A Note on the Application of the Double Sumudu–Generalized Laplace Decomposition Method and 1+1- and 2+1-Dimensional Time-Fractional Boussinesq Equations

Abstract

The current paper concentrates on discovering the exact solutions of the singular time-fractional Boussinesq equation and coupled time-fractional Boussinesq equation by presenting a new technique known as the double Sumudu–generalized Laplace and Adomian decomposition method. Here, two main theorems are addressed that are very useful in this work. Moreover, the mentioned method is effective in solving several problems. Some examples are presented to check the precision and symmetry of the technique. The outcomes show that the proposed technique is precise and gives better solutions as compared to existing methods in the literature.

1. Introduction

A famous hydrodynamics model used to explain the propagation of waves is the Boussinesq equation. It is suitable for problems involving the percolation of water in porous subsurface strata and is used in coastal and ocean engineering. Also, Boussinesq equations are the foundation of different models utilized to explain unconfined groundwater flow and subsurface drainage problems. These equations play a useful role in modeling diverse phenomena, such as long waves in shallow water [1]. The Boussinesq equation appears in different fields, such as a one-dimensional nonlinear lattice [2], shallow-water waves [3,4], and the propagation of longitudinal deformation waves in an elastic rod [5]. Throughout the past three decades, several methods have been developed and applied to solve Boussinesq equations, for example, the modified decomposition method (see [6,7,8]), homotopy analysis and homotopy perturbation methods [9,10,11], and the Laplace Adomian decomposition method [12,13,14]. A fractional Boussinesq equation is acquired by assuming power-law changes in flux in a control volume and applying a fractional Taylor series [15]. The space–time fractional Boussinesq equation was solved by using fractional variational principles with the semi-inverse technique [16]. The authors of [17] studied the solution of the Schrödinger–Boussinesq equation by implementing the Mittag Leffler kernel. A new generalized Laplace transform was studied in [18], and some properties of this transform are introduced in [19], while some ideas related to generalized Laplace transforms were employed to find the solutions of partial differential equations (PDEs) [20]. The double Sumudu transformation method was utilized in several papers (see [21,22,23]). Recently, the authors of [24,25] used Sumudu–generalized Laplace transform decomposition to solve fractional third-order dispersive partial differential equations and the 2+1-Pseudoparabolic Equation. The solution of the fractional-order Boussinesq equation was obtained by employing the Laplace transform and the Atangana–Baleanu fractional derivative operator in [26]. The solutions of linear and nonlinear singular conformable fractional Boussinesq equations were obtained by using the double conformable Laplace decomposition method [27]. The authors of [28] compared the solutions of the Caputo (with the singular kernel) and Caputo–Fabrizio (with a non-singular kernel) fractional operators of the linearized space–time fractional Boussinesq equation. Different solutions of the Boussinesq equation were obtained by using the Sardar Sub-Equation Technique (SSET) in [29]. The purpose of this work is to present a new hybrid of the double Sumudu–generalized Laplace transform to uncover the precise solutions of the time-fractional singular Boussinesq equation. Finally, three examples are provided to explain the proposed method. The remainder of this paper is organized in the following way: In Section 2, we begin with some basic definitions and theorems of the existence of the DSGLT of the function xy f(x, y, t). In Section 3, we examine two theorems that are regarded as the main contributions to this work. Section 3.1 shows how to obtain an approximate analytical solution for the 1+1-dimensional linear and nonlinear fractional singular Boussinesq equations using the DSGLTDM, and for each problem, we provide an example. In Section 3.2, we study the approximate analytical solution to the singular (2+1-D) linear fractional Boussinesq equation by using the DSGLTDM. In Section 4, we study the solution of the singular 2+1-dimensional coupled system Boussinesq equation using the DSGLTDM. Finally, concluding remarks are given in Section 5.

Some remarks: During this work, we utilize the following acronyms:

(1)

GLT in place of “generalized Laplace transform”;

(2)

DST in place of “double Sumudu transform”;

(3)

DSGLT in place of “double Sumudu–generalized Laplace transform”;

(4)

DSGLTDM in place of “double Sumudu–generalized Laplace transform decomposition method”.

2. Some Essential Ideas Related to the DSGLT

The definitions of the DST, GLT, DSGLT, and Caputo time-fractional derivative are as follows.

The DST of the function $f(x,y)$ is denoted by $F(u,v)$ in the following definition.

Definition 1

([30]). Let $f(t,x)$, $t,x$ $\in {\mathbb{R}}_{+}$, be a function that can be expressed as a convergent infinite series. Then, its double Sumudu transform is given by

$$F(u,v)=S_2\left[f(x,t)\right]=\underset{0}{\stackrel{\infty }{\int }}\underset{0}{\stackrel{\infty }{\int }}\frac{1}{uv}exp\left(-\frac{x}{u}-\frac{t}{v})\right)f(x,t)dtdx,$$

where $t\to u$ and $x\to v.$

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

Definition 2

([19]). 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 $s^{\alpha }exp\left(-\frac{t}{s}\right)$ from $t=0$ to ∞. It is a function of s, say $F\left(s\right)$, and is denoted by $G_{\alpha }\left(f\right)$; thus,

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

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

Definition 3

([31,32]). The Caputo time-fractional derivative operator of order $\beta >0$ is represented by

$$D_t^{\beta }f(x,t)=\left\{\begin{array}{c}\frac{1}{\Gamma \left(m-\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{m-\beta -1}\frac{{\partial }^{m}f(x,\tau )}{\partial {\tau }^m}d\tau ,m-1<\beta <m\\ \frac{{\partial }^{m}f(\chi ,t)}{\partial t^m},\mathit{for}m=\beta \in \mathbb{N}\end{array}\right.$$

Definition 4

([25]). If $f(x,y,t)$ is an integrable function defined for all $x,y,t\ge 0$, the definition of the DSGLT of the function $f(x,y,t)$ is defined by

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left(f\left(x,y,t\right)\right)& =& F\left({\xi }_1,{\xi }_2,s\right)=\frac{s^{\alpha }}{{\xi }_1{\xi }_2}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}exp\left(-\frac{x}{{\xi }_1}-\frac{y}{{\xi }_2}-\frac{t}{s}\right)f\left(x,y,t\right)dtdydx\hfill \end{array}$$

(1)

where the symbol $S_x{S}_y{G}_t$ is the DSGLT and the symbols ${\xi }_1,{\xi }_2$, and s are transforms of the variables $x,y$, and t in the double Sumudu transform and the generalized Laplace transform, respectively.

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 provided. 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 $t>T$,

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

(2)

for some $X,Y$, and T. Then, we write

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

and similarly,

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

(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 faster 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 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. 

