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Advances in Numerical Analysis
Volumeย 2011ย (2011), Article IDย 513148, 14 pages
http://dx.doi.org/10.1155/2011/513148
Research Article

3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

1Department of Mathematics, Faculty of Science, University of Lagos, Akoka, Yaba, Lagos, Nigeria
2Department of Computer Sciences, Faculty of Science, University of Lagos, Akoka, Yaba, Lagos, Nigeria

Received 3 February 2011; Revised 18 April 2011; Accepted 26 May 2011

Academic Editor: Istvรกnย Faragรณ

Copyright ยฉ 2011 J. O. Ehigie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form ๐‘ฆ๎…ž๎…ž=๐‘“(๐‘ฅ,๐‘ฆ,๐‘ฆ๎…ž),๐‘ฆ(๐‘ฅ0)=๐‘Ž,๐‘ฆ๎…ž(๐‘ฅ0)=๐‘. The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.

1. Introduction

In recent times, the integration of Ordinary Differential Equations (ODEs) are carried out using some kind of block methods. In particular, this paper discusses the general second-order ODEs which arise frequently in the area of science, engineering and mechanical systems and are generally written in the form, ๐‘ฆ๎…ž๎…ž๎€ท=๐‘“๐‘ฅ,๐‘ฆ,๐‘ฆ๎…ž๎€ธ๎€ท๐‘ฅ,๐‘ฆ0๎€ธ=๐‘Ž,๐‘ฆ๎…ž๎€ท๐‘ฅ0๎€ธ=๐‘.(1.1) This problem being second order is usually or sometimes solved by reducing the ordinary differential equation into systems of first-order ordinary differential equations. Thereafter, known numerical methods, such as Runge-Kutta methods and Linear Multistep Methods (LMMs), are used to solve them.

Development of LMM for solving ODEs can be generated using methods such as Taylor's Series, numerical integration, and collocation method, which are restricted by an assumed order of convergence. In this paper, we will follow suite from the previous papers of Okunuga and Ehigie [1] by deriving our new method in a multistep collocation technique introduced by Onumanyi et al. [2]. Some researchers have attempted the solution of (1.1) directly using linear multistep methods without reduction to systems of first-order ordinary differential equations, they include Brown [3], Onumanyi et al. [4], Ismail et al. [5], and Ehigie et al. [6].

Block methods for solving Ordinary Differential Equations have initially been proposed by Milne [7] who used them as starting values for predictor-corrector algorithm, Rosser [8] developed Milne's method in form of implicit methods, and Shampine and Watts [9] also contributed greatly to the development and application of block methods. Fatunla [10] gave a generalization to block methods using some definition in matrix form upon which the methods derived in this paper will follow.

Hybrid methods, using collocation technique, were discussed by Ehigie et al. [6] and the continuous linear multistep scheme (CLMS) generated was used to obtain block schemes that serve as predictor-corrector schemes which were of Stormer-Cowell type. This collocation method is preferred because it is self-starting and it is convenient for easy generation of block or parallel schemes. Also the paper will consider various properties and conditions for a convergent method.

2. Theoretical Procedure

The procedure for the derivation of our methods in a multistep collocation technique is discussed by the methods in previous papers by Okunuga and Ehigie [1] and Ehigie et al. [6].

Consider the second-order equation๐‘ฆ๎…ž๎…ž๎€ท=๐‘“๐‘ฅ,๐‘ฆ,๐‘ฆ๎…ž๎€ธ๎€ท๐‘ฅ,๐‘ฆ0๎€ธ=๐‘Ž,๐‘ฆ๎…ž๎€ท๐‘ฅ0๎€ธ=๐‘.(2.1)

The numerical solution to (2.1) can be obtained using a ๐‘˜-step explicit Linear Multistep Method (LMM) of the form๐‘˜๎“๐‘—=0๐›ผ๐‘—๐‘ฆ๐‘›+๐‘—=โ„Ž2๐‘˜๎“๐‘—=0๐›ฝ๐‘—๐‘“๐‘›+๐‘—,(2.2) where ๐‘ฆ๐‘›+๐‘—โ‰ˆ๐‘ฆ(๐‘ฅ๐‘›+๐‘—โ„Ž), ๐‘“๐‘›+๐‘—โ‰ก๐‘“(๐‘ฅ๐‘›+๐‘—โ„Ž,๐‘ฆ(๐‘ฅ๐‘›+๐‘—โ„Ž),๐‘ฆ๎…ž(๐‘ฅ๐‘›+๐‘—โ„Ž)), and ๐‘ฅ๐‘› is a discrete point at node point ๐‘›. Where, ๐›ผ๐‘— and ๐›ฝ๐‘— are parameters to be determined and usually ๐›ฝ๐‘˜=0 for an explicit scheme.

