Modified Variational Iteration Method for the Solution of a Class of Differential Equations

In this paper, a Modified Variational Iterat ion Method (MVIM) for the solution of a differential equation of Bratu-type is presented. The method converges to the exact solution after an iteration. This shows that the method is efficient for this class of init ial and boundary value problems.


Introduction
Differential equations as a subject are a deductive science and a branch of Mathematics which have strong roots in physical problems such as Physics and Engineering. This subject also derives much of its power and beauty from the variety of its applications This equations play a crucial role in applied mathematics and physics. The results of solving such equations can guide authors to know the described process deeply. But it is difficult to obtain the exact solution for some of these problems. In recent decades, there has been great development in the numerical methods for the solution of ordinary and partial linear and nonlinear d ifferential equations.
Bratu-type differential equation is an init ial and boundary value problem in one-dimensional p lanar coord inate. It is used to model a co mbustion problem in a slab such as fuel ignition of the thermal co mbustion theory and in the Chandrasekhar model of the expansion of the universe [11]. It also used to simulate a thermal reaction process in a rig id material where the process depends on the balance between chemically generated heat and heat transfer by conduction.
The Bratu-type is of the form: The exact solution is The equation has zero, one or t wo solution when It has been shown that see [1,11,12] Several authors have presented various numerical approach to the solution of Bratu-type differential equations [1,2,6,11,12].
The variational iteration method was proposed by J.H He [4][5]. In this paper a Modivied Variat ional Iteration Method proposed by Olayiwola M O [7][8][9] is presented for the solution of Bratu-type differential equation. MVIM is the combination of VIM and the Taylor's polynomial.

Variational Iteration Method
To illustrate the basic concept of the VIM, we consider the following general nonlinear part ial differential equation.
, , , Lu x t Ru x t Nu x t g x t + + = (5) where L is a linear t ime derivative operator, R is a linear operator which has partial derivative with respect to x, N is a nonlinear operator and g is an inho mogeneous term.
According to VIM, we can construct a correct fractional as follows: where λ is a Lagrange mult iplier which can be identified optimally via variational iteration method. The subscript n denote the nth approximat ion, In Modified Variat ional Iteration Method, equation (6.0) becomes:

Stationary Conditions
The simplest problem of the calculus of variation is to determine a function: ) (x f y = (10) for which the value of a g iven functional is a minimu m or maximu m. The extremu m condition (stationary condition) of the functional (11) requires that ( ) , : For arb itrary y δ , we have (13) and the boundary conditions (14) 4. Derivation of λ Consider (9) o f the form: (20) Making (21) stationary, we have:   x , then x ,then the limit exists.

U x t U x t e U x s U x s AU x s F s x ds s x
Since Tay lor's series converges and VIM converges, then MVIM converges.

Application of MVIM
We present a more stable and reliab le method for the solution of the form: 2 Where λ can take any value.

Conclusions
In this study, we have shown that the modified variational iteration method can be successfully applied for finding the solution of a class of differential eqution.
The above table is the comparison of the error results of MVIM result after two iterat ion, the Noor [6] after six iteration and Vahid [2] after five iterat ion.
The method does not involved the introduction of any set of algebraic equations that will be solved for another set of variables. The other two cases also converges to the exact solution.
The result shows that the MVIM is a novel approach to the solution of Bratu-type differential equations.