Through Equations (1) and (2), we obtain

$$\begin{array}{ccc}\hfill \left|F\left({\zeta }_1,{\zeta }_2,s\right)\right|& =& \left|\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}exp\left(-\frac{x}{{\zeta }_1}-\frac{y}{{\zeta }_2}-\frac{t}{s}\right)f\left(x,y,t\right)dxdydt\right|\hfill \\ & \le & \left|\frac{s^{\alpha }}{{\zeta }_1{\zeta }_2}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}exp\left(-\left(\frac{1}{{\xi }_1}-a\right)x-\left(\frac{1}{{\zeta }_2}-b\right)y-\left(\frac{1}{s}-c\right)t\right)dxdydt\right|\hfill \\ & =& \frac{R{s}^{\alpha +1}}{\left(1-a{\zeta }_1\right)\left(1-c{\zeta }_2\right)\left(1-bs\right)}.\hfill \end{array}$$

From the condition $Re\frac{1}{{\zeta }_1}>\frac{1}{{\lambda }_1},Re\frac{1}{{\zeta }_2}>\frac{1}{{\lambda }_2},Re\frac{1}{s}>\frac{1}{\eta }$ and Equation (3), we obtain

$$\underset{\begin{array}{c}x\to \infty \\ y\to \infty \\ t\to \infty \end{array}}{lim}\left|F({\zeta }_1,{\zeta }_2,s)\right|=0\mathrm{or}\underset{\begin{array}{c}x\to \infty \\ y\to \infty \\ t\to \infty \end{array}}{lim}F({\zeta }_1,{\zeta }_2,s)=0.$$

 □

The next theorem discusses the existence of the DSGLT.

Theorem 2

([25]). The function $f\left(x,y,t\right)$ is defined on $(0,X),\left(0,Y\right)$, and $(0,T)$ and of exponential order $\left(x,y,t\right)$. Then, the DSGLT of $f\left(x,y,t\right)$ exists for all $Re\frac{1}{{\xi }_1}>a,Re\frac{1}{{\xi }_2}>b,Re\frac{1}{s}>c$.

The inverse DSGLT $S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[S_x{S}_y{G}_t\left(f\left(x,y,t\right)\right)\right]=f\left(x,y,t\right)$ is denoted by the following formula:

$$f\left(x,y,t\right)=\frac{1}{{\left(2\pi i\right)}^3}\underset{\tau -i\infty }{\overset{\tau +i\infty }{\int }}\underset{\delta -i\infty }{\overset{\delta +i\infty }{\int }}\underset{\sigma -i\infty }{\overset{\sigma +i\infty }{\int }}exp\left(\frac{x}{{\xi }_1}+\frac{y}{{\xi }_2}+\frac{t}{s}\right)S_x{S}_y{G}_t\left[f\left(x,y,t\right)\right]dsd{\xi }_{2}d{\xi }_1.$$

Theorem 3.

If the DSGLT of the function $f\left(x,y,t\right)$ is represented by $S_x{S}_y{G}_t\left(f\left(x,y,t\right)\right)=F({\xi }_1,{\xi }_2,s)$, then the DSGLTs of the functions

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

is determined by

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

(4)

Proof. 

By applying the partial derivative according to ${\xi }_1$ for Equation (4), we obtain

$$\begin{array}{ccc}\hfill \frac{\partial F({\xi }_1,{\xi }_2,s)}{\partial {\xi }_1}& =& \underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\frac{s^{\alpha }}{{\xi }_2}exp\left(\left(\frac{y}{{\xi }_2}+\frac{t}{s}\right)\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{\partial }{\partial {\xi }_1}\frac{1}{{\xi }_1}exp\left(-\frac{x}{{\xi }_1}\right)f\left(x,y,t\right)dx\right)dydt,\hfill \end{array}$$

(5)

and by handling the partial derivative inside the brackets, we obtain

$$\begin{array}{ccc}\hfill \underset{0}{\overset{\infty }{\int }}\frac{\partial }{\partial {\xi }_1}\frac{1}{{\xi }_1}exp\left(-\frac{x}{{\xi }_1}\right)f\left(x,y,t\right)dx& =& \underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^3}xexp\left(-\frac{x}{{\xi }_1}\right)f\left(x,y,t\right)dx\hfill \\ & & -\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^2}exp\left(-\frac{x}{{\xi }_1}\right)f\left(x,y,t\right)dx.\hfill \end{array}$$

(6)

Substituting Equation (6) into Equation (5), one can obtain the following equation:

$$\begin{array}{ccc}\hfill \frac{{\partial }^{2}F({\xi }_1,{\xi }_2,s)}{\partial {\xi }_1\partial {\xi }_2}& =& \underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\frac{s^{\alpha }}{{\xi }_2}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^3}xexp\left(-\frac{x}{{\xi }_1}\right)f\left(x,y,t\right)dx\right)dydt\hfill \\ & & -\frac{s^{\alpha }}{{\xi }_2}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^2}exp\left(-\frac{x}{{\xi }_1}\right)f\left(x,y,t\right)dx\right)dydt.\hfill \end{array}$$

(7)

By taking the derivative according to ${\xi }_2$ for Equation (7), we achieve

$$\begin{array}{ccc}\hfill \frac{{\partial }^{2}F({\xi }_1,{\xi }_2,s)}{\partial {\xi }_1\partial {\xi }_2}& =& \frac{s^{\alpha }}{{\xi }_1^3}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}xeexp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(exp\left(-\frac{x}{{\xi }_1}\right)\left(\frac{1}{{\xi }_2^3}y-\frac{1}{{\xi }_2^2}\right)f\left(x,y,t\right)\right)dydydt\hfill \\ & & -\frac{s^{\alpha }}{{\xi }_1^2}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(exp\left(-\frac{x}{{\xi }_1}\right)\left(\frac{1}{{\xi }_2^3}y-\frac{1}{{\xi }_2^2}\right)f\left(x,y,t\right)\right)dxdydt.\hfill \end{array}$$

(8)

After rearrangement, Equation (8) becomes

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

By rearranging the above equation, we obtain

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

The proof is completed. □

The DSGLT of the function $\psi (x,y,t)$ is denoted by $S_x{S}_y{G}_t\left[\psi \left(x,y,t\right)\right]=\mathsf{\Psi }({\xi }_1,{\xi }_2,s)$. Then, the DSGLTs of $\frac{\partial \psi }{\partial x},$ $\frac{{\partial }^2\psi }{\partial x^2},\frac{\partial \psi }{\partial y},$ $\frac{{\partial }^2\psi }{\partial y^2}$ $D_t^{2\beta }\psi$ are represented by

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

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

$$S_x{S}_y{G}_t\left[D_t^{\beta }\psi \right]=\frac{\mathsf{\Psi }({\xi }_1,{\xi }_2,s)}{s^{\beta }}-s^{\alpha -\beta +1}\mathsf{\Psi }({\xi }_1,{\xi }_2,0).$$

and

$$S_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]=\frac{\mathsf{\Psi }({\xi }_1,{\xi }_2,s)}{s^{2\beta }}-s^{\alpha -2\beta +1}\mathsf{\Psi }({\xi }_1,{\xi }_2,0)-s^{\alpha -2\beta +2}{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0).$$

3. Main Results of Double Sumudu–Generalized Laplace Transform (DSGLT)

To solve the singular fractional Boussinesq equation, we prove the following two theorems of the fractional partial derivatives by using the DSGLT. For example, in Theorem 4, we discuss $x{D}_t^{2\beta }\psi$ and $y{D}_t^{2\beta }\psi$, and in Theorem 5, we study $xy{D}_t^{2\beta }\psi .$

Theorem 4.

