Exact Solution of Linear and Non-linear Goursat Problems

Jamshad Ahmad,Mariyam Mushtaq

doi:10.13189/ujcmj.2015.030103

Abstract

In this research paper, we applied the Adomian's decomposition method to determine the analytical exact solutions of linear and nonlinear Goursat problems which play very important part in applied and engineering sciences. The proposed technique is fully compatible with the complexity of these problems and obtained results are highly encouraging. Some examples with closed form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient. presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives. The Adomian's Decomposition Method has been shown to solve effectively, easily, and accurately a large class of linear and nonlinear problems, generally two or three iterations lead to high accurate solutions. The basic motivation of the present study is to extend the application of Adomian's decomposition method to linear and nonlinear Goursat problems.

Highlights

Most of the problems in natural and engineering sciences are modeled by differential equations
The basic motivation of the present study is to extend the application of Adomian’s decomposition method to linear and nonlinear Goursat problems
The Goursat partial differential equation arises in linear and non linear partial differential equations with mixed derivatives in the study of wave phenomena. This equation is a second order hyperbolic partial differential equation which occurs in various fields of study such as in engineering, physics, and applied mathematics

Summary

Introduction

Most of the problems in natural and engineering sciences are modeled by differential equations. The Goursat partial differential equation arises in linear and non linear partial differential equations with mixed derivatives in the study of wave phenomena This equation is a second order hyperbolic partial differential equation which occurs in various fields of study such as in engineering, physics, and applied mathematics. Adomian’s Decomposition Method was introduced by Adomian’s in [7,8] and used heavily in the literature to solve the wide class of physics and engineering problems such as non-linear differential equation, non-linear dynamic system, coupled non-Linear differential equations, linear and non-linear integro-differential equation and Airy’s equation successfully [9-16]. The exact solution may be obtained by using Eq (2)

Numerical Applications

Conclusions