partial differential containing mixed partial derivatives with respect to time and space. However, it is difficult to find clear solutions to linear and nonlinear pseudo-hyperbolic equations in general. There are several methods that have been discussed to solve some pseudo-hyperbolic equations, such as the residual power series method [1], the H1-Galerkin mixed finite element method in [2], the finite difference method [3], and the modified double Laplace transform method [4]. A sufficient condition for the nonexistence of weak solutions to the nonlinear pseudo-hyperbolic equation has been offered in the Heisenberg theory [5]. The definition of the parabolicity of systems of equations with partial derivatives, which generalizes the concept of Petrovsky parabolicity, has been developed in [6]. Established on this explanation, a priori evaluations of pseudo hyperbolic PDEs with constant coefficients are conducted and the solvability of the Cauchy problem is formulated in [7]. The modified Laplace Adomian decomposition is a famous method that was obtained by combining the Laplace transform and Adomian decomposition, see [8]. This variety of equations was discussed in general non-fixed waves in a layered and spinning liquid, see [9]. The approximate solution of multidimensional diffusion problems under an ABC fractional order derivative has been discussed by Nadeem et al. in [10]. The authors of [11] developed a new method which is called a fractional analytical scheme in order to obtain the approximate results of fourth-order parabolic fractional partial differential equations. The authors of [12] have examined the solution of singular linear and nonlinear one-dimensional pseudo-hyperbolic equations through the double Laplace decomposition method. Lately, a strong and useful method has been proposed to find approximate analytical solutions to partial differential equations, which is named the double Laplace decomposition method [13]. The Adomian decomposition method and series extension method have effectively been applied to obtain the solution for diffusion equations problems, see [14]. In [15], the authors discussed the solution of the heat equation of fractional order in two dimensions by recommending the so-called triple Laplace transform method. The approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation was obtained by using the double Sumudu transform iterative method in [16]. The authors of [17] studied the solution of nonlinear dynamical models with non-integer order derivatives by using a Laplace-type integral transform coupled with Adomian’s approach. Recently the double Sumudu-generalized Laplace decomposition method was applied to obtain the solution of a 2 + 1-pseudo parabolic equation, see [18]. The main aim of this work is to present a novel theorem for the double Sumudu-generalized Laplace transform. Moreover, we offer the application of this new transformation to partial derivatives. Finally, by examining examples, we discover that (DSGLTDM) is a strong tool for the solution of the linear, nonlinear, and coupled singular ($2+1$-D) pseudo-hyperbolic equations in comparison to the Adomian decomposition method (ADM), the homotopy analysis method (HAM), and the variational iteration method (VAM). This technique opens up new ways for solving complicated linear and nonlinear partial differential equations related to different scientific and engineering fields.

Some notes: During this work, we utilize the following notation:

(1)

(SGLT) in place of “Sumudu-generalized Laplace transform”.

(2)

(GLT) in place of “generalized Laplace transform”.

(3)

(DST) in place of “double Sumudu transform”.

(4)

(DSGLT) in place of “double Sumudu-generalized Laplace transform”.

(5)

(ADM) in place of “Adomian decomposition method”.

(6)

(DSGLTDM) in place of “double Sumudu-generalized Laplace transform decomposition method”.

The rest of this work is arranged as follows. In Section 2, we start with some basic definitions and theorems of the existence condition of the (DSGLT) and the (DSGLT) of function $xyf\left(x,y,t\right)$, respectively. In Section 3, we study two theorems that are considered the main contribution in this work. Section 4 presents how to obtain an approximate analytical solution for the 2 + 1-dimensional linear and nonlinear singular pseudo-hyperbolic equation using the (DSGLTDM), and for each problem we offer an example. In Section 5, we study the approximate analytical solution for the singular ($2+1$-dimensional) coupled pseudo-hyperbolic equation by using the (DSGLTDM). Finally, the concluding remarks are given in Section 6.

2. Some Fundamental Ideas of the (SGLT)

In this section, we introduce the essential definitions of the (DST), (GLT), (SGLT) and (DSGLT).

Definition 1

([19]). The double Sumudu transform of the function $f(x,t)$ in the positive quadrant of the $xt$-plane is given by

$$F(u,v)=S_x{S}_y\left[f(x,t)\right]=\frac{1}{uv}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{e}^{-(\frac{x}{u}+\frac{t}{v})}f(x,t)dxdt.$$

The (GLT) of the function $f\left(t\right)$ is denoted by $G_{\alpha }$ in the following definition.

Definition 2

([20]). If $f\left(t\right)$ is an integrable function defined for all $t\ge 0$, its (GLT) $G_{\alpha }$ is the integral of $f\left(t\right)$ times $s^{\alpha }{e}^{-\frac{t}{s}}$ from $t=0$ to ∞. It is a function of s, say $F\left(s\right)$, and is denoted by $G_{\alpha }\left(f\left(t\right)\right)$; thus

$$F\left(s\right)=G_{\alpha }\left(f\left(t\right)\right)=s^{\mathrm{\alpha }}\underset{0}{\overset{\infty }{\int }}f\left(t\right)e^{-\frac{t}{s}}dt,$$

where $s\in \mathbb{C}$ and $\alpha \in Z.$

The (SGLT) of the function $f(x,t)$ is determined by $F(u_1,s)$ in the following definition.

Definition 3

([21]). The (SGLT) of the function $f(x,t)$ of two variables x and t defined in the first quadrant of the $xt$-plane by the double integral in the form of

$$\begin{array}{ccc}\hfill S_x{G}_t\left(f\left(x,t\right)\right)& =& F\left(u_1,s\right)\hfill \\ & =& \frac{s^{\alpha }}{u_1}{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{x}{u_1}+\frac{t}{s}\right)}f\left(x,t\right)dtdx\hfill \end{array}$$

(1)

$S_x{G}_t$ indicates the (SGLT) and the symbols $u_1$ and s denote transforms of the variables x and t in Sumudu and generalized Laplace transforms, respectively.