The DSGLTs of the fractional partial derivatives $x{D}_t^{2\beta }\psi$ and $y{D}_t^{2\beta }\psi$ are achieved by

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[x{D}_t^{2\beta }\psi \right]& =& \frac{{\xi }_1}{s^{2\beta }}\frac{\partial }{\partial {\xi }_1}\left({\xi }_1\mathsf{\Psi }({\xi }_1,{\xi }_2,s)\right)\hfill \\ & & -{\xi }_1{s}^{\alpha -2\beta +1}\frac{\partial }{\partial {\xi }_1}\left({\xi }_1\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & -{\xi }_1{s}^{\alpha -2\beta +2}\frac{\partial }{\partial {\xi }_1}\left({\xi }_1{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill S_{\chi }{S}_{\sigma }{G}_t\left[y{D}_t^{2\beta }\psi \right]& =& \frac{{\xi }_2}{s^{2\beta }}\frac{\partial }{\partial {\xi }_2}\left({\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,s)\right)\hfill \\ & & -{\xi }_2{s}^{\alpha -2\beta +1}\frac{\partial }{\partial {\xi }_2}\left({\xi }_{12}\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & -{\xi }_2{s}^{\alpha -2\beta +2}\frac{\partial }{\partial {\xi }_2}\left({\xi }_2{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right),\hfill \end{array}$$

(9)

where $0<\beta \le 1.$

Proof. 

By utilizing partial derivatives according to ${\xi }_1$ for Equation (1), we can obtain

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

(10)

and the partial derivative inside the brackets can be computed as follows:

$$\begin{array}{ccc}\hfill \underset{0}{\overset{\infty }{\int }}\frac{\partial }{\partial {\xi }_1}\frac{1}{{\xi }_1}exp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx& =& \underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^3}xexp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx\hfill \\ & & -\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^2}exp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx.\hfill \end{array}$$

(11)

By inserting Equation (11) into Equation (10), we obtain

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\xi }_1}\left(S_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]\right)& =& \underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\frac{s^{\alpha }}{{\xi }_2}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^3}xexp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx\right)dydt\hfill \\ & & -\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\frac{s^{\alpha }}{{\xi }_2}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^2}eexp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx\right)dydt.\hfill \end{array}$$

(12)

Hence, Equation (12) becomes

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\xi }_1}\left(S_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]\right)& =& \frac{1}{{\xi }_1^2}\left(\frac{s^{\alpha }}{{\xi }_1{\xi }_2}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}exp\left(-\frac{x}{{\xi }_1}-\frac{y}{{\xi }_2}-\frac{t}{s}\right)x{D}_t^{2\beta }\psi dxdydt\right)\hfill \\ & & -\frac{1}{{\xi }_1}\left(\frac{s^{\alpha }}{{\xi }_1{\xi }_2}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}exp\left(-\frac{x}{{\xi }_1}-\frac{y}{{\xi }_2}-\frac{t}{s}\right)D_t^{2\beta }\psi dxdydt\right)\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\xi }_1}\left(S_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]\right)& =& \frac{1}{{\xi }_1^2}{S}_x{S}_y{G}_t\left[x{D}_t^{2\beta }\psi \right]-\frac{1}{{\xi }_1}{S}_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right].\hfill \end{array}$$

(13)

By rearranging the above equation, we obtain the proof of Equation (13) as follows:

$$\begin{array}{ccc}& & S_x{S}_y{G}_t\left[x{D}_t^{2\beta }\psi \right]\hfill \\ & =& \frac{{\xi }_1}{s^{2\beta }}\frac{\partial }{\partial {\xi }_1}\left({\xi }_1\mathsf{\Psi }({\xi }_1,{\xi }_2,s)\right)\hfill \\ & & -{\xi }_1{s}^{\alpha -2\beta +1}\frac{\partial }{\partial {\xi }_1}\left({\xi }_1\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & -{\xi }_1{s}^{\alpha -2\beta +2}\frac{\partial }{\partial {\xi }_1}\left({\xi }_1{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right).\hfill \end{array}$$

Similarly, we can prove Equation (9). □

The DSGLT of the fractional partial derivative is introduced in the next theorem.

Theorem 5.

The DSGLT of the fractional partial derivative $xy{D}_t^{2\beta }\psi$ is determined by

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[xy{D}_t^{2\beta }\psi \right]& =& \frac{{\xi }_1{\xi }_2}{s^{2\beta }}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,s)\right)\hfill \\ & & -{\xi }_1{\xi }_2{s}^{\alpha -2\beta +1}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & -{\xi }_1{\xi }_2{s}^{\alpha -2\beta +2}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right).\hfill \\ \hfill 0& <& \beta \le 1\hfill \end{array}$$

(14)

Proof. 

By taking the partial derivative according to ${\xi }_1$ for Equation (1), we have

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\xi }_1}{S}_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]& =& \underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\frac{s^{\alpha }}{{\xi }_2}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{\partial }{\partial {\xi }_1}\frac{1}{{\xi }_1}exp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx\right)dydt.\hfill \end{array}$$

(15)

We calculate the partial derivative inside the brackets as follows:

$$\begin{array}{ccc}\hfill \underset{0}{\overset{\infty }{\int }}\frac{\partial }{\partial {\xi }_1}\frac{1}{{\xi }_1}exp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx& =& \underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^3}xexp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx\hfill \\ & & -\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^2}exp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx.\hfill \end{array}$$

(16)

Putting Equation (16) into Equation (15), we obtain

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\xi }_1}{S}_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]& =& \underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\frac{s^{\alpha }}{{\xi }_2}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^3}xexp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx\right)dydt\hfill \\ & & -\frac{1}{{\xi }_1}\left(\frac{s^{\alpha }}{{\xi }_1{\xi }_2}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}exp\left(-\frac{x}{{\xi }_1}-\frac{y}{{\xi }_2}-\frac{t}{s}\right)D_t^{2\beta }\psi dxdydt\right).\hfill \end{array}$$

(17)

The partial derivative with respect to ${\xi }_2$ for Equation (17) is computed as follows:

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}{S}_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]& =& \frac{\partial }{\partial {\xi }_2}\left(\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\frac{s^{\alpha }}{{\xi }_2}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^3}xexp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx\right)dydt\right)\hfill \\ & & -\frac{\partial }{\partial {\xi }_2}\left(\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\frac{s^{\alpha }}{{\xi }_2}exp\left(-\frac{y}{{\xi }_2}-\frac{t}{s}\right)\left(\underset{0}{\overset{\infty }{\int }}\frac{1}{{\xi }_1^2}exp\left(-\frac{x}{{\xi }_1}\right)D_t^{2\beta }\psi dx\right)dydt\right),\hfill \end{array}$$

(18)

and therefore, Equation (18) becomes

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}{S}_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]& =& \frac{1}{{\zeta }_1^2{\xi }_2^2}{S}_x{S}_y{G}_t\left[xy{D}_t^{2\beta }\psi \right]+\frac{1}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[D_t^{2\beta }\psi \right]\hfill \\ & & -\frac{1}{{\xi }_1{\xi }_2^2}{S}_x{S}_y{G}_t\left[y{D}_t^{2\beta }\psi \right]-\frac{1}{{\xi }_1^2{\xi }_2}{S}_x{S}_y{G}_t\left[x{D}_t^{2\beta }\psi \right].\hfill \end{array}$$

