Solution of a Nonlinear Delay Differential Equation Using Adomian Decomposition Method with Accelerated Formula of Adomian Polynomial

I L El-Kalla,Ali R Elmahdy,Ahmed Y Sayed,Khaled M Abd Elgaber

doi:10.4236/ajcm.2019.94017

I L El-Kalla, Ali R Elmahdy + Show 2 more

Open Access

https://doi.org/10.4236/ajcm.2019.94017

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Abstract

The aim of this paper is to apply Adomian decomposition method (ADM) for solving some classes of nonlinear delay differential equations (NDDEs) with accelerated Adomian polynomial called El-kalla polynomial proposed by El-kalla [1]. The main advantages of El-kalla polynomials can be summarized in the following main three points: 1) El-kalla polynomials are recursive and do not have derivative terms so, El-kalla formula is easy in programming and save much time on the same processor compared with the traditional Adomian polynomials formula; 2) Solution using El-Kalla polynomials converges faster than the traditional Adomian polynomials; 3) El-Kalla polynomials used directly in estimating the maximum absolute truncated error of the series solution. Some convergence remarks are studied and some numerical examples are solved using the Adomian decomposition method using the two polynomials (Adomian polynomial and El-kalla polynomial). In all applied cases, we obtained an excellent performance that may lead to a promising approach for many applications.

Highlights

The basic motivation of this work is to compare between the solution of the NDDEs by the Adomian decomposition method using El-kalla polynomial and Adomian polynomial with the exact solution
We will talk about the Adomian decomposition method, we talk about the formulas that calculate the two polynomials, we will talk about the Convergence Remarks of the method, we solve some examples to show the importance of the new accelerated formula called El-kalla polynomial
For example the Adomian polynomials & El-Kalla polynomials of the nonlinear term y2 are showen in Table 1 we can see that the terms using El-Kalla polynomials appear faster than Adomian polynomial

Summary

Introduction

The Adomian decomposition method has been shown [8] [9] [10] to solve effectively, and accurately a large class of linear, nonlinear, ordinary and partial differential equations with approximate solutions which converge rapidly to accurate solutions. Al-Mazmumy and Al-Malki discussed Some Modifications of Adomian Decomposition Methods for Nonlinear Partial Differential Equations [12]. The basic motivation of this work is to compare between the solution of the NDDEs by the Adomian decomposition method using El-kalla polynomial and Adomian polynomial with the exact solution. We will talk about the Adomian decomposition method, we talk about the formulas that calculate the two polynomials, we will talk about the Convergence Remarks of the method, we solve some examples to show the importance of the new accelerated formula called El-kalla polynomial

The Method

Convergence Remarks

Solution Using El-Kalla Polynomials

Discussions