Optimal Homotopy Asymptotic Method for Solving Delay Differential Equations

N Ratib Anakira,I Hashim,A K Alomari

doi:10.1155/2013/498902

Abstract

We extend for the first time the applicability of the optimal homotopy asymptotic method (OHAM) to find the algorithm of approximate analytic solution of delay differential equations (DDEs). The analytical solutions for various examples of linear and nonlinear and system of initial value problems of DDEs are obtained successfully by this method. However, this approach does not depend on small or large parameters in comparison to other perturbation methods. This method provides us with a convenient way to control the convergence of approximation series. The results which are obtained revealed that the proposed method is explicit, effective, and easy to use.

Highlights

IntroductionDelay differential equation (DDE) is a form of differential equations in which derivative of the unknown function in a given time t is specified in terms of the values at an earlier point in time
Delay differential equation (DDE) is a form of differential equations in which derivative of the unknown function in a given time t is specified in terms of the values at an earlier point in time.delay differential equations (DDEs) have the general form ui󸀠 (x) = f (x, ui (x), ui (ξj (x))), (1)i = 1, 2, . . . , M, j = 1, 2, . . . , N, where ξj(x) = ajx + bj is the delay function
optimal homotopy asymptotic method (OHAM) is employed for the first time to propose a new analytic approximate solution of delay differential equations (DDEs)

Summary

Introduction

Delay differential equation (DDE) is a form of differential equations in which derivative of the unknown function in a given time t is specified in terms of the values at an earlier point in time. Patel et al [1] introduced an iterative scheme for the optimal control systems described by DDEs with a quadratic cost functional. A new approach of homotopy which is called optimal homotopy asymptotic method (OHAM) was proposed and developed by Marinca et al [20,21,22,23,24] for the approximate solution nonlinear problems of thin film flow of a fourth-grade fluid and for the study of the behavior of nonlinear mechanical vibration of electrical machines.

Description of the Method

Convergence Theorem

Applications

Conclusions