(19)

One can reorganize Equation (19) to prove Equation (14).

$$\begin{array}{ccc}& & S_x{S}_y{G}_t\left[xy{D}_t^{2\beta }\psi \right]\hfill \\ & =& \frac{{\xi }_1{\xi }_2}{s^{2\beta }}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,s)\right)\hfill \\ & & -{\xi }_1{\xi }_2{s}^{\alpha -2\beta +1}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & -{\xi }_1{\xi }_2{s}^{\alpha -2\beta +2}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right).\hfill \end{array}$$

 □

3.1. Sumudu–Generalized Laplace Transform Decomposition Method (SGLTDM) and 1+1-Dimensional Fractional Boussinesq Equation

The solution of the 1+1-dimensional fractional Boussinesq equation is considered by using the Sumudu–generalized Laplace transform decomposition method (SGLTDM). Here, we indicate the GLT of the function $\psi \left(x,t\right)$ by $\mathsf{\Psi }({\xi }_1,s).$ The linear fractional Boussinesq equation in one dimension with conditions is considered as follows:

$$D_t^{2\beta }\psi =a\frac{{\partial }^2\psi }{\partial x^2}+b\frac{{\partial }^{2}ln\psi }{\partial x^2}+c\frac{{\partial }^4\psi }{\partial x^4}+f\left(x,t\right),0<\beta \le 1$$

(20)

subject to

$$\psi \left(x,0\right)=f_1\left(x\right),\frac{\partial \psi \left(x,0\right)}{\partial t}=f_2\left(x\right),$$

(21)

where the functions $f_1\left(x\right)$, $f_2\left(x\right)$, and $f\left(x,t\right)$ are known, and a, b, and c are constants.

Firstly, the SGLT is used for Equation (20) and the single Sumudu transform for Equation (21), and we obtain

$$\begin{array}{ccc}\hfill \frac{\mathsf{\Psi }({\xi }_1,s)}{s^{2\beta }}& =& s^{\alpha -2\beta +1}\mathsf{\Psi }({\xi }_1,0)+s^{\alpha -2\beta +2}{\mathsf{\Psi }}_t({\xi }_1,0)+F\left({\xi }_1,s\right)\hfill \\ & & +S_x{G}_t\left[a\frac{{\partial }^2\psi }{\partial x^2}+b\frac{{\partial }^{2}ln\psi }{\partial x^2}+c\frac{{\partial }^4\psi }{\partial x^4}\right].\hfill \end{array}$$

(22)

By reorganizing and simplifying Equation (22), it becomes

$$\begin{array}{ccc}\hfill \mathsf{\Psi }({\xi }_1,s)& =& s^{\alpha +1}\mathsf{\Psi }({\xi }_1,0)+s^{\alpha +2}{\mathsf{\Psi }}_t({\xi }_1,0)+s^{2\beta }F\left({\xi }_1,s\right)\hfill \\ & & +s^{2\beta }{S}_x{G}_t\left[a\frac{{\partial }^2\psi }{\partial x^2}+b\frac{{\partial }^{2}ln\psi }{\partial x^2}+c\frac{{\partial }^4\psi }{\partial x^4}\right].\hfill \end{array}$$

(23)

The solution is obtained by utilizing the inverse SGLT for Equation (23):

$$\begin{array}{ccc}\hfill \psi \left(x,t\right)& =& f_1\left(x\right)+t{f}_2\left(x\right)+S_{{\xi }_1}^{-1}{S}_s^{-1}\left[s^{2\beta }F\left({\xi }_1,s\right)\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[a\frac{{\partial }^2\psi }{\partial x^2}+b\frac{{\partial }^{2}ln\psi }{\partial x^2}+c\frac{{\partial }^4\psi }{\partial x^4}\right]\right],\hfill \end{array}$$

(24)

where $S_{{\xi }_1}^{-1}{G}_s^{-1}$ indicates the inverse Sumudu–generalized Laplace transform. The Sumudu–generalized Laplace transform decomposition method (SGLTDM) defines the solutions $\psi (x,t)$ with the assistance of the following infinite series:

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

(25)

By inserting Equation (24) into Equation (25), we obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,t\right)& =& f_1\left(x\right)+t{f}_2\left(x\right)+S_{{\xi }_1}^{-1}{S}_s^{-1}\left[s^{2\beta }F\left({\xi }_1,s\right)\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[a\frac{{\partial }^2}{\partial x^2}\left(\sum _{n=0}^{\infty }{\psi }_n\left(x,t\right)\right)+b\frac{{\partial }^2}{\partial x^2}\sum _{n=0}^{\infty }ln{\psi }_n\left(x,t\right)\right]\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[c\frac{{\partial }^4}{\partial x^4}\left(\sum _{n=0}^{\infty }{\psi }_n\left(x,t\right)\right)\right]\right].\hfill \end{array}$$

(26)

By matching both sides of Equation (26), we obtain

$${\psi }_0\left(x,t\right)=f_1\left(x\right)+t{f}_2\left(x\right)+S_{{\xi }_1}^{-1}{S}_s^{-1}\left[s^{2\beta }F\left({\xi }_1,s\right)\right]$$

(27)

In general, the rest of the terms are given by

$$\begin{array}{ccc}\hfill {\psi }_{n+1}\left(x,t\right)& =& S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[a\frac{{\partial }^2}{\partial x^2}\left({\psi }_n\left(x,t\right)\right)+b\frac{{\partial }^2}{\partial x^2}\left(ln{\psi }_n\left(x,t\right)\right)\right]\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[c\frac{{\partial }^4}{\partial x^4}\left({\psi }_n\left(x,t\right)\right)\right]\right].\hfill \end{array}$$

(28)

Here, we show that the inverse exists for Equations (27) and (28).

For the purpose of discussing the advantages and the accuracy of the SGLTDM for solving one-dimensional Boussinesq equations, we use the method introduced in the next example.

Example 1.

Consider a Boussinesq equation in one dimension in the form of

$$D_t^{2\beta }\psi =\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^{2}ln\psi }{\partial x^2}+\frac{{\partial }^4\psi }{\partial x^4}-3e^{x}sint,0<\beta \le 1$$

(29)

with the initial conditions

$$\psi \left(x,0\right)=0,\frac{\partial \psi \left(x,0\right)}{\partial t}=e^x.$$

