﻿ Five Steps Block Predictor-Block Corrector Method for the Solution of <i>y''</i> = <i>f</i> (<i>x</i>,<i>y</i>,<i>y'</i>)

Applied Mathematics
Vol.5 No.8(2014), Article ID:45707,15 pages DOI:10.4236/am.2014.58117

Five Steps Block Predictor-Block Corrector Method for the Solution of

Mathew Remilekun Odekunle1, Michael Otokpa Egwurube1, Adetola Olaide Adesanya1, Mfon Okon Udo2

1Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria

2Department of Mathematics and Statistics, Cross River University of Technology, Calabar, Nigeria

Email: mfudo4sure@yahoo.com, remi_odekunle@yahoo.com

Received 15 February 2014; revised 15 March 2014; accepted 22 March 2014

Abstract

Theory has it that increasing the step length improves the accuracy of a method. In order to affirm this we increased the step length of the concept in [1] by one to get k = 5. The technique of collocation and interpolation of the power series approximate solution at some selected grid points is considered so as to generate continuous linear multistep methods with constant step sizes. Two, three and four interpolation points are considered to generate the continuous predictor-corrector methods which are implemented in block method respectively. The proposed methods when tested on some numerical examples performed more efficiently than those of [1] . Interestingly the concept of self starting [2] and that of constant order are reaffirmed in our new methods.

Keywords:Step Length, Power Series, Block Predictor, Block Corrector, Constant Order, Step Size, Grid Points, Self Starting, Efficiency

1. Introduction

In this paper we examine the solution to general second order initial value problem of the form

(1)

In literature, it has been stated clearly the journey of the development of direct methods to offset the burden of reduction [3] -[6] . Various methods have been proposed by scholars for solving higher order ordinary differential equation (ODE). Notable authors like [1] [7] -[11] have developed direct methods of solving general second order ODE’s to cater for the burden inherent in the method of reduction. Now writing computer code is less burdensome since it no longer requires special ways to incorporate the subroutine to supply the starting values. As a result, this leads to computer time and human effort conservation.

The new methods are continuous in nature with the advantage of possible evaluation at all points within the integration interval. We have taken advantage of the works of [7] [12] -[15] who proposed direct block methods as predictor in the form

(2)

where

matrix, identity martix.

And also the discrete block formula as corrector in the form

(3)

where identity matrix

with the aim to cater for some of the setbacks of predictor-corrector method [16] [17] . The fact that interpolation point cannot exceed the order of the differential equation for block methods is worrisome [9] . Also vital to this paper is the concept of block predictor-corrector method (Milne approach). This method formed a bridge between the predictor-corrector method and block method [4] [10] [13] . In [1] we stated that results generated at an overlapping interval affect the accuracy of the method and the nature of the model cannot be determined at the selected grid points.

In this paper as in [1] , we developed a method using the Milne approach but the corrector was implemented at a non overlapping interval. The numerical experiment compared the results generated at different step lengths, when k = 4 and when k = 5 respectively.

2. Methodology

2.1. Development of the Continuous Linear Multistep Methods

We consider a power series approximate solution in the form

(4)

where r and s are the number of interpolation and collocation points respectively.

The second derivative of (4) gives

(5)

Substituting (5) into (1) gives

(6)

Interpolating (4) and collocating (6) at some selected grid points gives a system of non linear equations in the form

(7)

where

Solving (7) for the unknown constants using Guassian elimination method and substituting back into (4) gives a continuous linear multistep method in the form

(8)

where and are polynomials,

2.1.1. Development of the Block Predictor

Interpolating (4) at and collocating (6) at the parameters in (7) becomes

Solving for the unknown constants using Guassian elimination method and substituting into (4), makes Equation (8) reduced to

(9)

where

Solving for the independent solution in (9) and simplifying gives

(10)

where

Evaluating (10) at the selected grid points, the parameters in (2) gives the following I) When

identity matrix

,

II) When

,

2.1.2. Development of the Block Corrector

Here there are three cases (I, II and III) to be considered.

Development of the Block Corrector for Case I

Interpolating (4) at and collocating (6) at makes Equation (7) reduced to

Solving for the unknown constants using Guassian elimination method and substituting into (4), makes Equation (8) reduced to

(11)

where

Evaluating (11) at gives the following

(12)

(13)

(14)

Evaluating the first derivatives of (11) at gives the following

(15)

(16)

Writing Equations (12) to (16) in block form, the parameters in (3) gives the following

identity matrix

,

,

In a similar way the results for cases II and III are summarized as:

Development of the Block Corrector for Case II

identity matrix

,

Development of Block Corrector case III

identity matrix

,

3. Analysis of the Properties of the Methods

3.1. Order of the Methods

3.1.1 Order of the Block Predictor

When if we take a Taylor series expansion, we get

Collecting coefficients in powers of h, we see that the order of the method is six and the error constant is

Also when

The order of the method is six and the error constant is

3.1.2. Order of the Block Corrector for Case I

Taking a Taylor series expansion gives

and the order of our method is seven with error constant as

In a similar way, we compute and summarize the order for cases II and III as follows.

3.1.3. Order of the Block Corrector for Case II

In this case the order of our method is eight with error constant as

3.1.4. Order of the Block Corrector for Case III

Also using the same approach, the order of our method is nine with error constant as

3.2. Consistency of the Method

A block method is said to be consistent if it has order [9] .

From the above, it clearly shows that our methods are consistent.

3.3. Zero Stability

A block method is said to be zero stable if the root of the first characteristics polynomials

that is satisfying must have multiplicity equal to unity [9] .

Applying this rule, we have that

where for each method. Hence the methods are zero stable

4. Numerical Experiment

4.1. Implementation

We implement the proposed methods to verify their efficacies over existing methods. To be considered are, two cases for k = 4 [1] and three cases for k = 5. Four examples were considered at h = 0.01 and h = 0.05. All computations were made with the usage of MATLAB (R2010a). An error (Err) is defined in this paper as the absolute value of the difference between the computed and expected values. The following keys are used in displaying our results on the tables for clearity.

CASE 1: Two interpolation points.

CASE II: Three interpolation points.

CASE III: Four interpolation points.

4.1.1. Test Problem I

Consider the non-linear ODE

Exact Solution:

4.1.2. Test Problem 2

Consider the non-linear initial value problem

;

Exact solution:.

4.1.3. Test Problem 3

Consider the initial value ODE

;

Exact solution:.

4.1.4. Test Problem 4

Consider the initial value problem

;

Exact solution:.

5. Discussion

We have considered two non-linear and two linear second order initial value problems in this paper as shown in Table 1 to Table 4. In [1] we compared our method with the existing methods like the block and block predictor-corrector and the results re-affirms the claim of [10] that though block predictor-corrector method takes longer time to implement, it gives better approximation than the block method. In this paper we extended the step length considered in [1] and considered varying the number of interpolation points to observe the effect on the performance of the method.

Table 1. Comparing results for different interpolation points.

Table 2. Comparing results for different interpolation points.

Table 3. Comparing results for different interpolation points.

Table 4. Comparing results for different interpolation points.

6. Conclusion/Recommendation

In this paper we have proposed the varying of the step length from k = 4 [1] to k = 5. Block methods which have the properties of evaluation at all points within the interval of integration are adopted to give independent solutions at non overlapping intervals as predictors to the correctors. The new method k = 5 performed better than that of k = 4. Thus it has been confirmed that varying the step length improves the accuracy of the method. However, increasing the number of interpolation points does not significantly improve the result. We therefore, recommend the block predictor-block corrector method for use in the quest for solutions to second order initial value problems of ordinary differential equations.

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