Application of double Laplace decomposition method to solve a singular one-dimensional pseudohyperbolic equation

In this work, the double Laplace decomposition method is applied to solve singular linear and nonlinear one-dimensional pseudohyperbolic equations. This method is based on double Laplace transform and decomposition methods. In addition, we prove the convergence of our method. This method is described and illustrated by some examples. These results show that the introduced method is highly accurate and easy to apply.


Introduction
The linear and nonlinear pseudohyperbolic equations are the important classes of evolution equations which have been developed in recent years, and there is an extensive application in chemistry, plasma physics, thermo-elasticity, and engineering. Many powerful methods have been developed to solve linear and nonlinear partial differential equations (PDEs), such as homotopy perturbation method, 1,2 combined Laplace transforms and decomposition method, 3 the transformed rational function method which presents exact traveling wave solutions to nonlinear integro-differential equations has been studied in Ma and Lee, 4 the bi-linear techniques 5 which present multiple wave solutions to nonlinear differential equations, and the integral transform method. [6][7][8][9] An auxiliary parameter method using Adomian polynomials and Laplace transformation have been powerfully combined 10 to study the nonlinear differential equation. The one-dimensional nonlinear hyperbolic equation with Bessel operator is one of the fundamental nonlinear wave equations having many applications in science. The energy-integral method is used to handle nonlinear singular one-dimensional hyperbolic equation. 11 The convergence of Adomian's method has been studied by several authors. [12][13][14][15][16][17][18] In this article, we are concerned with the following problem where a, b, c are constants and (1=x m )(∂=∂x)(x m (∂u=∂x)) is called Bessel's operator and f is a known function, where m = 1, 2, 3, . . .. Equation (1) is described by the following cases: In the general case when a 6 ¼ 0, b 6 ¼ 0, and c 6 ¼ 0, equation (1) is called singular one-dimensional pseudolike-wave equation.
The aim of this article is to use the double Laplace transform and domain decomposition method to obtain approximate solutions with high accuracy for a singular one-dimensional pseudohyperbolic equation and a singular one-dimensional pseudolike-wave equation. In addition, one of the main aims of this article is to provide a sufficient condition of convergence of the series. Now, we recall the following definitions which are given by previous studies. [19][20][21][22] The double Laplace transform is defined as where x, t.0 and p, s are complex values, and further double Laplace transform of the first-order partial derivative is given by Similarly, the double Laplace transform for secondorder partial derivative with respect to x and t are defined as follows The following Lemma is used in this article.
Lemma 1. Double Laplace transform of the nonconstant coefficient second-order partial derivative x r (∂ 2 u=∂t 2 ) and the function x r f (x, t) are given by L x L t x r ∂ 2 u ∂t 2 = À1 ð Þ r d r dp r s 2 U (p, s) À sU (p, 0) À and where r = 1, 2, 3, . . . . One can prove this lemma using the definition of double Laplace transform in equations (3)-(5).

Singular one-dimensional pseudohyperbolic equation
To illustrate the basic idea of the modified double Laplace decomposition method, we assume that c = 0 and m = 1 in equation (1), we obtain the singular onedimensional pseudohyperbolic equation where the term (1=x)(∂=∂x)(x(∂u=∂x)) is Bessel operator. In the following theorem, we apply modified double Laplace decomposition methods.
Theorem 1. We claim that the solution of the singular one-dimensional pseudohyperbolic equation given in equation (8) is denoted by where L x L t double Laplace transform with respect to x, t and L À1 p L À1 s double inverse Laplace transform with respect to p, s, the function F( p, s), F 1 ( p), and F 1 ( p) are Laplace transform of the functions f (x, t), f 1 (x), and f 2 (x), respectively. Here, we provided double inverse Laplace transform with respect to p and s exist for each term in the right-hand side of equation (10).
Proof. By multiplying equation (8) by x and using the definition of partial derivatives of the double Laplace transform, single Laplace transform, and the Lemma 1 for equation (8), respectively, we get Applying the integral for both sides of equation (11) from 0 to p with respect to p, we have The next step in double Laplace decomposition method is representing the solution of singular onedimensional pseudohyperbolic equation as u(x, t) by the infinite series By applying double inverse Laplace transform for equation (12) and use equation (13), we obtain In particular, we have and By extending equation (8) as follows On using the above theorem and Lemma 1, we have By taking double inverse Laplace transform for equation (18), we get Then, the solution of equation (17) is given by To illustrate our method for solving the singular one-dimensional pseudohyperbolic equation, in the case a = b = 1 in equation (8), we consider the following example.
subject to By taking double and single Laplace transform for equations (21) and (22) and applying theorem 1, we have dU p, s ð Þ dp = À 6 p 4 s 2 + 6 p 4 s 2 s 2 + 1 ð Þ Integrating both sides of equation (23) from 0 to p with respect p, we obtain On using double inverse Laplace transform, we have u x, t ð Þ= x 2 t + x 2 sin t À x 2 t + 4 sin t À 4t + 4 cos t À 4 Using equations (15) and (16), we get u 0 = x 2 t + x 2 sin t À x 2 t + 4 sin t À 4t + 4 cos t À 4 The other components are given by It is obvious that self-canceling some terms appear between various components and connected by coming terms, we have u x, t ð Þ= u 0 + u 1 + Á Á Á therefore, the exact solution is given by

Singular nonlinear one-dimensional pseudohyperbolic equation
In this section, we discuss the use of modified double Laplace to solve the singular one-dimensional pseudohyperbolic equation subject to where (1=x)(∂=∂x)(x(∂u=∂x)) is Bessel operator, and f (x, t) and a(x) are known functions. To obtain the solution of singular one-dimensional pseudohyperbolic equation (27), we apply our method as follows. Using the definition of partial derivatives of the double Laplace transform, single Laplace transform for equations (27) and (28), respectively and Lemma 1, we have dU p, s ð Þ dp = 1 s dF 1 p ð Þ dp By integrating both sides of equation (29) from 0 to p with respect to p, we have The double Laplace Adomian decomposition method (DLADM) defines the solution of equation (27) as u(x, t) by the infinite series By applying double inverse Laplace transform for equation (30) and use equation (31) we get The nonlinear operators can be defined as follows where A n and B n are denoted by and Some terms of Adomian's polynomials A n and B n are given by We can write equation (32) in the following way In particular dF p, s ð Þ dp dp 2 4 3

: ð38Þ
In general, we have Eltayeb et al.
where N 1 and N 2 are defined in equation (33). By calculating the terms u 0 , u 1 , . . . we obtain the solution as To illustrate the modified double Laplace decomposition method for solving the singular nonlinear onedimensional pseudohyperbolic equation, we let g(x) = x 2 , a = 8, h(x) = x=2, and f (u) = 0 in equation (27), hence we have the following example.

Convergence analysis of the method
In this section, we will discuss the convergence analysis of the modified double Laplace decomposition methods for the singular nonlinear one-dimensional pseudohyperbolic equation which is given by For L hemicontinuous operator, consider the following hypotheses:

(H1)
L u ð Þ À L v ð Þ, u À v ð Þ ! k u À v k k 2 ; k.0, 8u, v 2 H. Proof. To verify the convergence hypotheses (H1) for equation (44), we use the definition of our operator L, and we have the following form