Optimal Homotopy Asymptotic and Multistage Optimal Homotopy Asymptotic Methods for Solving System of Volterra Integral Equations of the Second Kind

Jafar Biazar,Roya Montazeri

doi:10.1155/2019/3037273

Abstract

In this paper, optimal homotopy asymptotic method (OHAM) and its implementation on subinterval, called multistage optimal homotopy asymptotic method (MOHAM), are presented for solving linear and nonlinear systems of Volterra integral equations of the second kind. To illustrate these approaches two examples are presented. The results confirm the efficiency and ability of these methods for such equations. The results will be compared to find out which method is more accurate. Advantages of applying MOHAM are also illustrated.

Highlights

IntroductionDifferential equations, integrodifferential equations, and system of such equations, linear and nonlinear, usually appeared in mathematical modeling of different phenomena in physics, biology, and engineering [1,2,3,4]
Yusufoglu employed homotopy pertubation method to solve a system of Fredholm–Volterra type integral equations [18]
We will consider a modified version of OHAM, that is called multistage optimal homotopy asymptotic method (MOHAM). This approach was introduced for the first time by Anakira et al to approximate the solutions of differential equations with initial-values [19]

Summary

Introduction

Differential equations, integrodifferential equations, and system of such equations, linear and nonlinear, usually appeared in mathematical modeling of different phenomena in physics, biology, and engineering [1,2,3,4]. Sezar et al applied Chebyshev polynomial method and Taylor collocation method for systems of linear differential equations and integrodifferential equations [1, 17]. We apply OHAM to solve systems of integral equations of the second kind. We will consider a modified version of OHAM, that is called multistage optimal homotopy asymptotic method (MOHAM). This approach was introduced for the first time by Anakira et al to approximate the solutions of differential equations with initial-values [19]. Illustrative examples are presented, and conclusion appeared in the last section

OHAM and MOHAM

Application of OHAM to Systems of Volterra Integral Equations

Application of MOHAM to Systems of Volterra Integral Equations

Convergence of the OHAM

Numerical Examples

Conclusion and Discussion