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

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


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

Properties of Double Sumudu-Generalized Laplace Transform
The definitions and existence conditions of the double Sumudu-generalized Laplace transform are presented.Here, we work with the double Sumudu-generalized Laplace transform, which is defined by Existence Condition for the Double Sumudu-Generalized Laplace Transform In the following, the conditions for the existence of the double Sumudu-generalized Laplace transform are offered.If f (x, y, t) is an exponential order a 1 , a 2 and b as x → ∞, y → ∞, t → ∞, and if ∃R > 0 similarly applies for all x > X , y > Y and t > T for some X, Y and T, then we write and similarly, lim Theorem 1.The function f (x, y, t) is defined on (0, X), (0, Y) and (0, T) and is of exponential order (x, y, t), then the double Sumudu-generalized Laplace transform of f (x, y, t) exists for all Proof.By putting Equation (1) into Equation (2), we obtain From using the condition ( 1 Proof.By applying a partial derivative according to ζ 1 for Equation (1), we obtain and by computing the partial derivative within the brackets, we obtain then substituting Equation ( 7) into Equation ( 6), we obtain and by taking the derivative according to ζ 2 for Equation (8), we achieve Then, Equation ( 9), becomes by arranging the above equation, we achieve and thus, through simplifying, we obtain The proof is finished.
The following theorem offers the double Sumudu-generalized Laplace transform of the partial derivatives xyψ t .
Theorem 3. The double Sumudu-generalized Laplace transform of the partial derivatives xyψ t is presented by Proof.By taking the partial derivative according to ζ 1 for Equation ( 22), we have we calculate the partial derivative inside brackets as shown below: substituting Equation ( 13) into Equation ( 12), we obtain by taking the partial derivative according to ζ 2 for Equation ( 14), and therefore, Equation (15) becomes and hence, by reordering Equation ( 17), we prove Equation ( 11): . Moreover, the Sumudu-generalized Laplace transform of ψ x , ψ xx , ψ tt and ψ t is determined by and s); therefore, the double Sumudu -generalized Laplace transform of ψ x , ψ xx , ψ t and ψ tt is provided by and

Double Sumudu-Generalized Laplace Decomposition Method and 2+1-Dimensional Linear Pseudoparabolic Equation
The following is the procedure showing two problems that are concerning to the linear and nonlinear singular 2+1-D pseudoparabolic equation.
The general singular 2+1-D pseudo parabolic equation is considered as follows: dependent on the initial condition where the functions f 1 (x, y) and f (x, y, t).Firstly, we obtain the product of both sides of Equation ( 21) by xy, and implementing double Sumudu-generalized Laplace, we yield Equation ( 21), and implementing double Sumudu transform for Equation ( 22), we obtain and by arranging Equation ( 23), it becomes and therefore, Equation (24) becomes By taking the integral for Equation (25), from 0 to ζ 1 and 0 to ζ 2 according to ζ 1 and ζ 2 , we have The solution is obtained by using the inverse double Sumudu-generalized Laplace for Equation (26): where S −1 s indicates the inverse double Sumudu-generalized Laplace.The double Sumudu-generalized Laplace decomposition method (DSGLTDM) defines the solutions ψ(x, y, t) with the help of infinite series as By substituting Equation (28) into Equation ( 27), we receive By matching both sides of Equation (29), we obtain In general, the remaining terms are given by where the inverse double Sumudu-generalized Laplace transform is given by S −1 Here, we assume that the inverse exists for Equations ( 30) and (31).In order to demonstrate the advantages and the preciseness of the (DSGLTDM) for solving singular 2+1-D parabolic equations, we employed the method proposed in the example below.
Example 2. Consider the singular 2+1-D pseudoparabolic equation given by For the purpose of keep on with our method for Equation (35), we obtain the exponential function formula is defined by where at n = 0, we have and therefore, In a similar way, at n = 1, we have In the same way, at n = 1, we obtain Therefore, the approximate solution of Equation ( 35) is presented as follows:

Double Sumudu-Generalized Laplace Decomposition Method and 2+1-Dimensional Nonlinear Pseudoparabolic Equation
We must illustrate the double Sumudu-generalized Laplace decomposition method to solve the singular 2+1-dimensional nonlinear pseudoparabolic equation: Consider the following general form of the singular 2+1-dimensional nonlinear pseudoparabolic equation of the model: with initial condition ψ(x, y, 0) = h 1 (x, y), where the functions µ(x) and ν(x) are arbitrary.With a view to procure the solution of Equation (37), first, obtain the product of both sides of Equation (37) by xy, and implementing double the Sumudu-generalized Laplace transform, we find Second, applying theorem 1 and 2 into Equation (39), we obtain By taking the integral for Equation (40), from 0 to ζ 1 and 0 to ζ 2 according to ζ 1 and ζ 2 , we have x S y G t y(xψ x ) x + x yψ y y dζ 1 dζ 2 The third step, now using inverse double Sumudu-generalized Laplace for both sides of Equation ( 41), the solution of Equation (37) can be written as By substituting Equation (28) into Equation (42), we obtain where n = 0, 1, 2, . . . .Hence, from Equation (43) above, we have where nonlinear terms A n and B n are addressed as where the nonlinear terms ψ x ψ xx and (ψ x ) 2 are in the following forms: and To ascribe this method to a linear singular 2+1-dimensional nonlinear pseudoparabolic equation, we give the next example.

Conclusions
In this study, we established a new hybrid method, which is named the double Sumudu-generalized Laplace transform decomposition method (DSGLTDM).This approach is successfully applied in singular 2+1-D pseudoparabolic equations.The (DS-GLTDM) is an analytical process and works by utilizing the initial conditions only.Three examples are offered to examine the accuracy of the method.
Funding: The author would like to extend their sincere appreciation to Researchers Supporting Project number (RSPD 2023R948), King Saud University, Riyadh, Saudi Arabia.

Theorem 2 .
If the double Sumudu-generalized Laplace transform of the function f (x, y, t) is presented by S x S y G t ( f (x, y, t)) = F(ζ 1 , ζ 2 , s), then the double Sumudu-generalized Laplace