Equivalence of ( A.D.M, H.P.M, A.P.M ) for Solving Functional Equations

J Biazar,P Shams Derakhsh

doi:10.13189/ujes.2014.020101

J Biazar, P Shams Derakhsh

Open Access

https://doi.org/10.13189/ujes.2014.020101

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Abstract

The Adomian's decomposition method, the Homotopy perturbation method, and lyapunov's method are three powerful methods which consider an approximate solution of linear and non-linear equations, as an infinite series. In this paper, we show that these three methods are equivalent in solving functional equations. To illustrate the capability and reliability of the methods two examples are provided. Numerical solutions obtained by these methods are compared with the exact solutions we see that usually converging to an exact solution.

Highlights

Adomian has developed a numerical technique for solving functional equations
In this method an artificial parameter is as a factor to non-linear part of equation [9,10]. in this paper we show that the ADM, APM and homotopy perturbation method (HPM) are equivalence for functional equations
It is worth to mention that the solution in all of these methods are as a power series. let us consider the non-linear functional equation: A (u) - f (r) = 0, r ∈Ω, (1)

Summary

Introduction

Adomian has developed a numerical technique for solving functional equations. The homotopy perturbation method (HPM) was established by Ji-Huan He in 1999 In this method, the solution is considered as the sum of an infinite series, which converges rapidly to accurate solutions .It is a powerful and efficient technique for solving non-linear functional equations,without the need of linearization process. Artificial small parameter method has been introduced by lyapunov, which is usually applied to solve non-linear functional equations. In this method an artificial parameter is as a factor to non-linear part of equation [9,10]. In this method an artificial parameter is as a factor to non-linear part of equation [9,10]. in this paper we show that the ADM, APM and HPM are equivalence for functional equations

Illustrate the Equivalence of Methods

Examples

Example2

Discussion