The inverse (SGLT) $S_{u_1}^{-1}{G}_s^{-1}\left[S_x{G}_t\left(f\left(x,t\right)\right)\right]=f\left(x,t\right)$ is determined by the following formula:

$$f\left(x,t\right)=\frac{1}{{\left(2\pi i\right)}^2}{\int }_{\tau -i\infty }^{\tau +i\infty }{\int }_{\nu -i\infty }^{\nu +i\infty }{e}^{\frac{1}{u_1}x+\frac{1}{s}t}{S}_x{G}_t\left[f\left(x,t\right)\right]dsd{u}_1,\tau ,\nu \ge 0$$

The definition of the (DSGLT) is given in the following.

Definition 4

([18]). Let $f(x,y,t)$ be a continuous function of three variables x, y and t. Then, the (DSGLT) is defined by

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left(f\left(x,y,t\right)\right)& =& F\left(u_1,u_2,s\right)=\frac{s^{\alpha }}{u_1{u}_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{x}{u_1}+\frac{y}{u_2}+\frac{t}{s}\right)}f\left(x,y,t\right)dtdydx\hfill \end{array}$$

(2)

where $S_x{S}_y{G}_t$ denotes the (DSGLT) and $u_{1,}{u}_2$ and s represent transforms of the variables x, y and t in the (DSGLT), respectively.

The existence condition of the (DSGLT) is given in the next theorem.

Theorem 1

([18]). The function $f\left(x,y,t\right)$ is defined on $(0,X),\left(0,Y\right)$ and $(0,T)$ 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 $\mathrm{Re}\frac{1}{u_1}>\frac{1}{{\lambda }_1},\mathrm{Re}\frac{1}{u_2}>\frac{1}{{\lambda }_2},\mathrm{Re}\frac{1}{s}>\frac{1}{\eta }$.

The following theorem is useful in this study.

Theorem 2

([18]). If the (DSGLT) of the function $f\left(x,y,t\right)$ is presented by $S_x{S}_y{G}_t\left(f\left(x,y,t\right)\right)=F(u_1,u_2,s)$, then the (DSGLT) of the functions

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

determined by

$$S_x{S}_y{G}_t\left[xyf\left(x,y,t\right)\right]=u_1{u}_2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right).$$

(3)

In the following, we address the (DSGLT) of the function ${\psi }_x,$${\psi }_{xx},$${\psi }_{tt},$ which is defined in [18] as follows

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[{\psi }_x\right]& =& \frac{\Psi (u_1,u_2,s)-\Psi (0,u_2,s)}{u_1},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left({\psi }_{xx}\right)& =& \frac{\Psi (u_1,u_2,s)}{u_1^2}-\frac{\psi \left(0,u_2,s\right)}{u_1^2}-\frac{{\psi }_x(0,u_2,s)}{u_1}\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[{\psi }_y\right]& =& \frac{\Psi (u_1,u_2,s)-\Psi (u_1,0,s)}{u_2},\hfill \\ \hfill S_x{S}_y{G}_t\left({\psi }_{yy}\right)& =& \frac{\Psi (u_1,u_2,s)}{u_2^2}-\frac{\Psi (u_1,0,s)}{u_2^2}-\frac{{\psi }_y(u_1,0,s)}{u_2}\hfill \end{array}$$

(6)

$$S_x{S}_y{G}_t\left[{\psi }_{tt}\right]=\frac{\Psi (u_1,u_2,s)}{s^2}-s^{\alpha -1}\Psi (u_1,u_2,0)-s^{\alpha }{\Psi }_t(u_1,u_2,0).$$

(7)

3. The Idea of the (DSGLT) to the Functions $\mathit{y}{\mathit{\psi }}_{\mathit{tt}}$, $\mathit{x}{\mathit{\psi }}_{\mathit{tt}}$ and $\mathit{xy}{\mathit{\psi }}_{\mathit{tt}}$

In this part, we study two theorems which are the main contributions of this paper; in the first theorem, we discuss how to apply the (DSGLT) to the functions $y{\psi }_{tt}$ and $x{\psi }_{tt},$ and in the second theorem, we apply the (DSGLT) to the function $xy{\psi }_{tt}.$

Theorem 3.

The (DSGLTs) of the partial derivatives $y{\psi }_{tt}$ and $x{\psi }_{tt}$ are denoted by

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[y{\psi }_{tt}\right]& =& \frac{u_2}{s^2}\frac{\partial }{\partial u_2}\left(u_2\Psi (u_1,u_2,s)\right)\hfill \\ & & -u_2{s}^{\alpha -1}\frac{\partial }{\partial u_2}\left(u_2\Psi (u_1,u_2,0)\right)\hfill \\ & & -u_2{s}^{\alpha }\frac{\partial }{\partial u_2}\left(u_2{\Psi }_t(u_1,u_2,0)\right),\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[x{\psi }_{tt}\right]& =& \frac{u_1}{s^2}\frac{\partial }{\partial u_1}\left(u_1\Psi (u_1,u_2,s)\right)\hfill \\ & & -u_1{s}^{\alpha -1}\frac{\partial }{\partial u_1}\left(u_1\Psi (u_1,u_2,0)\right)\hfill \\ & & -u_2{s}^{\alpha }\frac{\partial }{\partial u_1}\left(u_1{\Psi }_t(u_1,u_2,0)\right)\hfill \end{array}$$

(9)

Proof. 

On utilizing the partial derivatives with respect to $u_2$ for Equation (2), one can obtain

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial u_2}\left(S_x{S}_y{G}_t\left[{\psi }_{tt}\right]\right)& =& \frac{\partial }{\partial u_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_1{u}_2}{e}^{-\left(\frac{1}{u_1}x+\frac{1}{u_2}y+\frac{1}{s}t\right)}{\psi }_{tt}dxdydt,\hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_1}{e}^{-\left(\frac{1}{u_1}x+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{\partial }{\partial u_2}\frac{1}{u_2}{e}^{-\frac{1}{u_2}y}{\psi }_{tt}dy\right)dxdt,\hfill \end{array}$$

(10)

calculating the derivative in the above equation according to $u_2,$ we yield

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }\frac{\partial }{\partial u_2}\frac{1}{u_2}{e}^{-\frac{1}{u_2}y}{\psi }_{tt}dy& =& {\int }_0^{\infty }\left(\frac{1}{u_2^3}y-\frac{1}{u_1^2}\right)e^{-\frac{1}{u_2}y}{\psi }_{tt}dy\hfill \\ & =& {\int }_0^{\infty }\frac{1}{u_2^3}y{e}^{-\frac{1}{u_2}y}{\psi }_{tt}dy\hfill \\ & & -{\int }_0^{\infty }\frac{1}{u_1^2}{e}^{-\frac{1}{u_2}y}{\psi }_{tt}dy\hfill \end{array}$$

