Homotopy Analysis Method for Solving Initial Value Problems of Second Order with Discontinuities

Waleed Al-Hayani,Rasha Fahad

doi:10.4236/am.2019.106030

Abstract

In this paper, the standard homotopy analysis method was applied to initial value problems of the second order with some types of discontinuities, for both linear and nonlinear cases. To show the high accuracy of the solution results compared with the exact solution, a comparison of the numerical results was made applying the standard homotopy analysis method with the iteration of the integral equation and the numerical solution with the Simpson rule. Also, the maximum absolute error, , the maximum relative error, the maximum residual error and the estimated order of convergence were given. The research is meaningful and I recommend it to be published in the journal.

Highlights

Liao Shijun [1] [2] [3] proposed in 1992 in his Ph.D. dissertation a new and fruitful method (Homotopy Analysis Method (HAM)) for solving linear and nonlinear equations
The standard homotopy analysis method was applied to initial value problems of the second order with some types of discontinuities, for both linear and nonlinear cases
To show the high accuracy of the solution results compared with the exact solution, a comparison of the numerical results was made applying the standard homotopy analysis method with the iteration of the integral equation and the numerical solution with the Simpson rule

Summary

Introduction

Liao Shijun [1] [2] [3] proposed in 1992 in his Ph.D. dissertation a new and fruitful method (Homotopy Analysis Method (HAM)) for solving linear and nonlinear (ordinary differential, partial differential, integral, etc.) equations. Al-Hayani and Casasùs [4] [5] applied the Adomian decomposition method (ADM) to the initial value problems (IVPs) with discontinuities. Ji-Huan [6] used the homotopy perturbation method (HPM) solving for nonlinear oscillators with discontinuities. In the consulted bibliography we have not found any results of the application of the HAM to differential problems with discontinuities. For this reason, this paper systematically analyzes its application to IVPs of ODEs of second order with independent non-continuous term. We give a brief review of the HAM

Basic Idea of HAM

HAM Applied to an IVP of the Second Order

Linear Case

Non-Linear Case

Conclusions