Most of the problems encountered in solving the general second-order equation (2.1) is in the evaluation of the derivative term ๐‘ฆโ€ฒ present in the equation. This often makes different authors to either reduce the second-order equation to system of first-order ordinary differential equations or are restricted to solve the equation of the form ๐‘ฆ๎…ž๎…ž=๐‘“(๐‘ฅ,๐‘ฆ), while ๐‘ฆโ€ฒ is set to zero. However, by the introduction a of continuous scheme, this is easily taken care of. Thus if ๐‘ฆ(๐‘ฅ) is a basis polynomial of the form๐‘ฆ(๐‘ฅ)=๐‘๎“๐‘—=0๐‘Ž๐‘—๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚๐‘—.(2.3)

To derive an ๐‘š point block method, where ๐‘š is a positive integer, we set ๐‘=2(๐‘šโˆ’1) for an explicit scheme or ๐‘=2๐‘š+1 for an implicit scheme, interpolating (2.3) at points ๐‘ฅ๐‘›+๐‘—, ๐‘—=0,1,2,โ€ฆ,๐‘˜, and collocating ๐‘ฆ๎…ž๎…ž(๐‘ฅ) at points x๐‘›+๐‘—, ๐‘—=0,1,2,โ€ฆ,๐‘˜, will result to a (๐‘+1) system of equation for arbitrary ๐‘˜, Okunuga and Ehigie [1], ๐‘ฆ๎€ท๐‘ฅ๐‘›+๐‘—๎€ธ=๐‘ฆ๐‘›+๐‘—๐‘“๎€ท๐‘ฅ,๐‘—=0,1,2,โ€ฆ,๐‘˜โˆ’1,๐‘›+๐‘—๎€ธ=๐‘“๐‘›+๐‘—,๐‘—=0,1,2,โ€ฆ,๐‘˜.(2.4)

The coefficients ๐‘Ž0,๐‘Ž1,๐‘Ž2,โ€ฆ,๐‘Ž๐‘ are obtained and substituted in (2.3) to obtain the Continuous Linear Multistep Scheme (CLMS) of the form ๐‘Œ(๐‘ฅ)=๐‘˜โˆ’1๎“๐‘—=0๐›ผ๐‘—(๐‘ฅ)๐‘ฆ๐‘›+๐‘—+โ„Ž2๐‘˜๎“๐‘—=0๐›ฝ๐‘—(๐‘ฅ)๐‘“๐‘›+๐‘—.(2.5)

This is evaluated for at ๐‘ฅ๐‘›+๐‘–, ๐‘–=0,1,2,โ€ฆ๐‘š to obtain an ๐‘š-point block method generally represented by Fatunla [10]. With the ๐‘š-vector ๐‘Œ๐‘š and ๐น๐‘š specified as, ๐‘Œ๐‘›=๎€บ๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+2,๐‘ฆ๐‘›+3,โ€ฆ,๐‘ฆ๐‘›+๐‘š๎€ป๐‘‡,๐‘Œ๐‘›โˆ’1=๎€บ๐‘ฆ๐‘›,๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’2,โ€ฆ,๐‘ฆ๐‘›โˆ’๐‘š+1๎€ป๐‘‡,๐น๐‘›โˆ’1=๎€บ๐‘“๐‘›,๐‘“๐‘›โˆ’1,๐‘“๐‘›โˆ’2,โ€ฆ,๐‘“๐‘›โˆ’๐‘š+1๎€ป๐‘‡,๐น๐‘›=๎€บ๐‘“๐‘›+1,๐‘“๐‘›+2,๐‘“๐‘›+3,โ€ฆ,๐‘“๐‘›+๐‘š๎€ป๐‘‡.(2.6)

The ๐‘Ÿ-block, ๐‘š-point EBM will be generally represented as๐‘Œ๐‘›=๐‘Ÿ๎“๐‘–=0๐ด(๐‘–)๐‘Œ๐‘›โˆ’1+โ„Ž2๐‘Ÿ๎“๐‘–=0๐ต(๐‘–)๐น๐‘›โˆ’1,(2.7) where ๐ด(๐‘–) and ๐ต(๐‘–), ๐‘–=0,1,2,โ€ฆ,๐‘Ÿ, are ๐‘šร—๐‘š square matrix with elements ๐‘Ž๐‘–๐‘™๐‘—, ๐‘๐‘–๐‘™๐‘— for ๐‘™,๐‘—=1,2,โ€ฆ,๐‘š. The block scheme (2.7) is explicit if the coefficient Matrix ๐ต(0) is a null matrix.