(11)

By placing Equation (11) into Equation (10), we get

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial u_2}\left(S_x{S}_y{G}_t\left[{\psi }_{tt}\right]\right)& =& {\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_1}{e}^{-\left(\frac{1}{u_1}x+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{u_2^3}y{e}^{-\frac{1}{u_2}y}{\psi }_{tt}dy\right)dxdt\hfill \\ & & -{\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_1}{e}^{-\left(\frac{1}{u_1}x+\frac{1}{s}t\right)}\left({\int }_0^{\infty }{\int }_0^{\infty }\frac{1}{u_2^2}{e}^{-\frac{1}{u_2}y}{\psi }_{tt}dy\right)dxdt,\hfill \end{array}$$

(12)

Equation (12) becomes

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial u_2}\left(S_x{S}_y{G}_t\left[{\psi }_{tt}\right]\right)& =& \frac{1}{u_2^2}\left(\frac{s^{\alpha }}{u_1{u}_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{u_1}x+\frac{1}{u_2}y+\frac{1}{s}t\right)}y{\psi }_{tt}dxdydt\right)\hfill \\ & & -\frac{1}{u_2}\left(\frac{s^{\alpha }}{u_1{u}_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{u_1}x+\frac{1}{u_2}y+\frac{1}{s}t\right)}{\psi }_{tt}dxdydt\right)\hfill \end{array}$$

(13)

consequently

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial u_2}\left(S_x{S}_y{G}_t\left[{\psi }_{tt}\right]\right)& =& \frac{1}{u_2^2}{S}_x{S}_y{G}_t\left[y{\psi }_{tt}\right]-\frac{1}{u_2}{S}_x{S}_y{G}_t\left[{\psi }_{tt}\right]\hfill \end{array}$$

(14)

By organizing Equation (14), we will obtain the proof for Equation (8) as follows

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[y{\psi }_{tt}\right]& =& \frac{u_2}{s^2}\frac{\partial }{\partial u_2}\left(u_2\Psi (u_1,u_2,s)\right)\hfill \\ & & -u_2{s}^{\alpha -1}\frac{\partial }{\partial u_2}\left(u_2\Psi (u_1,u_2,0)\right)\hfill \\ & & -u_2{s}^{\alpha }\frac{\partial }{\partial u_2}\left(u_2{\Psi }_t(u_1,u_2,0)\right)\hfill \end{array}$$

In the same way, we can prove Equation (8). □

Theorem 4.

The (DSGLT) of the partial derivatives $xy{\psi }_{tt}$ is specified by

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[xy{\psi }_{tt}\right]& =& \frac{u_1{u}_2}{s^2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,s)\right)\hfill \\ & & -u_1{u}_2{s}^{\alpha -1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,0)\right)\hfill \\ & & -u_1{u}_2{s}^{\alpha }\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Psi }_t(u_1,u_2,0)\right)\hfill \end{array}$$

(15)

Proof. 

By taking the partial derivative according to $u_1$ for Equation (2), we have

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial u_1}\left(S_x{S}_y{G}_t\left[{\psi }_{tt}\right]\right)& =& \frac{\partial }{\partial u_1}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_1{u}_2}{e}^{-\left(\frac{1}{u_1}x+\frac{1}{u_2}y+\frac{1}{s}t\right)}{\psi }_{tt}dxdydt,\hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_2}{e}^{-\left(\frac{1}{u_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{\partial }{\partial u_1}\frac{1}{u_1}{e}^{-\frac{1}{u_1}x}{\psi }_{tt}dx\right)dydt,\hfill \end{array}$$

(16)

we calculate the partial derivative inside the brackets as follows:

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }\frac{\partial }{\partial u_1}\frac{1}{u_1}{e}^{-\frac{1}{u_1}x}{\psi }_{tt}dx& =& {\int }_0^{\infty }\left(\frac{1}{u_1^3}x-\frac{1}{u_1^2}\right)e^{-\frac{1}{u_1}x}{\psi }_{tt}dx\hfill \\ & =& {\int }_0^{\infty }\frac{1}{u_1^3}x{e}^{-\frac{1}{u_1}x}{\psi }_{tt}dx\hfill \\ & & -{\int }_0^{\infty }\frac{1}{u_1^2}{e}^{-\frac{1}{u_1}x}{\psi }_{tt}dx\hfill \end{array}$$

(17)

Placing Equation (17) into Equation (16), we yield

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial u_1}\left(S_x{S}_y{G}_t\left[{\psi }_{tt}\right]\right)& =& {\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_2}{e}^{-\left(\frac{1}{u_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{u_1^3}x{e}^{-\frac{1}{u_1}x}{\psi }_{tt}dx\right)dydt\hfill \\ & & -{\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_2}{e}^{-\left(\frac{1}{u_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{u_1^2}{e}^{-\frac{1}{u_1}x}{\psi }_{tt}dx\right)dydt,\hfill \end{array}$$

(18)

the partial derivative with respect to $u_2$ for Equation (18) is computed as the following

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial u_1\partial u_2}\left(S_x{S}_y{G}_t\left[{\psi }_{tt}\right]\right)& =& \frac{\partial }{\partial u_2}\left({\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_2}{e}^{-\left(\frac{1}{u_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{u_1^3}x{e}^{-\frac{1}{u_1}x}{\psi }_{tt}dx\right)dydt\right)\hfill \\ & & -\frac{\partial }{\partial u_2}\left({\int }_0^{\infty }{\int }_0^{\infty }\frac{s^{\alpha }}{u_2}{e}^{-\left(\frac{1}{u_2}y+\frac{1}{s}t\right)}\left({\int }_0^{\infty }\frac{1}{u_1^2}{e}^{-\frac{1}{u_1}x}{\psi }_{tt}dx\right)dydt\right)\hfill \end{array}$$

(19)

