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We present the optimal homotopy asymptotic method (OHAM) to find the numerical solution of the second order initial value problems of Bratu-type. We solve some examples to illustrate the validity and efficiency of the method.

Herişanu et al. [

On the other hand, the standard Bratu problem is used in a large variety of applications, such as the fuel ignition model of the theory of thermal combustion, the thermal reaction process model, the Chandrasekhar model of the expansion of the universe, radiative heat transfer, nanotechnology and theory of chemical reaction, for more information see [

The Bratu initial value problems have been studied extensively because of its mathematical and physical properties. In [

The main goal of this paper is to extend OHAM method to solve the initial value problems of second order differential equations of Bratu-type. The OHAM is very useful to get an approximate solution of the initial value problems of second order differential equations of Bratu-type. Our numerical examples of OHAM are compared with exact ones.

In this section we start by describing the basic formulation of OHAM, see for example [

where

By means of OHAM one constructs a homotopy

where

Therefore, the unknown function

In the sequel, we choose auxiliary function

where

In order to obtain an approximate solution, we expand

Now, substituting by Equation (2.5) into Equation (2.2) and equating the coefficients of like powers of

and

respectively. Also, the general governing problems of

where

where

Observe that the convergence of the series (2.5) depends upon the auxiliary constants

The m-th order approximations are given by

By substituting Equation (2.11) into Equation (2.1), we get the following expression for residual

If

where

After knowing those constants, the approximate solution of order

Example 1 Consider the second order initial value problem of Bratu type

The initial value problem (3.1) has

Next, we apply the OHAM method to the initial value problem (3.1). We have

we have

Problem of zero order:

which has a solution

Problem of first order:

Problem (3.3) has a solution

The problem of second order

The solution of Problem (3.5) is given by

Third order problem is

and its solution is given in the form

Finally, fourth order problem is

which has a solution in the form

Now, by using equations (3.4), (3.6), (3.8) and (3.10), the fourth order approximate solution, using OHAM with

Next, we follow the procedure presented in Section 2, we obtain the following values of

Example 2 In this example, let us consider the Bratu initial value problem

which has

Now, we apply the OHAM method presented in previous section. In this example, we have

Problem of zero order:

Problem (3.13) has a solution

Problem of first order:

The solution of Problem (3.14) is given by

The problem of second order

and its solution is given by

Third order problem is

The solution of Problem (3.18) is given by

In the end, the fourth order problem is given by

which has a solution in the form

Now, by using equations (3.4), (3.6), (3.8) and (3.10), the fourth order approximate solution, using OHAM with

Next, we follow the procedure presented in Section 0.2, we obtain the following values of

Throughout this paper, an technique for obtaining a numerical solution for second order initial value problems of Bratu-type, is optimal homotopy asymptotic method (OHAM). The main advantage of the used technique is achieving high accurate approximate solutions. In the numerical tables and graphics, our numerical results are compared with the exact ones.

Mohamed AbdallaDarwish,Bothayna S.Kashkari, (2014) Numerical solutions of second order initial value problems of Bratu-type via optimal homotopy asymptotic method. American Journal of Computational Mathematics,04,47-54. doi: 10.4236/ajcm.2014.42005