The determination process for the numerical solution of initial value problems in ordinary differential equations can be classified into two categories-single step methods and multistep methods. Single step methods are those in which the approximation for the point involves information from only one of the previous points. Methods using the approximation at more than one previous points to determine the approximation at the next point are called multistep methods. Thus a k-step method requires information about the solution at k points to compute the solution at the point. Finite difference methods for boundary value problems are discussed in [1] . Linear multi step methods of second order differential equations are discussed in [2] . The methods based on numerical differentiation for first-order differential equations have been shown to be stiffly stable by Gear [3] . A detailed study of the single step and multistep methods has been carried out by Gear [3] , Gragg and Statter [4] and Henrici [5] . Gear [3] and Peter Henrici [5] have derived special multistep methods based on numerical integration and numerical differentiation for solving first-order differential equations. Jain [6] has considered high order stiffly stable methods. Further information can be had from [7] and [8] . Special multistep methods based on numerical differentiation for solving the initial value problem have been derived in Rama Chandra Rao [9] . The methods now to be discussed are based on replacing the function which is unknown, by an interpolating polynomial having the values on a set of points x_n where y_n has already been computed. The methods discussed in this paper are essentially based on the idea that the solution is best approximated by polynomials. The motivation for the work carried out in this paper arises from the methods based on numerical differentiation for the first-order differential equations, special multistep methods based on numerical integration for the solution of the special second-order differential equations by Henrici [5] and Special multistep methods based on numerical differentiation for solving the initial value problem by Rama Chandra Rao [9] . In Henrici [5] methods based on Numerical Integration have been derived by integrating twice and replacing the function by an interpolating polynomial. Special multistep methods have been derived by replacing on the left hand side of by an interpolating polynomial and differentiating it four times. We have investigated a class of implicit methods. It is found that the implicit methods have order. Some local truncation errors are provided. The regions of absolute stability of the methods are derived. Numerical tests of the performance of the methods are established by solving differential equation and compared with the exact solution. The numerical results reported show the validity of our methods.

The special fourth order differential equation

occurs frequently in many number of problems of science and engineering.

A general linear multistep method of step number k for the numerical solution of equation (1) is given by

where a_j, b_j are constants and “h” is the step length.

Introducing the polynomials

Equation (2) can be written as

In Equation (4), “E” is the shift operator defined by

Applying (4) to, we get the characteristic equation

The roots of the characteristic Equation (5) and are in general, complex and the region of absolute stability is defined to be the region of the complex -plane such that the roots of the characteristic Equation (5) lie within the unit circle whenever lies in the interior of the region. Denoting the region of absolute stability of R and its boundary by, the locus of is given by

Let be the backward difference interpolating polynomial of at abscissas_. Then is given by

Differentiating (7) four times with respect to x, we get

Replacing by in Equation (1) and putting i.e., we get,

where

Taking r = 0 in (8), a class of methods can be attained which are given by

The coefficients are shown in Table 1.

Differences in (10) are expressed in terms of function values.

After simplification, the Equation (10) will turn out into the form

The coefficients are shown in Table 2.

The local truncation error of the formula (11) is given by

Table 1. Coefficients of;.

Table 1Table 1. Coefficients of;.

Table 1. Coefficients of;.

M	0	1	2	3	4	5	6	7	8	9
	0	0	0	0	1	2

Table 2. Coefficients of;,.

Table 2Table 2. Coefficients of;,.

Table 2. Coefficients of;,.

K	J
	0	1	2	3	4	5	6	7	8	9
4	1	−4	6	−4	1
5	3	−14	26	−24	11	−2
6
7
8
9

It follows that the k-step method (14) has the order, which is absolutely stable for

For the method (13), we have

The regions of absolute stability of the method for k = 4, 5, 6, 7, 8 and 9 are shown in Figure 1 and Figure 2 (Taking real part on x-axis and imaginary part on y-axis). The region of absolute stability is the region lying outside the boundary.

In this section, we have applied ND methods to solve the differential equation

in the interval with h = 0.01 and h = 0.02 and the results are shown in Table 3 and Table 4.

The fourth order numerical differentiation method derived in this paper for k = 6 is

The methods based on numerical integration are found to be closed regions of absolute stability; the methods

based on numerical differentiation are found to be absolutely stable outside some closed boundaries. We have obtained the solution by numerical differentiation methods which are derived in this paper and are more accurate. The absolute errors are very small.