Therefore, we rewrite Equation (19) as follows

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial u_1\partial u_2}\left(S_x{S}_y{G}_t\left[{\psi }_{tt}\right]\right)& =& \frac{1}{u_1^2{u}_2^2}\left(\frac{s^{\alpha }}{u_1{u}_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{u_1}x+\frac{1}{u_2}y+\frac{1}{s}t\right)}xy{\psi }_{tt}dxdydt\right)\hfill \\ & & +\frac{1}{u_1{u}_2}\left(\frac{s^{\alpha }}{u_1{u}_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{u_1}x+\frac{1}{u_2}y+\frac{1}{s}t\right)}{\psi }_{tt}dxdydt\right)\hfill \\ & & -\frac{1}{u_1{u}_2^2}\left(\frac{s^{\alpha }}{u_1{u}_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{u_1}x+\frac{1}{u_2}y+\frac{1}{s}t\right)}y{\psi }_{tt}dxdydt\right)\hfill \\ & & -\frac{1}{u_1^2{u}_2}\left(\frac{s^{\alpha }}{u_1{u}_2}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\left(\frac{1}{u_1}x+\frac{1}{u_2}y+\frac{1}{s}t\right)}x{\psi }_{tt}dxdydt\right),\hfill \end{array}$$

(20)

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

(21)

One can rearrange Equation (21) and apply Equation (7) to prove Equation (15)

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[xy{\psi }_{tt}\right]& =& \frac{u_1{u}_2}{s^2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,s)\right)\hfill \\ & & -u_1{u}_2{s}^{\alpha -1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,0)\right)\hfill \\ & & -u_1{u}_2{s}^{\alpha }\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Psi }_t(u_1,u_2,0)\right),\hfill \end{array}$$

□

4. Singular (2 + 1-D) Pseudo-Hyperbolic Equation and the (DSGLTDM)

Here, the (DSGLTDM) is considered for solving a 2 + 1-dimensional singular pseudo-hyperbolic equation.

Problem 1.

(DSGLTDM) is a helpful method for solving linear singular $2+1$-dimensional pseudo-hyperbolic equations.

Allow us to consider a general singular ($2+1$-dimensional) pseudo-hyperbolic equation in the following form

$$\begin{array}{ccc}\hfill {\psi }_{tt}& =& \frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}{\left(y{\psi }_y\right)}_y+\frac{1}{x}{\left(x{\psi }_x\right)}_{xt}\hfill \\ & & +\frac{1}{y}{\left(y{\psi }_y\right)}_{yt}+f\left(x_1,x_2,t\right),\hfill \end{array}$$

(22)

to the following initial conditions:

$$\begin{array}{ccc}\hfill \psi \left(x,y,0\right)& =& f_1\left(x,y\right)\hfill \\ \hfill {\psi }_t\left(x,y,0\right)& =& f_2\left(x,y\right),\hfill \end{array}$$

(23)

where $\frac{1}{x}{\left(x{\psi }_x\right)}_x$ and $\frac{1}{y}{\left(y{\psi }_y\right)}_y$ are known as Bessel’s operators and $f\left(x,y,t\right)$, $f_1\left(x,y\right)$ and $f_2\left(x,y\right)$ are familiar functions. The solution of Equation (22), is given by the following steps:

• Step 1: Multiplying Equation (22) by $xy$, we obtain

  $$\begin{array}{ccc}\hfill xy{\psi }_{tt}& =& y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y\hfill \\ & & +y{\left(x{\psi }_x\right)}_{xt}+x{\left(y{\psi }_y\right)}_{yt}+xyf\left(x,y,t\right),\hfill \end{array}$$

  (24)
• Step 2: By employing the (DSGLT) first for both sides of Equation (24), applying the differentiation property of the (DST) and the initial condition stated in Equation (23) and employing Theorems 1 and 3, we gain

  $$\begin{array}{cc}\hfill \frac{u_1{u}_2}{s^2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,s)\right)& =u_1{u}_2{s}^{\alpha -1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,0)\right)\hfill \\ \hfill & +u_1{u}_2{s}^{\alpha }\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Psi }_t(u_1,u_2,0)\right)\hfill \\ \hfill & +u_1{u}_2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\hfill \\ \hfill & +S_2{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y\right],\hfill \\ \hfill & +S_2{G}_t\left[y{\left(x{\psi }_x\right)}_{xt}+x{\left(y{\psi }_y\right)}_{yt}\right]\hfill \end{array}$$

  (25)
• Step 3: By multiplying Equation (25) by $\frac{s^2}{u_1{u}_2}$, using integral, from 0 to $u_1$ and 0 to $u_2$ with respect to $u_1$ and $u_2$, respectively, and dividing the new equation by $u_1{u}_2,$ we have

  $$\begin{array}{cc}\hfill \Psi (u_1,u_2,s)& =\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Psi }_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y\right]\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_x\right)}_{xt}+x{\left(y{\psi }_y\right)}_{yt}\right]\right)d{u}_{1}d{u}_2,\hfill \end{array}$$

  (26)
• Step 4: The series solution of the singular ($2+1$-dimensional) pseudo-hyperbolic equation denoted by:

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

  (27)

  where $\Psi (u_1,u_2,s),$ $F_1\left(u_1,u_2\right)$ and $F_2\left(u_1,u_2\right)$ are the (DSGLT) and (DST) of the functions $\psi \left(x,y,t\right),$ $f_1\left(x,y\right)$ and $f_2\left(x,y\right)$, respectively.
• Step 5: On using the inverse (DSGLT) for both sides of Equation (26), and with the assistance of Equation (27), we find

  $$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right)& =S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Psi }_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\psi }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y\sum _{n=0}^{\infty }{\psi }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\psi }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y\sum _{n=0}^{\infty }{\psi }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \end{array}$$

In light of the first approximation, we have

$$\begin{array}{ccc}\hfill {\psi }_0& =& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Psi }_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

(28)

the rest of the components, ${\psi }_{n+1}$, $n\ge 0,$ are described by

$$\begin{array}{cc}\hfill & {\psi }_{n+1}=\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

(29)

Assume that the inverse of the (DSGLT) with respect to $u_1,u_2,$ and s for Equations (28) and (29) exists. To offer an obvious outline of this method, we provide the following example.

Example 1.