3. Derivation of Explicit Block Methods

To derive a 1 block 3-point Explicit Block Method (EBM) that is, ๐‘š=3, we set ๐‘=4. Let ๐‘ฆ(๐‘ฅ) be a basis function so that๐‘ฆ(๐‘ฅ)=๐‘๎“๐‘—=0๐‘Ž๐‘—๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚๐‘—(3.1) while we interpolate (3.1) at point ๐‘ฅ=๐‘ฅ๐‘› and ๐‘ฅ๐‘›โˆ’1 and collocate ๐‘ฆ๎…ž๎…ž(๐‘ฅ) at ๐‘ฅ๐‘›, ๐‘ฅ๐‘›โˆ’1, and ๐‘ฅ๐‘›โˆ’2 to obtain a system of equationsโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘Ž100001โˆ’11โˆ’1100200002โˆ’612002โˆ’12480๐‘Ž1๐‘Ž2๐‘Ž3๐‘Ž4โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ฆ๐‘›๐‘ฆ๐‘›โˆ’1โ„Ž2๐‘“๐‘›โ„Ž2๐‘“๐‘›โˆ’1โ„Ž2๐‘“๐‘›โˆ’2โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(3.2) Solving the matrix equation above, we obtain๐‘Ž0=๐‘ฆ๐‘›,๐‘Ž1=๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1+7โ„Ž242๐‘“๐‘›+14โ„Ž2๐‘“๐‘›โˆ’1โˆ’1โ„Ž242๐‘“๐‘›โˆ’2,๐‘Ž2=12โ„Ž2๐‘“๐‘›,๐‘Ž3=14โ„Ž2๐‘“๐‘›โˆ’13โ„Ž2๐‘“๐‘›โˆ’1+1โ„Ž122๐‘“๐‘›โˆ’2,๐‘Ž4=1โ„Ž242๐‘“๐‘›โˆ’1โ„Ž122๐‘“๐‘›โˆ’1+1โ„Ž242๐‘“๐‘›โˆ’2.(3.3)

Substituting the values ๐‘Ž0,๐‘Ž1,๐‘Ž2,๐‘Ž3, and ๐‘Ž4 in (3.1), we obtain the CLMS๎‚€๐‘ฆ(๐‘ฅ)=๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚๐‘ฆ๐‘›โˆ’1โˆ’๎‚€๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚๎‚๐‘ฆ+1๐‘›+โ„Ž2๎‚ต7๎‚€24๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚+12๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚2+14๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3+1๎‚€24๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚4๎‚ถ๐‘“๐‘›+โ„Ž2๎‚ต14๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚โˆ’13๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3โˆ’1๎‚€12๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚4๎‚ถ๐‘“๐‘›โˆ’1+โ„Ž2๎‚ตโˆ’1๎‚€24๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚+1๎‚€24๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3+1๎‚€24๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚4๎‚ถ๐‘“๐‘›โˆ’2.(3.4)

On evaluating (3.4) at points ๐‘ฅ=๐‘ฅ๐‘›+๐‘–,๐‘–=1,2,3, we obtained the convergent explicit 3-point EBM as๐‘ฆ๐‘›+1=โˆ’๐‘ฆ๐‘›โˆ’1+2๐‘ฆ๐‘›+โ„Ž2๎‚€13๐‘“12๐‘›โˆ’16๐‘“๐‘›โˆ’1+1๐‘“12๐‘›โˆ’2๎‚,๐‘ฆ๐‘›+2=โˆ’2๐‘ฆ๐‘›โˆ’1+3๐‘ฆ๐‘›+โ„Ž2๎‚€214๐‘“๐‘›โˆ’72๐‘“๐‘›โˆ’1+54๐‘“๐‘›โˆ’2๎‚,๐‘ฆ๐‘›+3=โˆ’3๐‘ฆ๐‘›โˆ’1+4๐‘ฆ๐‘›+โ„Ž2๎‚€312๐‘“๐‘›โˆ’15๐‘“๐‘›โˆ’1+112๐‘“๐‘›โˆ’2๎‚.(3.5)