Using the above method, Equation (29) becomes

$$\begin{array}{ccc}\hfill \psi \left(x,t\right)& =& t{e}^x-S_{{\xi }_1}^{-1}{G}_s^{-1}\left(\frac{3}{1-u}\left(s^{\alpha +2\beta +2}-s^{\alpha +2\beta +4}+s^{\alpha +2\beta +6}-s^{\alpha +2\beta +8}+\dots \right)\right)\hfill \\ & & +S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[\frac{{\partial }^2}{\partial x^2}\left({\psi }_n\left(x,t\right)\right)+\frac{{\partial }^2}{\partial x^2}\left(ln{\psi }_n\left(x,t\right)\right)\right]\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[\frac{{\partial }^4}{\partial x^4}\left({\psi }_n\left(x,t\right)\right)\right]\right],\hfill \end{array}$$

and by arranging the above equation, one can obtain

$${\psi }_0=t{e}^x-3e^x\left(\frac{t^{2\beta +1}}{\left(2\beta +1\right)!}-\frac{t^{2\beta +3}}{\left(2\beta +3\right)!}+\frac{t^{2\beta +5}}{\left(2\beta +5\right)!}-\frac{t^{2\beta +7}}{\left(2\beta +7\right)!}+\dots \right),$$

and

$$\begin{array}{ccc}\hfill {\psi }_{n+1}\left(x,t\right)& =& S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[\frac{{\partial }^2}{\partial x^2}\left({\psi }_n\left(x,t\right)\right)\right]\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[\frac{{\partial }^2}{\partial x^2}\left(ln{\psi }_n\left(x,t\right)\right)\right]\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[\frac{{\partial }^4}{\partial x^4}\left({\psi }_n\left(x,t\right)\right)\right]\right],\hfill \end{array}$$

for $n=0,1,2,\dots$. Hence, at $n=0$,

$$\begin{array}{ccc}\hfill {\psi }_1& =& S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[\frac{{\partial }^2}{\partial x^2}\left({\psi }_0\left(x,t\right)\right)+\frac{{\partial }^2}{\partial x^2}\left(ln{\psi }_0\left(x,t\right)\right)+\frac{{\partial }^4}{\partial x^4}\left({\psi }_0\left(x,t\right)\right)\right]\right],\hfill \\ \hfill {\psi }_1& =& S_{{\xi }_1}^{-1}{S}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[2t{e}^x-6e^x\left(\frac{t^{2\beta +1}}{\left(2\beta +1\right)!}-\frac{t^{2\beta +3}}{\left(2\beta +3\right)!}+\frac{t^{2\beta +5}}{\left(2\beta +5\right)!}-\frac{t^{2\beta +7}}{\left(2\beta +7\right)!}+\dots \right)\right]\right],\hfill \end{array}$$

and the above equation becomes

$$\begin{array}{ccc}\hfill {\psi }_1& =& S_{{\xi }_1}^{-1}{G}_s^{-1}\left[\frac{2}{1-u}{s}^{\alpha +2\beta +2}-\frac{6}{1-u}\left(s^{\alpha +4\beta +2}-s^{\alpha +4\beta +4}+s^{\alpha +4\beta +6}-s^{\alpha +4\beta +8}+\dots \right)\right],\hfill \\ \hfill {\psi }_1& =& 2e^x\frac{t^{2\beta +1}}{\left(2\beta +1\right)!}-6e^x\left(\frac{t^{4\beta +1}}{\left(4\beta +1\right)!}-\frac{t^{4\beta +3}}{\left(4\beta +3\right)!}+\frac{t^{4\beta +5}}{\left(4\beta +5\right)!}-\frac{t^{4\beta +7}}{\left(4\beta +7\right)!}+\dots \right).\hfill \end{array}$$

At $n=1$,

$$\begin{array}{ccc}\hfill {\psi }_2& =& S_{{\xi }_1}^{-1}{G}_s^{-1}\left[s^{2\beta }{S}_x{G}_t\left[\frac{{\partial }^2}{\partial x^2}\left({\psi }_1\left(x,t\right)\right)+\frac{{\partial }^2}{\partial x^2}\left(ln{\psi }_1\left(x,t\right)\right)+\frac{{\partial }^4}{\partial x^4}\left({\psi }_1\left(x,t\right)\right)\right]\right],\hfill \\ \hfill {\psi }_2& =& S_{{\xi }_1}^{-1}{G}_s^{-1}\left[\frac{4}{1-u}{s}^{\alpha +4\beta +2}-\frac{6}{1-u}\left(s^{\alpha +6\beta +2}-s^{\alpha +6\beta +4}+s^{\alpha +6\beta +6}-s^{\alpha +6\beta +8}+\dots \right)\right].\hfill \end{array}$$

By computing the inverse, we obtain

$${\psi }_2=4e^x\frac{t^{4\beta +1}}{\left(4\beta +1\right)!}-12e^x\left(\frac{t^{6\beta +1}}{\left(6\beta +1\right)!}-\frac{t^{6\beta +3}}{\left(6\beta +3\right)!}+\frac{t^{6\beta +5}}{\left(6\beta +5\right)!}-\frac{t^{6\beta +7}}{\left(6\beta +7\right)!}+\dots \right)$$

and at $n=2$,

$${\psi }_3=8e^x\frac{t^{6\beta +1}}{\left(6\beta +1\right)!}-24e^x\left(\frac{t^{8\beta +1}}{\left(8\beta +1\right)!}-\frac{t^{8\beta +3}}{\left(8\beta +3\right)!}+\frac{t^{8\beta +5}}{\left(8\beta +5\right)!}-\frac{t^{8\beta +7}}{\left(8\beta +7\right)!}+\dots \right).$$

By applying Equation (25), we obtain

$$\sum _{n=0}^{\infty }{\psi }_n\left(x,t\right)={\psi }_0+{\psi }_1+{\psi }_2+\dots ,$$

and hence,

$$\begin{array}{ccc}\hfill \psi \left(x,t\right)& =& t{e}^x-3e^x\left(\frac{t^{2\beta +1}}{\left(2\beta +1\right)!}-\frac{t^{2\beta +3}}{\left(2\beta +3\right)!}+\frac{t^{2\beta +5}}{\left(2\beta +5\right)!}-\frac{t^{2\beta +7}}{\left(2\beta +7\right)!}+\dots \right)\hfill \\ & & +2e^x\frac{t^{2\beta +1}}{\left(2\beta +1\right)!}-6e^x\left(\frac{t^{4\beta +1}}{\left(4\beta +1\right)!}-\frac{t^{4\beta +3}}{\left(4\beta +3\right)!}+\frac{t^{4\beta +5}}{\left(4\beta +5\right)!}-\frac{t^{4\beta +7}}{\left(2\beta +7\right)!}+\dots \right)\hfill \\ & & +4e^x\frac{t^{4\beta +1}}{\left(4\beta +1\right)!}-12e^x\left(\frac{t^{6\beta +1}}{\left(6\beta +1\right)!}-\frac{t^{6\beta +3}}{\left(6\beta +3\right)!}+\frac{t^{6\beta +5}}{\left(6\beta +5\right)!}-\frac{t^{6\beta +7}}{\left(6\beta +7\right)!}+\dots \right)\hfill \\ & & +8e^x\frac{t^{6\beta +1}}{\left(6\beta +1\right)!}-24e^x\left(\frac{t^{8\beta +1}}{\left(8\beta +1\right)!}-\frac{t^{8\beta +3}}{\left(8\beta +3\right)!}+\frac{t^{8\beta +5}}{\left(8\beta +5\right)!}-\frac{t^{8\beta +7}}{\left(8\beta +7\right)!}+\dots \right).\hfill \end{array}$$

The exact solution of Equation (29) is obtained at $\beta =1$:

