1. 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 differential equations.

Bratu-type differential equation is an initial and boundary value problem in one-dimensional planar coordinate. It is used to model a combustion problem in a slab such as fuel ignition of the thermal combustion theory and in the Chandrasekhar model of the expansion of the universe[11]. It also used to simulate a thermal reaction process in a rigid material where the process depends on the balance between chemically generated heat and heat transfer by conduction.

The Bratu-type is of the form:

(1)

The exact solution is

(2)

Where

(3)

The equation has zero, one or two solution when respectively. The is the critical value which satisfies

(4)

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 Variational Iteration Method proposed by Olayiwola M O[7-9] is presented for the solution of Bratu-type differential equation. MVIM is the combination of VIM and the Taylor’s polynomial.

2. Variational Iteration Method

To illustrate the basic concept of the VIM, we consider the following general nonlinear partial differential equation.

(5)

where L is a linear time derivative operator, R is a linear operator which has partial derivative with respect to x, N is a nonlinear operator and g is an inhomogeneous term. According to VIM, we can construct a correct fractional as follows:

(6)

where is a Lagrange multiplier which can be identified optimally via variational iteration method. The subscript n denote the nth approximation, is considered as a restricted variation i.e, . The successive approximation of the solution will be readily obtained upon using the determined Lagrange multiplier and any selective function , consequently, the solution is given by:

(7)

In Modified Variational Iteration Method, equation (6.0) becomes:

(8)

(9)

where can be found by substituting for in (5.0) when .

3. Stationary Conditions

The simplest problem of the calculus of variation is to determine a function:

(10)

for which the value of a given functional

(11)

is a minimum or maximum.

The extremum condition (stationary condition) of the functional (11) requires that

(12)

For arbitrary , we have

(13)

and the boundary conditions

(14)

4. Derivation of

Consider (9) of the form:

Making (15) stationary

(16)

(17)

(18)

Solving (17-18), we have

(19)

Consider equation (9) of the form:

(20)

(21)

Making (21) stationary, we have:

(22)

This yields the following stationary condition

(23)

(24)

(25)

Solving (23-25), we have

(26)

Equation (21) becomes

(27)

5. Convergence of the MVIM

Let

(28)

Theorem 1: Let be the exact solution of (9) and be the solution of the sequence

With

If then the functional sequence converges to [10].

Theorem 2: Taylor’s Theorem:

A polynomial is a function of the form:

Where and are constants. The polynomial is said to be written in powers of , and is of degree if

Lemma 1: If is differentiable at , then

where is defined on a neighborhood of and

Theorem 3: If exists for some integer and is the Taylor polynomial of about ,then the limit exists.

Since Taylor’s series converges and VIM converges, then MVIM converges.

6. Application of MVIM

We present a more stable and reliable method for the solution of the form:

(29)

Where can take any value.

Applying (8.0 - 9.0) in ( 10.0) we have

(30)

Using Maple to implement MVIM, we obtained

(31)

Test Case 1: Vahidi and Hasanzade[2] considered. The problem becomes:

(32)

Vahidi and Hasanzade[2]. (11.0) becomes:

(33)

Table 1. Error comparison of the methods

The table above shows the comparison of result obtained by Vahidi[2] , Noor and Mohyud-Din[6] and .with the exact solution

Test Case 2:Noor and Mohyud-Din[6], considerd:

(34)

where

This gives:

(35)

Test Case 3:Noor and Mohyud-Din[6], considerd:

(36)

where

This gives:

(37)

The exact solutions for (13.0, 15.0, 17.0) are:

(38)

(39)

(40)

7. 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 iteration, the Noor[6] after six iteration and Vahid[2] after five iteration.

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.

ACKNOWLEDGEMENTS

Dr. M.O Olayiwola is highly grateful to the Tertiary Education Trust Funds-Education Trust Funds through Osun State University, Osogbo, Nigeria for the provision of grants for this research and also to Prof. Dr. Ji-Huan He, Soochow University China for his assistance in various forms.