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In this paper, we developed a new numerical scheme which aimed to solve some initial value problems of ordinary differential equations. The full breakdown of this new numerical scheme derivation is presented. While in our subsequent research, we shall fully examine the characteristics of the scheme such as consistency, convergence and stability. Also, the implementation of this new numerical scheme shall be worked-on and comparison shall also be made with some existing methods.

Many numerical analysts such as: S. O. Fatunla [

Ogunrinde, R. B. [

In this paper, a new numerical scheme was developed with the above mentioned characteristics in mind to solve some initial value problems of ordinary differential equations which was based on the local representation of the theoretical solution to initial value problem of the form:

Suppose we have the initial value problem:

Let us assume that the theoretical solution

where

We shall assume that

Therefore, from (2), we proceed to the scheme derivation as follows:

from (2),

from (3),

from (4),

from (5),

putting (8) into (9), we have:

multiply through by

putting (11) into (10), we obtain:

putting (12) into (11), we obtained:

putting (12) and (13) into (8), we have:

Now,

Let

Therefore,

Now, imposing the following constraints on the interpolating function (2) in the following order:

1) The interpolating function (2) must coincide with the theoretical solution at

2) The derivative of

from conditions (1) and (2) above, it follows that:

if

Collecting like-terms

So,

Now, suppose:

Also,

from (15), we have:

Similarly,

by factorization, we have:

Putting (16) through (20) into (15), we have the new scheme follows:

Equation (21) is the proposed scheme.

We aim to develop a new numerical scheme which can favourably agree with the existing ones for solving some initial value problems of ordinary differential equations. Clearly, this paper has been able to show the development of the new numerical scheme as proposed.

In our subsequent research, we shall pay more attention on the implementation of this new scheme to solve some initial value problems (ivp) of the form (1) and also compare the results with the existing methods and thereafter we examine the characteristics properties such as the stability, convergence, accuracy and consistency of the scheme.

Yu-Wen Chen,Der-Shing Lee,R. B. Ogunrinde,T. E. Olaosebikan, (2016) Development of a Numerical Scheme. American Journal of Computational Mathematics,06,49-54. doi: 10.4236/ajcm.2016.61006