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

3.2. Double Sumudu–Generalized Laplace Transform Decomposition and Singular 2+1-Dimensional Boussinesq Equation

We demonstrate the essential idea of the DSGLTDM for the general singular 2+1-dimensional fractional Boussinesq equation of the form

$$\begin{array}{ccc}& & D_t^{2\beta }\psi -\frac{1}{x}{\left(x{\psi }_x\right)}_x-\frac{1}{y}{\left(y{\psi }_y\right)}_y+a\left(x,y\right){\psi }_{xxxx}+b\left(x,y\right){\psi }_{yyyy}\hfill \\ & & +c\left(x,y\right){\psi }_{xxtt}+d\left(x,y\right){\psi }_{yytt}=f\left(x,y,t\right),0<\beta \le 1\hfill \end{array}$$

(30)

with the initial condition

$$\psi \left(x,y,0\right)=g_1\left(x,y\right),{\psi }_t\left(x,y,0\right)=g_2\left(x,y\right),$$

where the functions $a(x,y),$ $b\left(x,y\right),$ $c\left(x,y\right)$, and $d\left(x,y\right)$ are arbitrary. In order to acquire the solution of Equation (30), we perform the following steps. First, multiplying both sides of Equation (30) by $xy$ and the DSGLT, we have

$$\begin{array}{ccc}\hfill S_x{S}_y{G}_t\left[xy{D}_t^{2\beta }\psi \right]& =& S_x{S}_y{G}_t\left[y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y-axy{\psi }_{xxxx}-b\left(x\right)xy{\psi }_{yyyy}\right]\hfill \\ & +& S_x{S}_y{G}_t\left[-cxy{\psi }_{xxtt}-dxy{\psi }_{yytt}+xyf\left(x,y,t\right)\right].\hfill \end{array}$$

Second, employing Equation (14), we obtain

$$\begin{array}{ccc}& & \frac{{\xi }_1{\xi }_2}{s^{2\beta }}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,s)\right)\hfill \\ & =& {\xi }_1{\xi }_2{s}^{\alpha -2\beta +1}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & +{\xi }_1{\xi }_2{s}^{\alpha -2\beta +2}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & +S_x{S}_y{G}_t\left[\Delta +\Pi +xyf\left(x,y,t\right)\right].\hfill \end{array}$$

(31)

where

$$\begin{array}{ccc}\hfill \Delta & =& y{\left(x{\psi }_x\right)}_x+x{\left(y{\psi }_y\right)}_y-axy{\psi }_{xxxx}-b\left(x\right)xy{\psi }_{yyyy}\hfill \\ \hfill \Pi & =& -cxy{\psi }_{xxtt}-dxy{\psi }_{yytt},\hfill \end{array}$$

and by rewriting Equation (31),

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,s)\right)& =& s^{\alpha +1}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & +s^{\alpha +2}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right)\hfill \\ & & +\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[\Delta +\Pi +xyf\left(x,y,t\right)\right].\hfill \end{array}$$

(32)

Using the integral with respect to ${\xi }_1$ and ${\xi }_2$ from 0 to ${\xi }_1$ and 0 to ${\xi }_2$ for Equation (32), respectively, we have

$$\begin{array}{ccc}\hfill \mathsf{\Psi }({\xi }_1,{\xi }_2,s)& =& \frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}{s}^{\alpha +1}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)d{\xi }_{1}d{\xi }_2\hfill \\ & & +\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}{s}^{\alpha +2}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right)d{\xi }_{1}d{\xi }_2\hfill \\ & & +\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[\Delta +\Pi +xyf\left(x,y,t\right)\right]d{\xi }_{1}d{\xi }_2,\hfill \end{array}$$

In the third step, using the inverse DSGLT for both sides of Equation (33), the solution of Equation (30) can be written in the form

$$\begin{array}{ccc}& & \psi \left(x,y,t\right)=S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}{s}^{\alpha +1}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}{s}^{\alpha +2}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right)d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[\Delta +\Pi +xyf\left(x,y,t\right)\right]d{\xi }_{1}d{\xi }_2\right].\hfill \end{array}$$

(33)

By substituting Equation (25) into Equation (33), we obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}{s}^{\alpha +1}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2\mathsf{\Psi }({\xi }_1,{\xi }_2,0)\right)d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}{s}^{\alpha +2}\frac{{\partial }^2}{\partial {\xi }_1\partial {\xi }_2}\left({\xi }_1{\xi }_2{\mathsf{\Psi }}_t({\xi }_1,{\xi }_2,0)\right)d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xyf\left(x,y,t\right)\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[{\left(x\left(\sum _{n=0}^{\infty }{\psi }_{nx}\right)\right)}_x\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & +S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[x{\left(y\left(\sum _{n=0}^{\infty }{\psi }_{ny}\right)\right)}_y\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\sum _{n=0}^{\infty }a{\psi }_{nxxxx}+b{\psi }_{nyyyy}\right)\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\sum _{n=0}^{\infty }cxy{\psi }_{nxxtt}+dxy{\psi }_{nyytt}\right)\right]d{\xi }_{1}d{\xi }_2\right],\hfill \end{array}$$

(34)

where $n=0,1,2,\dots$. Hence, from Equation (34) above, we have

$${\psi }_0\left(x,y,t\right)=g_1\left(x,y\right)+t{g}_2\left(x,y\right)+S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{1}{u_1{u}_2}{S}_x{S}_y{G}_t\left[xyf\left(x,y,t\right)\right]d{\xi }_{1}d{\xi }_2\right]$$

and

$$\begin{array}{ccc}\hfill {\psi }_{n+1}\left(x,y,t\right)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[{\left(x\left({\psi }_{nx}\right)\right)}_x+{\left(y\left({\psi }_{ny}\right)\right)}_y\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\left(a\left(x\right){\psi }_{nxxxx}+b{\psi }_{nyyyy}\right)\right)\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\left(cxy{\psi }_{nxxtt}+dxy{\psi }_{nyytt}\right)\right)\right]d{\xi }_{1}d{\xi }_2\right],\hfill \end{array}$$

To clarify this method for the linear singular Boussinesq equation, we give the following example. We let $a=b=c=d=1$ and $f\left(x,y,t\right)=0$.

Example 2.

The singular fractional Boussinesq equation in one dimension is denoted by

$$\begin{array}{ccc}& & D_t^{2\beta }\psi -\frac{1}{x}{\left(x{\psi }_x\right)}_x-\frac{1}{y}{\left(y{\psi }_y\right)}_y+{\psi }_{xxxx}+{\psi }_{yyyy}\hfill \\ & & +{\psi }_{xxtt}+{\psi }_{yytt}-\psi =0,0<\beta \le 1\hfill \end{array}$$

(35)

with the initial conditions

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

By utilizing the indicated method for Equation (35), we can obtain

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

and

$$\begin{array}{ccc}\hfill {\psi }_{n+1}\left(x,y,t\right)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[{\left(x\left({\psi }_{nx}\right)\right)}_x+{\left(y\left({\psi }_{ny}\right)\right)}_y\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\left(a\left(x\right){\psi }_{nxxxx}+b{\psi }_{nyyyy}\right)\right)\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\left(cxy{\psi }_{nxxtt}+dxy{\psi }_{nyytt}+{\psi }_n\right)\right)\right]d{\xi }_{1}d{\xi }_2\right].\hfill \end{array}$$