Differentiating (3.4) and evaluating again at the same 3 discrete points of ๐‘–, we obtain a block of first-order derivatives which can be used to determine the derivative term in the initial value problem (2.1).๐‘ฆ๎…ž๐‘›+1=1โ„Ž๎€ท๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1๎€ธ๎‚€+โ„Ž53๐‘“24๐‘›โˆ’13๐‘“12๐‘›โˆ’1+38๐‘“๐‘›โˆ’2๎‚,๐‘ฆ๎…ž๐‘›+2=1โ„Ž๎€ท๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1๎€ธ๎‚€+โ„Ž538๐‘“๐‘›โˆ’77๐‘“12๐‘›โˆ’1+55๐‘“24๐‘›โˆ’2๎‚,๐‘ฆ๎…ž๐‘›+3=1โ„Ž๎€ท๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1๎€ธ๎‚€+โ„Ž349๐‘“24๐‘›โˆ’714๐‘“๐‘›โˆ’1+161๐‘“24๐‘›โˆ’2๎‚.(3.6)

Expressing the schemes (3.5) as block using previous definition (2.7), we obtainโŽ›โŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ฆ100010001๐‘›+1๐‘ฆ๐‘›+2๐‘ฆ๐‘›+3โŽžโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ฆ0โˆ’120โˆ’230โˆ’34๐‘›โˆ’2๐‘ฆ๐‘›โˆ’1๐‘ฆ๐‘›โŽžโŽŸโŽŸโŽŸโŽ +โ„Ž2โŽ›โŽœโŽœโŽœโŽœโŽœโŽ1โˆ’1126135124โˆ’72214112โˆ’15312โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘“๐‘›โˆ’2๐‘“๐‘›โˆ’1๐‘“๐‘›โŽžโŽŸโŽŸโŽŸโŽ .(3.7)

Equation (3.7) is therefore said to be of the form (2.7). Thus (3.7) is represented notationally as ๐‘Œ๐‘›=๐ด(1)๐‘Œ๐‘›โˆ’1+โ„Ž2๐ต(1)๐น๐‘›โˆ’1.

4. Derivation of Implicit Block Methods

To derive a 1 block 3-point Implicit Block Method (IBM), we also define the following terms: ๐‘Œ๐‘›=๎€บ๐‘ฆ๐‘›+1,๐‘ฆ๐‘›+2,๐‘ฆ๐‘›+3๎€ป๐‘‡,๐‘Œ๐‘›โˆ’1=๎€บ๐‘ฆ๐‘›,๐‘ฆ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’2๎€ป๐‘‡,๐น๐‘›=๎€บ๐‘“๐‘›+1,๐‘“๐‘›+2,๐‘“๐‘›+3๎€ป๐‘‡,๐น๐‘›โˆ’1=๎€บ๐‘“๐‘›,๐‘“๐‘›โˆ’1,๐‘“๐‘›โˆ’2๎€ป๐‘‡.(4.1)