The singular $\left(2+1\text{-}\mathrm{D}\right)$ pseudo-hyperbolic equation is given by:

$$\begin{array}{cc}\hfill D_{tt}^{\alpha }\psi & =\frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}\frac{\partial }{\partial y}\left(y{\psi }_y\right)\hfill \\ \hfill & +\frac{1}{x}{\left(x{\psi }_x\right)}_{xt}+\frac{1}{y}{\left(y{\psi }_y\right)}_{yt}-\psi \hfill \\ \hfill 0& \le x_1,x_2,t<\infty ,0<\alpha \le 1,\hfill \end{array}$$

(30)

subject to the initial

$$\psi \left(x,y,0\right)=0,{\psi }_t\left(x,y,0\right)=x^2-y^2$$

(31)

By performing the above-mentioned method, the elements of the sequence are computed as follows:

$${\psi }_0=\left(x^2-y^2\right)t$$

(32)

$$\begin{array}{ccc}\hfill {\psi }_{n+1}& =& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & & -S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\psi }_n\right]\right)d{u}_{1}d{u}_2\right]\hfill \end{array}$$

(33)

As stated by the (DSGLTDM), we gain the following elements at $n=0$:

$$\begin{array}{ccc}\hfill {\psi }_1& =& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{0x}\right)}_x+x{\left(y{\psi }_{0y}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{0x}\right)}_{xt}+x{\left(y{\psi }_{0y}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & & -S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\psi }_0\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & =& -S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[\left(x^{3}y-x{y}^3\right)t\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & =& -S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(3!u_1^3{u}_2-3!u_1{u}_2^3\right)s^{\alpha +4}d{u}_{1}d{u}_2\right]\hfill \\ & =& -S_2^{-1}{G}_s^{-1}\left[\left(2!u_1^2-2!u_2^2\right)s^{\alpha +4}\right]\hfill \\ \hfill {\psi }_1& =& -\left(\left(x^2-y^2\right)\frac{t^3}{3!}\right).\hfill \end{array}$$

Likewise, at $n=1,$ we obtain that

$$\begin{array}{ccc}\hfill {\psi }_2& =& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{1x}\right)}_x+x{\left(y{\psi }_{1y}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{1x}\right)}_{xt}+x{\left(y{\psi }_{1y}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & & -S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\psi }_1\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & =& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[\left(x^{3}y-x{y}^3\right)\frac{t^3}{3!}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ & =& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(3!u_1^3{u}_2-3!u_1{u}_2^3\right)s^{\alpha +6}d{u}_{1}d{u}_2\right]\hfill \\ & =& S_2^{-1}{G}_s^{-1}\left[\left(2!u_1^2-2!u_2^2\right)s^{\alpha +6}\right]\hfill \\ \hfill {\psi }_2& =& \left(\left(x^2-y^2\right)\frac{t^5}{5!}\right).\hfill \end{array}$$

In the same way, let $n=2$

$$\begin{array}{ccc}\hfill {\psi }_3& =& -S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(3!u_1^3{u}_2-3!u_1{u}_2^3\right)s^{\alpha +8}d{u}_{1}d{u}_2\right]\hfill \\ & =& -S_2^{-1}{G}_s^{-1}\left[\left(2!u_1^2-2!u_2^2\right)s^{\alpha +7}\right]\hfill \\ \hfill {\psi }_3& =& -\left(\left(x^2-y^2\right)\frac{t^7}{7!}\right).\hfill \end{array}$$

As the sum of all terms, we have

$$\psi \left(x_1,x_2,t\right)={\psi }_0+{\psi }_1+{\psi }_2+{\psi }_3+...,$$

hence, the approximation solution of Equation (30) is confirmed by

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

We obtain the exact solution:

$$\begin{array}{ccc}\hfill \psi \left(x,y,t\right)& =& \left(x^2-y^2\right)\left(t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+\frac{t^9}{9!}-...\right)\hfill \\ \hfill \psi \left(x_1,x_2,t\right)& =& \left(x^2-y^2\right)sint.\hfill \end{array}$$

In the forthcoming problem, we apply the proposed method to a nonlinear pseudo-hyperbolic equation.

Problem 2.

Consider the next nonlinear singular ($2+1$-D) pseudo-hyperbolic equation:

$$\begin{array}{ccc}\hfill {\psi }_{tt}& =& \frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}{\left(y{\psi }_y\right)}_y\hfill \\ & & +\frac{1}{x}{\left(x{\psi }_x\right)}_{xt}+\frac{1}{y}{\left(y{\psi }_y\right)}_{yt}\hfill \\ & & +y\psi {\psi }_x+x\psi {\psi }_y+f\left(x,y,t\right),\hfill \end{array}$$

(34)

with initial conditions

$$\begin{array}{ccc}\hfill \psi \left(x,y,0\right)& =& f_1\left(x,y\right)\hfill \\ \hfill {\psi }_t\left(x,y,0\right)& =& f_2\left(x,y\right),\hfill \end{array}$$

(35)

By performing the previous analysis, the primary approximation is described by

$$\begin{array}{ccc}\hfill {\psi }_0& =& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Psi (u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Psi }_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ & & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

(36)

the remaining terms are identified by

$$\begin{array}{cc}\hfill & {\psi }_{n+1}=\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x\frac{\partial }{\partial x}{\psi }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{y}^2{A}_n+x^{2}y{B}_n\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

(37)

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

$$A_n=\sum _{n=0}^{\infty }{\psi }_n\frac{\partial {\psi }_n}{\partial x},B_n=\sum _{n=0}^{\infty }{\psi }_n\frac{\partial {\psi }_n}{\partial y},$$

(38)

and the nonlinear terms $\psi \frac{\partial \psi }{\partial x_1}$ and $\psi \frac{\partial \psi }{\partial x_2}$ are in the following forms:

$$\begin{array}{cc}\hfill A_0& ={\psi }_0{\psi }_{0x}\hfill \\ \hfill A_1& ={\psi }_0{\psi }_{01x}+{\psi }_1{\psi }_{0x},\hfill \\ \hfill A_2& ={\psi }_0{\psi }_{2x}+{\psi }_1{\psi }_{1x}+{\psi }_2{\psi }_{0x},\hfill \\ \hfill A_3& ={\psi }_0{\psi }_{3x}+{\psi }_1{\psi }_{2x}+{\psi }_2{\psi }_{1x}+{\psi }_3{\psi }_{0x}\hfill \end{array}$$

(39)

and

$$\begin{array}{cc}\hfill B_0& ={\psi }_0{\psi }_{0y}\hfill \\ \hfill B_1& ={\psi }_0{\psi }_{1y}+{\psi }_1{\psi }_{0y},\hfill \\ \hfill B_2& ={\psi }_0{\psi }_{2y}+{\psi }_1{\psi }_{1y}+{\psi }_2{\psi }_{0y},\hfill \\ \hfill B_3& ={\psi }_0{\psi }_{3y}+{\psi }_1{\psi }_{2y}+{\psi }_2{\psi }_{1y}+{\psi }_3{\psi }_{0y}.\hfill \end{array}$$

(40)

To show the applicability of this method for nonlinear problems, we study the following example.

