APPLICATION OF VARIATIONAL ITERATION METHOD TO FIND THE SOLUTION OF COUPLED SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS

Present work is devoted to the application of variational iteration method to obtain the solutions of coupled system of first order differential equations. Numerical examples are taken to test the efficiency of this method. We have shown that the successive approximations of each problem are converging to their exact solution. Further we have shown graphically the third, fourth, fifth and sixth approximation values and exact values.


INTRODUCTION
Many problems in various fields of sciences and engineering yield partial differential equations. 4119

APPLICATION OF VARIATIONAL ITERATION METHOD
For their physical interpretation we need their solutions. It may not possible to find the exact solution of certain partial differential equations. There are many methods to find the approximate solutions of such equations. For example, numerical methods, Adomain decomposition method, Homotopy analysis method etc. J.H.He [3][4] developed the variational iteration method to solve linear and non-linear ordinary and partial differential equations. Several researchers [1,2,5] are working on application of this method to find the solutions of linear and nonlinear partial differential equations. E.Rama, K.Somaiah and K.Sambaiah [6] have used this method for obtaining solution of various types of problems. In this paper an attempt is made to find the successive approximate solutions of system of first order coupled differential equations using Variational iteration method. Further it was shown that these solutions are converging to their exact solutions.

DESCRIPTION OF VARIATIONAL ITERATION METHOD
Let L be a linear operator, N be a non-linear operator and g(x) is known continuous function.
Let ) (t y n be the n th approximate solution and ) ( t y n denotes restricted variation, that is The successive approximations yn+1(t) will be computed by applying the obtained
The correctional functional for the above system of differential equations are given by …………………….
are Lagrangian multipliers which are to be obtained using the variational theory. yi,j denotes jth approximation of dependent variable yi and n ỹ denotes restricted variation.

APPLICATIONS
We consider few examples to know the efficiency and accuracy of the variational iteration method for obtaining the solution of coupled system of equations of first order differential equations.

Example:
Consider the following coupled system of first order differential equations with the initial conditions y1(0)=2 and y2(0)=0. Note that y1 and y2 are dependent variables and t is the independent variable. The exact solution of this system is y1(t)=e 4t +e -2t and y2(t)= e 4t -e -2t .
These y1(t) and y2(t) in terms of series are as follows.
To solve (4.1) and (4.2) by variational iteration method we consider the terms involving y 1 and y 2 as restricted variation except their derivative terms. Hence the correction functional of (4.1) and   Continuing this process for n=1,2,3 we have the following approximations of y1 and y2.
The exact values and the 3 rd ,4 th ,5 th and 6 th approximations of y 1 are shown in the Fig.1. and that of y 2 are shown in the Fig.2. with the initial conditions y1(0)=1 and y2(0)= -1. The exact solution this system is y1(t)=cos(t)+sin(t) and y2(t)= -2sin(t)-cos(t). These y1(t) and y2(t) in terms of series are as follows.
Proceeding as in the above example it can be shown that the Lagrangian multipliers are -1 and -1.
In view of (4.15) and (4.16) we have