Abstract
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.
Highlights
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]
The Modified double Laplace decomposition methods applied to the nonlinear singular one-dimensional pseudohyperbolic equation (44) with homogeneous initial condition converges toward a solution
Summary
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. 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. The 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 uΓ°x, tΓ = LΓp 1LΓs 1 2 Γ°p. 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 sΓ =. By applying double inverse Laplace transform for equation (30) and use equation (31) we get u+Γ°xL, tΓpΓ1=LΓs L124Γp 1s1L2 ΓsΓ°p1FdF1sΓ°dΓ°pppΓ,
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