The first reiteration at $n=0$ is given by

$$\begin{array}{ccc}\hfill {\psi }_1\left(x,y,t\right)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[{\left(x\left({\psi }_{0x}\right)\right)}_x+{\left(y\left({\psi }_{0y}\right)\right)}_y\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\left(a\left(x\right){\psi }_{0xxxx}+b{\psi }_{0yyyy}\right)\right)\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\left(cxy{\psi }_{0xxtt}+dxy{\psi }_{0yytt}\right)\right)-xy{\psi }_0\right]d{\xi }_{1}d{\xi }_2\right],\hfill \\ \hfill {\psi }_1\left(x,y,t\right)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}\left[S_x{S}_y{G}_t\left[\left(x^{3}y-x{y}^3\right)+\left(x^{3}y-x{y}^3\right)t\right]\right]d{\xi }_{1}d{\xi }_2\right],\hfill \\ \hfill {\psi }_1\left(x,y,t\right)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\left(2!{\xi }_1^2-2!{\xi }_2^2\right)s^{\alpha +2\beta +1}+\left(2!{\xi }_1^2-2!{\xi }_2^2\right)s^{\alpha +2\beta +2}\right],\hfill \\ \hfill {\psi }_1\left(x,y,t\right)& =& \left(x^2-y^2\right)\frac{t^{2\beta }}{\left(2\beta \right)!}+\left(x^2-y^2\right)\frac{t^{2\beta +1}}{\left(2\beta +1\right)!}.\hfill \end{array}$$

At $n=1$, we have

$$\begin{array}{ccc}\hfill {\psi }_2\left(x,y,t\right)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[{\left(x\left({\psi }_{1x}\right)\right)}_x+{\left(y\left({\psi }_{1y}\right)\right)}_y\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\left(a\left(x\right){\psi }_{1xxxx}+b{\psi }_{1yyyy}\right)\right)\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & & -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}{S}_x{S}_y{G}_t\left[xy\left(\left(cxy{\psi }_{1xxtt}+dxy{\psi }_{1yytt}\right)\right)-xy{\psi }_0\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\frac{1}{{\xi }_1{\xi }_2}{\int }_0^{{\xi }_1}{\int }_0^{{\xi }_2}\frac{s^{2\beta }}{{\xi }_1{\xi }_2}\left[S_x{S}_y{G}_t\left[\left(x^2-y^2\right)\frac{t^{2\beta }}{\left(2\beta \right)!}+\left(x^2-y^2\right)\frac{t^{2\beta +1}}{\left(2\beta +1\right)!}\right]\right]d{\xi }_{1}d{\xi }_2\right]\hfill \\ & =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[\left(2!{\xi }_1^2-2!{\xi }_2^2\right)s^{\alpha +4\beta +1}+\left(2!{\xi }_1^2-2!{\xi }_2^2\right)s^{\alpha +4\beta +2}\right]\hfill \\ & =& \left(x^2-y^2\right)\frac{t^{4\beta }}{\left(4\beta \right)!}+\left(x^2-y^2\right)\frac{t^{4\beta +1}}{\left(4\beta +1\right)!}.\hfill \end{array}$$

In the same way, let $n=2$. We arrive at

$${\psi }_3\left(x,y,t\right)=\left(x^2-y^2\right)\frac{t^{6\beta }}{\left(4\beta \right)!}+\left(x^2-y^2\right)\frac{t^{6\beta +1}}{\left(4\beta +1\right)!}.$$

Therefore, by using Equation (25), the series solutions are obtained by

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

At $\beta =1$, we obtain the exact solution as

$$\begin{array}{ccc}\hfill \psi \left(x,y,t\right)& =& \left(x^2-y^2\right)\left(1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\frac{t^4}{4!}+\frac{t^5}{5!}\dots \right)\hfill \\ & =& \left(x^2-y^2\right)e^t.\hfill \end{array}$$

4. Double Sumudu–Generalized Laplace Transform Decomposition and Singular 2+1-Dimensional Coupled System Boussinesq Equation

In this section, the DSGLTDM is used to obtain the solution of the singular 2+1-dimensional coupled system Boussinesq equation. The general formula of the singular 2+1-dimensional coupled system Boussinesq equation is given by

$$\begin{array}{ccc}\hfill D_t^{\beta }w& =& a\left(x,y\right){\left(w\psi \right)}_x+b\left(x,y\right){\psi }_{xxx}+c\left(x,y\right){\psi }_{yyy}\hfill \\ \hfill D_t^{\beta }\psi & =& d\left(x,y\right)w_x+e\left(x,y\right)w_y-\psi {\psi }_x,0<\beta \le 1\hfill \end{array}$$

(36)

with the following conditions:

$$w\left(x,y,0\right)=f_1\left(x,y\right),\psi \left(x,y,0\right)=f_2\left(x,y\right),$$

(37)

where the functions $a(x,y),$ $b\left(x,y\right),$ $c\left(x,y\right),d\left(x,y\right)$, and $e\left(x,y\right)$ are arbitrary. For the purpose of obtaining the solution of Equation (36), we use the double Sumudu–generalized Laplace transforms for Equation (36) and the DST for Equation (37). We have

$$\begin{array}{ccc}\hfill \frac{W({\xi }_1,{\xi }_2,s)}{s^{\beta }}& =& s^{\alpha -\beta +1}W({\xi }_1,{\xi }_2,0)+S_x{S}_y{G}_t\left[a{\left(w\psi \right)}_x+b{\psi }_{xxx}+c{\psi }_{yyy}\right]\hfill \\ \hfill \frac{\mathsf{\Psi }({\xi }_1,{\xi }_2,s)}{s^{\beta }}& =& s^{\alpha -\beta +1}\mathsf{\Psi }({\xi }_1,{\xi }_2,0)+S_x{S}_y{G}_t\left[d{w}_x+e{w}_y-\psi {\psi }_x\right]\hfill \end{array}$$

(38)

By reorganizing Equation (38), we can obtain

$$\begin{array}{ccc}\hfill W({\xi }_1,{\xi }_2,s)& =& s^{\alpha +1}{F}_1\left({\xi }_1,{\xi }_2\right)+s^{\beta }{S}_x{S}_y{G}_t\left[a{\left(w\psi \right)}_x+b{\psi }_{xxx}+c{\psi }_{yyy}\right]\hfill \\ \hfill \mathsf{\Psi }({\xi }_1,{\xi }_2,s)& =& s^{\alpha +1}{F}_2\left({\xi }_1,{\xi }_2\right)+s^{\beta }{S}_x{S}_y{G}_t\left[d{w}_x+e{w}_y-\psi {\psi }_x\right]\hfill \end{array}$$

Applying the inverse DSGLT, we obtain

$$\begin{array}{ccc}\hfill w(x,y,t)& =& f_1\left(x,y\right)+S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[a{\left(w\psi \right)}_x+b{\psi }_{xxx}+c{\psi }_{yyy}\right]\right]\hfill \\ \hfill \psi (x,y,t)& =& f_2\left(x,y\right)+S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[d{w}_x+e{w}_y-\psi {\psi }_x\right]\right].\hfill \end{array}$$

(39)

