Adomian method for solving some coupled systems of two equations

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Abstract

Coupled systems of two linear and nonlinear differential equations for second- and first-orders, respectively, can be solved with some techniques and Adomian decomposition method. A few simple examples are also studied to show with analytical results how the (ADM) works efficiently.

Introduction

Coupled system of two equations for second-order are needed in the formulation of various physical situations [3], [4], [5], [6], [7], [8], [9], [11]. For comments on their importance, we refer the reader to the above papers. As an example of such systems is the coupled system of two Schrödinger linear equations which has been given extensive attention in recent years both analytically and numerically [3], [4], [5], [6], [7], [8]. The consideration of this system is motivated by a number of physical problem in various fields. Also, systems of nonlinear differential equations for first-order are encountered when studying mathematical models for certain natural, physical and biological processes. As an example of such systems, is the system was proposed for the model of Lotka–Volterra [10]. This mathematical model is based on nonlinear system of two ordinary differential equations.

The plan of the present paper is as follows: In Section 1 we formulate a coupled system of linear equations for second-order which contains as a special case the system of Schrödinger equations and we prove a result which governs the separation of a system of two coupled equations. The Adomian decomposition method is investigated in Section 2. It is shown that (ADM) can be an effective scheme to obtain the analytical and approximate solutions. In Section 3 we propose a new transformation for solving a system of nonlinear differential equations for first-order and the coupled system can be solved by (ADM) and back-substitution.

Section snippets

Coupled system of two equations

Consider the following coupled system of two ordinary differential equations for second-order [3], [4], [5], [6], [7], [8]:d2udx2+p(x)dudx-f1(x)u=b1(x)v+F1(x),d2vdx2+p(x)dvdx-f2(x)v=b2(x)u+F2(x),u(0)=α1,dudx(0)=β1,v(0)=α2,dvdx(0)=β2,where p(x), bi(x), fi and Fi(x), (i = 1, 2) are assumed to be analytic functions.

As announced in the Introduction, we first begin with the following result on the separation of this system in which the two equations are decoupled.

Lemma 1

The system (2.1), (2.2), (2.3), (2.4)

Adomian’s decomposition method

Adomian [1], [2] has presented and developed a so- called decomposition method for solving linear or nonlinear problems such as ordinary differential equations. It consists of splitting the given equation into linear and nonlinear parts, inverting the highest-order derivative operator contained in the linear operator on both sides, identifying the initial and/or boundary conditions and the terms involving the independent variable alone as initial approximation, decomposing the unknown function

A system of nonlinear differential equations for first-order

Mathematical modelling of many frontier physical leads to systems of nonlinear ordinary differential equations. Motivated by this, we propose a class of coupled system of two nonlinear differential equations with first-order similar to model of Lotka–Volterra [10] which may be written in the following form:dudx+f1(x)u+b1(x)v=α(x)(u2+uv),dvdx+f2(x)v+b2(x)u=α(x)(v2+uv).Subject to the initial conditionsu(0)=u0,v(0)=v0,where fi, bi, i = 1, 2 and α(x) are being arbitrary analytic functions.

The

Conclusion

The Adomian decomposition method was used successfully to solve a class of coupled systems of two linear and nonlinear differential equations for second- and first-orders, respectively. Some examples with closed form solutions are studied carefully in order to illustrate the possible practical use of this method, and the results obtained are just the same as those given from applying the ADM.

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