Example 2.

Consider the following nonlinear singular pseudo-hyperbolic equation

$$\begin{array}{ccc}\hfill {\psi }_{tt}& =& \frac{1}{x}{\left(x{\psi }_x\right)}_x+\frac{1}{y}{\left(y{\psi }_y\right)}_y\hfill \\ & & +\frac{1}{x}{\left(x{\psi }_x\right)}_{xt}+\frac{1}{y}{\left(y{\psi }_y\right)}_{yt}\hfill \\ & & +y\psi {\psi }_x+x\psi {\psi }_y+\left(x^2-y^2\right)e^{-t}\hfill \\ \hfill 0& \le & x,y,t<\infty ,0<\alpha \le 1,\hfill \end{array}$$

(41)

with the following conditions

$$\psi \left(x,y,0\right)=x^2-y^2,{\psi }_t\left(x,y,0\right)=-\left(x^2-y^2\right)$$

(42)

By utilizing the stated method above, we have

$$\begin{array}{ccc}\hfill {\psi }_0& =& \left(x^2-y^2\right)-\left(x^2-y^2\right)t\hfill \\ & & +\left(x^2-y^2\right)\left(\frac{t^2}{2!}-\frac{t^3}{3!}+\frac{t^4}{4!}-\frac{t^5}{5!}+...\right)\hfill \end{array}$$

(43)

and

$$\begin{array}{cc}\hfill & {\psi }_{n+1}=\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{y}^2{A}_n+x^{2}y{B}_n\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

(44)

where $A_n$ and $B_n$ are defined in Equations (39) and (40). The subsequent terms are denoted by

$$\begin{array}{cc}\hfill & {\psi }_1=S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{0x}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{0y}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\psi }_{0x}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\psi }_{0y}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{y}^2{A}_0+x^{2}y{B}_0\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill {\psi }_1& =0,\hfill \end{array}$$

By following the same method, we have

$${\psi }_2=0,{\psi }_3=0,{\psi }_4=0,...$$

Hence, as stated in Equation (27), we get

$${\psi }_0=\left(x^2-y^2\right)-\left(x^2-y^2\right)t+\left(x^2-y^2\right)\left(\frac{t^2}{2!}-\frac{t^3}{3!}+\frac{t^4}{4!}-\frac{t^5}{5!}+...\right),$$

and the exact solution of Equation (41) is demonstrated by

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

5. Singular (2 + 1-D) Coupled Pseudo-Hyperbolic Equation and the (DSGLTDM)

The fundamental evaluation of the (DSGLTDM) is clearly shown to indicate its ability and high accuracy by examining the general coupled singular ($2+1$-D) pseudo-hyperbolic equation of the form

$$\begin{array}{ccc}\hfill {\phi }_{tt}& =& \frac{1}{x}{\left(x{\phi }_x\right)}_x+\frac{1}{y}{\left(y{\phi }_y\right)}_y\hfill \\ & & +\frac{1}{x}{\left(x{\phi }_x\right)}_{xt}+\frac{1}{y}{\left(y{\phi }_y\right)}_{yt}+\omega +f\left(x,y,t\right)\hfill \\ \hfill {\omega }_{tt}& =& \frac{1}{x}{\left(x{\omega }_x\right)}_x+\frac{1}{y}{\left(y{\omega }_y\right)}_y\hfill \\ & & +\frac{1}{x}{\left(x{\omega }_x\right)}_{xt}+\frac{1}{y}{\left(y{\omega }_y\right)}_{yt}+\phi +g\left(x,y,t\right),\hfill \\ & & \hfill \end{array}$$

(45)

with the following initial conditions

$$\begin{array}{ccc}\hfill \phi \left(x,y,0\right)& =& f_1\left(x,y\right),{\phi }_t\left(x,y,0\right)=f_2\left(x,y\right)\hfill \\ \hfill \omega \left(x,y,0\right)& =& g_1\left(x,y\right),{\omega }_t\left(x,y,0\right)=g_2\left(x,y\right),\hfill \\ & & \hfill \end{array}$$

(46)

where $f\left(x,y,t\right)$, $g\left(x,y,t\right), f_1\left(x,y\right), f_2\left(x,y\right), g_1\left(x,y\right)$ and $g_1\left(x,y\right)$ are given functions. By using the $\left(\mathrm{DSGLTDM}\right)$, this method contains the following steps.

• Step 1: Multiply both sides of Equation (45) by $xy$, resulting in the following equation,

  $$\begin{array}{ccc}\hfill xy{\phi }_{tt}& =& y{\left(x{\phi }_x\right)}_x+x{\left(y{\phi }_y\right)}_y+y{\left(x{\phi }_x\right)}_{xt}\hfill \\ & & +x{\left(y{\phi }_y\right)}_{yt}+xy\omega +xyf\left(x,y,t\right),\hfill \end{array}$$

  (47)

  and

  $$\begin{array}{ccc}\hfill xy{\omega }_{tt}& =& y{\left(x{\omega }_x\right)}_x+x{\left(y{\omega }_y\right)}_y+y{\left(x{\omega }_x\right)}_{xt}\hfill \\ & & +x{\left(y{\omega }_y\right)}_{yt}+xy\phi +xyf\left(x,y,t\right),\hfill \end{array}$$

