A Numerical Solution of Fractional Lienard’s Equation by Using the Residual Power Series Method

Muhammed Syam

doi:10.3390/math6010001

Abstract

In this paper, we investigate a numerical solution of Lienard’s equation. The residual power series (RPS) method is implemented to find an approximate solution to this problem. The proposed method is a combination of the fractional Taylor series and the residual functions. Numerical and theoretical results are presented.

Highlights

The ordinary Lienard’s equation is given by: y00 ( x ) + f (y)y0 ( x ) + g(y) = r ( x ). (1)Different choices of f, g, and r will produce different models
Liao studied an analytical method termed as the homotopy analysis method (HAM) [9,10,11] to examine nonlinear problems
We have investigated the analytical solution of Lienard’s equation based on the residual power series (RPS)

Summary

Introduction

Liao studied an analytical method termed as the HAM [9,10,11] to examine nonlinear problems. The HAM is used by many researchers to solve various types of nonlinear problems such as fractional Black–Scholes equation [12], natural convective heat and mass transfer in a steady 2-D MHD fluid flow over a stretching vertical surface via porous media [16], micropolar flow in a porous channel in the presence of mass injection [17], etc. The power rule of the Caputo derivative is given as follows. The Caputo fractional derivative of the power function is given by:.

The Residual Power Series Method for Fractional Lienard’s Equation

Convergence Analysis

Numerical Results

Conclusions

Methods