Novel techniques for solving Goursat partial differential equations in the linear and nonlinear regime

Tahir Naseem

doi:10.54938/ijemdm.2022.01.1.7

Abstract

The Goursat problem, which is related to hyperbolic partial differential equations, occurs in a variety of branches of physics and engineering. We studied the solution of the Goursat partial differential equation utilizing the reduced differential transform (RDT) and Adomian decomposition (AD) techniques in this inquiry. The problem's analytical solution is found in series form, which converges to exact solutions. The approaches' reliability and efficiency were evaluated using the Goursat problems (linear and non-linear). Additionally, the accuracy of the findings obtained demonstrates the reduced differential approach's superiority over the Adomian decomposition method and other numerical methods previously applied to the Goursat problem.

Highlights

IntroductionThere are numerous uses of Zhou [11]'s differential transform in electric circuit analysis, including solving both linear and nonlinear initial value problems
French mathematician Goursat [1] coined the name for a linear problem of the kind = (, ) + (, ) + (, ) + (, ) + (, )
The Goursat problem, which is related to hyperbolic partial differential equations, occurs in a variety of branches of physics and engineering

Summary

Introduction

There are numerous uses of Zhou [11]'s differential transform in electric circuit analysis, including solving both linear and nonlinear initial value problems. This method generates a polynomial-based analytical answer. Gorge Adomian's [12,13,14] introduced a strategy for solving partial differential equations of any type (algebraic, differential, integro-differential, and integral) as well as stochastic situations Since this procedure has been referred to as Adomian's Decomposition Method (ADM). We will consider the following nonlinear differential equation in order to demonstrate the solution procedure of the Adomian's Decomposition method (ADM). The exact answer can be derived by employing Eq (6)

Implementation of Reduced Differential Transform Method

Implementation of Adomian’s Decomposition Method

Conclusion