  (48)
• Step 2: Using the $\left(\mathrm{DSGLT}\right)$ for both sides of Equations (47) and (48) and the $\left(\mathrm{DST}\right)$ for Equation (46), we obtain

  $$\begin{array}{cc}\hfill \frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Phi (u_1,u_2,s)\right)& =s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Phi (u_1,u_2,0)\right)\hfill \\ \hfill & +s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Phi }_t(u_1,u_2,0)\right)\hfill \\ \hfill & +s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\hfill \\ \hfill & +\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_x\right)}_x+x{\left(y{\phi }_y\right)}_y\right],\hfill \\ \hfill & +\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_x\right)}_{xt}+x{\left(y{\phi }_y\right)}_{yt}+xy\omega \right]\hfill \end{array}$$

  (49)

  and

  $$\begin{array}{cc}\hfill \frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\rm Y}(u_1,u_2,s)\right)& =s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\rm Y}(u_1,u_2,0)\right)\hfill \\ \hfill & +s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{{\rm Y}}_t(u_1,u_2,0)\right)\hfill \\ \hfill & +s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}G(u_1,u_2,s)\right)\hfill \\ \hfill & +\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_x\right)}_x+x{\left(y{\omega }_y\right)}_y\right],\hfill \\ \hfill & +\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_x\right)}_{xt}+x{\left(y{\omega }_y\right)}_{yt}+xy\phi \right]\hfill \end{array}$$

  (50)
• Step 3: By operating the integral for both sides for Equations (50) and (49), from 0 to $u_1$ and 0 to $u_2$ with respect to $u_1$ and $u_2$, respectively, we have

  $$\begin{array}{cc}\hfill \Phi (u_1,u_2,s)& =\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Phi (u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Phi }_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_x\right)}_x+x{\left(y{\phi }_y\right)}_y\right]\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_x\right)}_{xt}+x{\left(y{\phi }_y\right)}_{yt}+xy\omega \right]\right)d{u}_{1}d{u}_2,\hfill \end{array}$$

  (51)

  and

  $$\begin{array}{cc}\hfill {\rm Y}(u_1,u_2,s)& =\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\rm Y}(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{{\rm Y}}_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}G(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_x\right)}_x+x{\left(y{\omega }_y\right)}_y\right]\right)d{u}_{1}d{u}_2\hfill \\ \hfill & +\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_x\right)}_{xt}+x{\left(y{\omega }_y\right)}_{yt}+xy\phi \right]\right)d{u}_{1}d{u}_2,\hfill \end{array}$$

  (52)

The series solution of Equation (45) is thus entirely stated by

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

(53)

Step 4: By applying the inverse (DSGLT) for both sides of Equations (51) and (52) and applying Equation (53), one can obtain

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{\phi }_n\left(x,y,t\right)& =S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Phi (u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Phi }_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\phi }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & ++S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y\sum _{n=0}^{\infty }{\phi }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\phi }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y\sum _{n=0}^{\infty }{\phi }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy\sum _{n=0}^{\infty }{\omega }_n\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{\omega }_n\left(x,y,t\right)& =S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\rm Y}(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{{\rm Y}}_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}G(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\omega }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & ++S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y\sum _{n=0}^{\infty }{\omega }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x\sum _{n=0}^{\infty }{\omega }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y\sum _{n=0}^{\infty }{\omega }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy\sum _{n=0}^{\infty }{\phi }_n\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

From the above equations, we equate terms on both sides to produce recurrence relations as

$$\begin{array}{cc}\hfill {\omega }_0\left(x,y,t\right)& =S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\rm Y}(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{{\rm Y}}_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}G(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\right]\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\phi }_0\left(x,y,t\right)& =S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +1}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2\Phi (u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^{\alpha +2}\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_2{\Phi }_t(u_1,u_2,0)\right)\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(s^2\frac{{\partial }^2}{\partial u_1\partial u_2}\left(u_1{u}_{2}F(u_1,u_2,s)\right)\right)d{u}_{1}d{u}_2\right]\hfill \end{array}$$

(55)

The remaining terms ${\phi }_{n+1}$ and ${\omega }_{n+1}$, $n\ge 0,$ are described by

$$\begin{array}{cc}\hfill {\phi }_{n+1}=& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\phi }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\phi }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\omega }_n\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill {\omega }_{n+1}=& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\omega }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\omega }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\phi }_n\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

Thus, we verify the applicability of our method for solving coupled pseudo-hyperbolic equations. The next example is now considered.

Example 3.

Let us consider the coupled pseudo-hyperbolic equations presented by

$$\begin{array}{ccc}\hfill {\phi }_{tt}& =& \frac{1}{x}{\left(x{\phi }_x\right)}_x+\frac{1}{y}{\left(y{\phi }_y\right)}_y\hfill \\ & & +\frac{1}{x}{\left(x{\phi }_x\right)}_{xt}+\frac{1}{y}{\left(y{\phi }_y\right)}_{yt}+\omega \hfill \\ \hfill {\omega }_{tt}& =& \frac{1}{x}{\left(x{\omega }_x\right)}_x+\frac{1}{y}{\left(y{\omega }_y\right)}_y\hfill \\ & & +\frac{1}{x}{\left(x{\omega }_x\right)}_{xt}+\frac{1}{y}{\left(y{\omega }_y\right)}_{yt}+\phi ,\hfill \end{array}$$

(57)

where

$$0\le x,y,t<\infty ,$$

with initial condition

$$\begin{array}{ccc}\hfill \phi \left(x,y,0\right)& =& x^2-y^2,{\phi }_t\left(x_1,x_2,0\right)=-\left(x^2-y^2\right)\hfill \\ \hfill \omega \left(x,y,0\right)& =& x^2-y^2,{\omega }_t\left(x,y,0\right)=-\left(x^2-y^2\right),\hfill \end{array}$$

(58)

By using our method above, we acquire the following components:

$${\phi }_0=\left(x^2-y^2\right)-\left(x^2-y^2\right)t,{\omega }_0=\left(x^2-y^2\right)-\left(x^2-y^2\right)t,$$

and the remaining terms ${\phi }_{n+1}$ and ${\omega }_{n+1}$, $n\ge 0,$ are presented by

$$\begin{array}{cc}\hfill {\phi }_{n+1}=& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\phi }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\phi }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\omega }_n\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\omega }_{n+1}=& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_{nx}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & ++S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\omega }_{ny}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_{nx}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\omega }_{ny}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\phi }_n\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