By inserting Equation (25) into Equation (39), one can obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{w}_n(x,y,t)& =& f_1\left(x,y\right)+S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[a{\left(\sum _{n=0}^{\infty }{w}_n{\psi }_n\right)}_x+b\sum _{n=0}^{\infty }{\psi }_{nxxx}+c\sum _{n=0}^{\infty }{\psi }_{nyyy}\right]\right],\hfill \\ \hfill \sum _{n=0}^{\infty }{\psi }_n(x,y,t)& =& f_2\left(x,y\right)+S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[d\sum _{n=0}^{\infty }{w}_{nx}+e\sum _{n=0}^{\infty }{w}_{ny}-\sum _{n=0}^{\infty }{\psi }_n{\psi }_{nx}\right]\right],\hfill \end{array}$$

(40)

and $w_0\left(x,y,t\right),{\psi }_0\left(x,y,t\right),w_{n+1}\left(x,y,t\right)$, and ${\psi }_{n+1}\left(x,y,t\right)$ are given by

$$w_0(x,y,t)=f_1\left(x,y\right),{\psi }_0(x,y,t)=f_2\left(x,y\right)$$

and

$$\begin{array}{ccc}\hfill w_{n+1}(x,y,t)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[a{\left(w_n{\psi }_n\right)}_x+b{\psi }_{nxxx}+c{\psi }_{nyyy}\right]\right]\hfill \\ \hfill {\psi }_{n+1}(x,y,t)& =& S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[d{w}_{nx}+e{w}_{ny}-{\psi }_n{\psi }_{nx}\right]\right]\hfill \end{array}$$

(41)

Now, we assume that the inverse DSGLTs with respect to ${\xi }_1,{\xi }_2$, and s exist for Equation (41).

To verify the pertinence of the above-described method for the 2+1-dimensional coupled system Boussinesq equation, we provide the upcoming example at $a=b=c=d=e=-1$.

Example 3.

The 2+1-dimensional coupled system Boussinesq equation is given by

$$\begin{array}{ccc}\hfill D_t^{\beta }w& =& -\frac{1}{2}{\left(w\psi \right)}_x-{\psi }_{xxx}-{\psi }_{yyy}\hfill \\ \hfill D_t^{\beta }\psi & =& -w_x-w_y-\psi {\psi }_x,0<\beta \le 1\hfill \end{array}$$

(42)

with the initial conditions

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

As suggested by the above method, the zeroth components $w_0$ and ${\psi }_0$ are given by the Adomian method,

$$w_0=2x-2y,{\psi }_0=2x-2y.$$

The remaining components $w_{n+1}$ and ${\psi }_{n+1},$ $n\ge 0$, are represented by using the relations

$$\begin{array}{ccc}\hfill w_{n+1}& =& -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[{\left(w_n{\psi }_n\right)}_x+{\psi }_{nxxx}+{\psi }_{nyyy}\right]\right],\hfill \\ \hfill {\psi }_{n+1}& =& -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[w_{nx}+w_{ny}+{\psi }_n{\psi }_{nx}\right]\right].\hfill \end{array}$$

(43)

By letting $n=0$ in Equation (43), we have

$$\begin{array}{ccc}\hfill w_1& =& -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[\frac{1}{2}{\left(w_0{\psi }_0\right)}_x+{\psi }_{0xxx}+{\psi }_{0yyy}\right]\right],\hfill \\ \hfill w_1& =& -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[4x-4y\right]\right]=-S_x{S}_y{S}_t\left[\left(4{\xi }_1-4{\xi }_2\right)s^{\alpha +\beta +1}\right],\hfill \\ \hfill w_1& =& -\left(4x-4y\right)\frac{t^{\beta }}{\beta !},\hfill \\ \hfill {\psi }_1& =& -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[w_{0x}+w_{0y}+{\psi }_0{\psi }_{0x}\right]\right],\hfill \\ \hfill {\psi }_1& =& -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[4x-4y\right]\right]=-S_x{S}_y{S}_t\left[\left(4{\xi }_1-4{\xi }_2\right)s^{\alpha +\beta +1}\right],\hfill \\ \hfill {\psi }_1& =& -\left(4x-4y\right)\frac{t^{\beta }}{\beta !},\hfill \end{array}$$

and at $n=1,$

$$\begin{array}{ccc}\hfill w_2& =& -S_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[\frac{1}{2}\left(w_{0x}{\psi }_1+w_{1x}{\psi }_0+w_1{\psi }_{0x}+w_0{\psi }_{1x}\right)+{\psi }_{1xxx}+{\psi }_{1yyy}\right]\right],\hfill \\ \hfill w_2& =& 4\left(2x-2y\right)\frac{t^{2\beta }}{\left(2\beta \right)!},\hfill \\ \hfill {\psi }_2& =& -S{S}_{{\xi }_1}^{-1}{S}_{{\xi }_2}^{-1}{G}_s^{-1}\left[s^{\beta }{S}_x{S}_y{G}_t\left[w_{1x}+w_{1y}+{\psi }_0{\psi }_{1x}+{\psi }_1{\psi }_{0x}\right]\right]\hfill \\ & =& 4\left(2x-2y\right)\frac{t^{2\beta }}{\left(2\beta \right)!}.\hfill \end{array}$$

In a similar way, we obtain

$$\begin{array}{ccc}\hfill w_3& =& -8\left(2x-2y\right)\frac{t^{3\beta }}{\left(3\beta \right)!},\hfill \\ \hfill {\psi }_3& =& -8\left(2x-2y\right)\frac{t^{3\beta }}{\left(3\beta \right)!},\hfill \end{array}$$

and hence, by using Equation (25), the series solutions are defined by

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{w}_n\left(x,y,t\right)& =& w_0+w_1+w_2+\dots \hfill \\ & =& \left(2x-2y\right)\left(1-\frac{2t^{\beta }}{\beta !}+4\frac{t^{2\beta }}{\left(2\beta \right)!}-8\frac{t^{3\beta }}{\left(3\beta \right)!}+16\frac{t^{4\beta }}{\left(4\beta \right)!}-\dots \right),\hfill \\ \hfill \sum _{n=0}^{\infty }{\psi }_n\left(x,y,t\right)& =& {\psi }_0+{\psi }_1+{\psi }_2+\dots \hfill \\ & =& \left(2x-2y\right)\left(1-\frac{2t^{\beta }}{\beta !}+4\frac{t^{2\beta }}{\left(2\beta \right)!}-8\frac{t^{3\beta }}{\left(3\beta \right)!}+16\frac{t^{4\beta }}{\left(4\beta \right)!}-\dots \right).\hfill \end{array}$$

Thus, the solution of Equation (42) is provided by $w\left(x,y,t\right)=\frac{2x-2y}{1+2t}$ and $\psi \left(x,y,t\right)=\frac{2x-2y}{1+2t}$.

5. Conclusions

The DSGLT is a hybrid method of the DST and GLT. In this paper, we successfully proved two main theorems by using a double Sumudu–generalized Laplace transform (DSGLT). Then, we combined this method and the ADM in order to solve the 1+1- and 2+1-dimensional fractional Boussinesq equations and the 2+1-dimensional coupled system Boussinesq equation. Moreover, we provided three examples to check the accuracy and significance of our method. We expect to examine and solve several novel and scientific phenomena in the future by employing our technique to extend the horizons of modeling in our research area.