A Five-Step P-Stable Method for the Numerical Integration of Third Order Ordinary Differential Equations

In this paper we derived a continuous linear multistep method (LMM) with step number k = 5 through collocation and interpolation techniques using power series as basis function for approximate solution. An order nine p-stable scheme is developed which was used to solve the third order initial value problems in ordinary differential equation without first reducing to a system of first order equations. Taylor’s series algorithm of the same order was developed to implement our method. The result obtained compared favourably with existing methods.


Introduction
Linear multistep methods (LMM) for solving first order initial value problems (ivps) is of the form 0 0 k k j n j j n j j j y h f where j α and j β are uniquely determined and 0 0, 1 j k a system first order has some serious drawback which includes wastage of human effort and computer time [6].
The LMM in (1) generates discrete schemes which are used to solve first order odes.Various forms of this LMM have been developed [1]- [4].Other researchers have introduced the continuous LMM using the continuous collocation and interpolation technique.This has led to the development of continuous LMM of form ( ) ( ) ( ) (2) j α and j β are expressed as continuous functions of t and are at least differentiable once.
The introduction of continuous collocation methods as against the discrete schemes enhances better global error estimation and ability to approximate solution at all interior points [6]- [10].In this study, we shall develop continuous multistep collocation method for the solution of third order ordinary differential equations using power series as the basis function.

Power Series Collocation
In [6] [8] [9], some continuous LMM of Type (2) were developed using power series of form ( ) In [10] Chebyshev polynomial function of the form ( ) ( ) The approximate function ( ) p x reduces to ( ) Y x as n → ∞ In this study we proposed the polynomial function of the form in [7]: which is of Type (3) to develop a continuous LMM for the solution of initial value problem of the form: This paper is organized as follows: Section 1 consists of introduction and background of study; Section 2, we derive a continuous approximation to ( ) Y x for exact solution ( ) y x , and specific methods; Section 3 consists of the analysis and implementation followed by numerical examples.

Derivation of the Method
Consider the third order differential Equation (7), we proposed an approximate solution of the form: where The derivative of (8) up to the third order yield And a x b ≤ ≤ where j a s ′ are the parameters to be determined.By substituting (8) and ( 9) into (7) we have x y The above equations are solved to obtain the values of j a s ′ which when substituted into Equation ( 6) yield a method of the form in Equation ( 13) The continuous polynomial obtained when the values of j a s ′ are substituted into (6) and simplified is as follows        x

Basic Properties of the Method
The method (15) is a specific member of the conventional LMM which can expressed as 3 0 0 k k j n j j n j j j y h y Following [1] [2], we define the local truncation error associated with (16) by the difference operator where ( ) y x is assumed to have continuous derivatives of sufficiently high order.Therefore expanding (23) in Taylor series about the point x to obtain the expression

L y x h C y x C hy x C hy x C h y x C h y x C h y x
where the 0 1 2 2 , , , , In the sense of [1], we say that the method (20) is of order p and error constant Using the concept above, the method (19) has order 9 p = and error constant given by 0.005403

Zero-Stability of the 5-Step Method
Considering the first characteristics polynomial of the method of Equation (15) given as ( ) 5

Region of Absolute Stability of the 5-Step Scheme
Applying the boundary locus method, we have that ( ) 5  D f where p is the order of the method.

Numerical Experiments
Our methods of order 9 p = were used to solve some initial value problems of both general and special nature using Taylor's series.Our results were compared with the results of other researchers in this area as seen in Table 1.
In Table 2 and Table 3, the accuracy of our method is seen in the small error values.
The following initial value problems were used as our test problems:

T
x are some Chebyshev function used to develop continuous LMM.The use power series as basis function for derivation of continuous LMM are based on the property of analytic function that given the Taylor's polynomial of the form

1 r
− is a factor.Therefore solving the polynomial it is found that In the spirit ofLambert (1973),

Table 1 .
Showing the result of test problem 1.

Table 2 .
Showing the result of test problem 2.

Table 3 .
Showing the result of test problem 3.