At $n=0,$

$$\begin{array}{cc}\hfill {\phi }_1=& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_{0x}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\phi }_{0y}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\phi }_{0x}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\phi }_{0y}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\omega }_0\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\omega }_1=& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_{0x}\right)}_x\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & ++S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\omega }_{0y}\right)}_y\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[y{\left(x{\omega }_{0x}\right)}_{xt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[x{\left(y{\omega }_{0y}\right)}_{yt}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[xy{\phi }_0\right]\right)d{u}_{1}d{u}_2\right],\hfill \end{array}$$

hence

$$\begin{array}{cc}\hfill & {\phi }_1=S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[\left(x^{3}y-x{y}^3\right)\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & -S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[\left(x^{3}y-x{y}^3\right)t\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & =S_2^{-1}{G}_s^{-1}\left[s^{\alpha +3}\left(2!u_1^2-2!u_2^2\right)-s^{\alpha +4}\left(2!u_1^2-2!u_2^2\right)\right]\hfill \\ \hfill & {\phi }_1=\left(x^2-y^2\right)\frac{t^2}{2!}-\left(x^2-y^2\right)\frac{t^3}{3!},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\omega }_1=S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[\left(x^{3}y-x{y}^3\right)\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & -S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[\left(x^{3}y-x{y}^3\right)t\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & =S_2^{-1}{G}_s^{-1}\left[s^{\alpha +3}\left(2!u_1^2-2!u_2^2\right)-s^{\alpha +4}\left(2!u_1^2-2!u_2^2\right)\right]\hfill \\ \hfill & {\omega }_1=\left(x^2-y^2\right)\frac{t^2}{2!}-\left(x^2-y^2\right)\frac{t^3}{3!},\hfill \end{array}$$

At $n=1$,

$$\begin{array}{cc}\hfill {\phi }_2=& S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[\left(x^{3}y-x{y}^3\right)\frac{t^2}{2!}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill & +S_2^{-1}{G}_s^{-1}\left[\frac{1}{u_1{u}_2}{\int }_0^{u_1}{\int }_0^{u_2}\left(\frac{s^2}{u_1{u}_2}{S}_2{G}_t\left[\left(x^{3}y-x{y}^3\right)\frac{t^3}{3!}\right]\right)d{u}_{1}d{u}_2\right]\hfill \\ \hfill {\phi }_2& =S_2^{-1}{G}_s^{-1}\left[s^{\alpha +5}\left(2!u_1^2-2!u_2^2\right)-s^{\alpha +6}\left(2!u_1^2-2!u_2^2\right)\right]\hfill \\ \hfill & =\left(x^2-y^2\right)\frac{t^4}{4!}-\left(x^2-y^2\right)\frac{t^5}{5!}\hfill \end{array}$$

and similarly

$${\omega }_2=\left(x^2-y^2\right)\frac{t^4}{4!}-\left(x^2-y^2\right)\frac{t^5}{5!}$$

In the same way, at $n=2$, we obtain

$${\phi }_3=\left(x^2-y^2\right)\frac{t^6}{6!}-\left(x^2-y^2\right)\frac{t^7}{7!},$$

$${\omega }_3=\left(x^2-y^2\right)\frac{t^6}{6!}-\left(x^2-y^2\right)\frac{t^7}{7!}.$$

Using Equation (53), the other components of the decomposition series can be fixed in the same way; we can handle the solution of $\phi \left(x,y,t\right)$ and $\omega \left(x,y,t\right)$ in a Taylor series, which denotes the closed-form solutions as thus,

$$\begin{array}{ccc}\hfill \phi \left(x,y,t\right)& =& \left(x^2-y^2\right)-\left(x^2-y^2\right)t+\left(x^2-y^2\right)\frac{t^2}{2!}-\left(x^2-y^2\right)\frac{t^3}{3!}\hfill \\ & & +\left(x^2-y^2\right)\frac{t^4}{4!}-\left(x^2-y^2\right)\frac{t^5}{5!}\hfill \\ & & +\left(x^2-y^2\right)\frac{t^6}{6!}-\left(x^2-y^2\right)\frac{t^7}{7!}+...\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \omega \left(x,y,t\right)& =& \left(x^2-y^2\right)-\left(x^2-y^2\right)t+\left(x^2-y^2\right)\frac{t^2}{2!}-\left(x^2-y^2\right)\frac{t^3}{3!}\hfill \\ & & +\left(x^2-y^2\right)\frac{t^4}{4!}-\left(x^2-y^2\right)\frac{t^5}{5!}\hfill \\ & & +\left(x^2-y^2\right)\frac{t^6}{6!}-\left(x^2-y^2\right)\frac{t^7}{7!}+...\hfill \end{array}$$

The solution becomes

$$\begin{array}{cc}\hfill \phi \left(x,y,t\right)& ={\phi }_0+{\phi }_1+{\phi }_2+{\phi }_3+...=\left(1-t+\frac{t^2}{2!}-\frac{t^3}{3!}+\frac{t^4}{4!}-\dots \right)\left(x^2-y^2\right)\hfill \\ \hfill \omega \left(x,y,t\right)& ={\omega }_0+{\omega }_1+{\omega }_2+{\omega }_3+\dots =\left(1-t+\frac{t^2}{2!}-\frac{t^3}{3!}+\frac{t^4}{4!}-\dots \right)\left(x^2-y^2\right),\hfill \end{array}$$

and hence the exact solution becomes

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

6. Conclusions

The (DSGLTDM) is a hybrid method that consists of the (DSGLT) and the (ADM). This method has some advantages over the previously studied methods. In this research work, we have utilized the (DSGLTDM) in order to obtain the approximate and series solutions of linear, nonlinear, and coupled singular ($2+1$-D) pseudo-hyperbolic equations. By investigating examples, we found that the (DSGLTDM) is a powerful tool for the solution of linear, nonlinear, and coupled singular ($2+1$-D) pseudo-hyperbolic equations and compared it with the Adomian decomposition method (ADM), the homotopy analysis method (HAM), and the variational iteration method (VAM). Nevertheless, until now, there has been an open problem to investigate the rate of convergence to the exact solution for these types of problems. It can be studied by the (DSGLTDM) using an analytical solution to the other singular $2+1$-dimensional partial differential equations, which occur in science as well as engineering, and may deliver a better knowledge of the real-world problems that are described by singular $2+1$-dimensional partial differential equations.