1. Introduction

One of the problems that had been attacked by several authors is Bratu’s boundary value problem which is given by

(1)

(2)

where A typical example occurs in the theory of the electric charge around a hot wire and also in certain problems of solid mechanics. The Bratu’s problem in one-dimensional planner coordinates has two known, bifurcated solutions for values of , no solution for and a unique solution when .The value of is related to the fixed point of hyperbolic contangent function and satisfies the equation

The exact solution of (1) and (2) is given by

provided that is the solution of With given boundary conditions it is not possible to solve (1) by elementary methods.Various kinds of analytical methods and numerical methods were used to solve this equation[1-6]. For example, Bellman and Kalaba find substantial agreement of to exact solution by applying quasilinearization method [5] and S. A. Khuri used a Laplace transform numerical technique for solving this problem [3].

It been shown that the variational iteration method is a very efficient tool for solving various kinds of nonlinear ordinary and partial differential equations[7-12]. Itbeen used to solve the Fokker Planck equation, the Lane-Emden differential equation, the Klein-Gordon partial differential equations, the Cauchy reaction-diffusion problem, the biological population model. For more applications of the method the interested reader is referred to [7]. It also is useful to solve integral equations [13-15].

To illustrate the basic idea of the method, we consider the differential equation

(3)

Where ℒ and N are linear and nonlinear operators respectively, and is the source inhomogeneous term.

The variational iteration method presents a correction functional for Eq. (3)in the form

Where is a general Lagrange multiplier[3], which can be optimally found via variational theory and is a restricted variation which means,

2. Procedure

RecallBratu’s boundary value problem

(1)

(2)

We use Taylor series expansion for nonlinear part and write

Thus

Impose (one may use and apply to find k), we obtain

Integrating by parts leads to

We have

Therefore,

Again, we use integration by parts. We obtain

(4)

Since with a given value for we can find four values for kWith appropriate choose of k, can be approximated for

3. Examples

In this section we examine the above procedure for . Applying to Eq. (4) and simplify it, we come to

Solving this equation by a numerical method, gives

Since with two decimal points we may choose, then

We tested it for values of , respectively and the results are given in Table 1.

For to Eq. (4), we come to

(5)

If we apply to Eq. (5), we obtain

Roots of this equation are

Our appropriate choose will be . We tested it similar to case of The results are given in Table 2.

Table 1.

Table 2.

4. Conclusions

Results in this paper show the efficiency of variational iterative method and is quite reliable.It can be observed that our approximations are accurate using only two iterates and avoid the much computation using Adomian polynomials[1, 3] or quasilinearization method [5]. There is no doubt, if more terms of Taylor series expansion, or more iterates were used, we obtain much better results. This approach also can be used for Poisson-Boltzman’s equation in one-dimensional planar coordinates if we choose .