The 3- point IBM will be generally represented as๐‘Œ๐‘›=๐ด(1)๐‘Œ๐‘›โˆ’1+โ„Ž2๎€ท๐ต(1)๐น๐‘›โˆ’1+๐ต(0)๐น๐‘›๎€ธ,(4.2) where ๐ด(1), ๐ต(0), and ๐ต(1) are 3ร—3 square matrix. Let ๐‘ฆ(๐‘ฅ) be a basis function so that ๐‘ฆ(๐‘ฅ)=๐‘๎“๐‘—=0๐‘Ž๐‘—๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚๐‘—.(4.3) Setting ๐‘=7 for an implicit 3-point block scheme, we will interpolate (4.3) at points ๐‘ฅ๐‘› and ๐‘ฅ๐‘›โˆ’1 and collocate ๐‘ฆ๎…ž๎…ž(๐‘ฅ) at 6 points ๐‘ฅ๐‘›+๐‘–, ๐‘–=โˆ’2,โˆ’1,0,1,2,3, to obtain a system of equations represented by the matrixโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘Ž1โˆ’11โˆ’11โˆ’11โˆ’110000000002โˆ’1248โˆ’160480โˆ’1344002โˆ’612โˆ’2030โˆ’420020000000261220304200212481604801344002181085402430102060๐‘Ž1๐‘Ž2๐‘Ž3๐‘Ž4๐‘Ž5๐‘Ž6๐‘Ž7โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ฆ๐‘›โˆ’1๐‘ฆ๐‘›โ„Ž2๐‘“๐‘›โˆ’2โ„Ž2๐‘“๐‘›โˆ’1โ„Ž2๐‘“๐‘›โ„Ž2๐‘“๐‘›+1โ„Ž2๐‘“๐‘›+2โ„Ž2๐‘“๐‘›+3โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(4.4) Solving the matrix equation, we obtain ๐‘Ž0=๐‘ฆ๐‘›,๐‘Ž1=๐‘ฆ๐‘›+271โ„Ž20162๐‘“๐‘›โˆ’1+1111โ„Ž25202๐‘“๐‘›โˆ’๐‘ฆ๐‘›โˆ’1โˆ’451โ„Ž50402๐‘“๐‘›+1+131โ„Ž50402๐‘“๐‘›+2โˆ’41โ„Ž50402๐‘“๐‘›โˆ’2,โˆ’37โ„Ž100802๐‘“๐‘›+3,๐‘Ž2=12๐‘“๐‘›,๐‘Ž31=โˆ’โ„Ž182๐‘“๐‘›โˆ’1โ„Ž242๐‘“๐‘›+2+1โ„Ž1202๐‘“๐‘›โˆ’2+16โ„Ž2๐‘“๐‘›+1โˆ’1โ„Ž122๐‘“๐‘›โˆ’1+1โ„Ž1802๐‘“๐‘›+3,๐‘Ž45=โˆ’โ„Ž482๐‘“๐‘›โˆ’1โ„Ž2882๐‘“๐‘›+2โˆ’1โ„Ž2882๐‘“๐‘›โˆ’2+1โ„Ž182๐‘“๐‘›+1+1โ„Ž182๐‘“๐‘›โˆ’1,๐‘Ž5=1โ„Ž482๐‘“๐‘›+7โ„Ž4802๐‘“๐‘›+2โˆ’1โ„Ž4802๐‘“๐‘›โˆ’2โˆ’7โ„Ž2402๐‘“๐‘›+1โˆ’1โ„Ž4802๐‘“๐‘›โˆ’1โˆ’1โ„Ž4802๐‘“๐‘›+3,๐‘Ž6=1โ„Ž1202๐‘“๐‘›+1โ„Ž7202๐‘“๐‘›+2+1โ„Ž7202๐‘“๐‘›โˆ’2+1โ„Ž1802๐‘“๐‘›+1โˆ’1โ„Ž1802๐‘“๐‘›โˆ’1,๐‘Ž71=โˆ’โ„Ž5042๐‘“๐‘›โˆ’1โ„Ž10082๐‘“๐‘›+2+1โ„Ž50402๐‘“๐‘›โˆ’2+1โ„Ž5042๐‘“๐‘›+1โˆ’1โ„Ž10802๐‘“๐‘›โˆ’1โˆ’1โ„Ž50402๐‘“๐‘›+3.(4.5)

Substituting the ๐‘Ž๐‘–,๐‘–=0,1,โ€ฆ,7 in (4.3), we obtain the CLMS,๎‚€๐‘ฆ(๐‘ฅ)=โˆ’๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚๐‘ฆ๐‘›โˆ’1+๎‚€๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚๎‚๐‘ฆ+1๐‘›+โ„Ž2๎‚ตโˆ’451๎‚€5040๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚+16๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3+1๎‚€18๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚4โˆ’7๎‚€240๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚5โˆ’1๎‚€180๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚6+1๎‚€504๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚7๎‚ถ๐‘“๐‘›+1+โ„Ž2๎‚ตโˆ’131๎‚€5040๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚โˆ’1๎‚€24๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3โˆ’1๎‚€288๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚4+7๎‚€480๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚5+1๎‚€720๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚6โˆ’1๎‚€1008๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚7๎‚ถ๐‘“๐‘›+2+โ„Ž2๎‚ตโˆ’37๎‚€10080๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚+1๎‚€180๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3โˆ’1๎‚€480๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚5+1๎‚€5040๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚7๎‚ถ๐‘“๐‘›+3+โ„Ž2๎‚ตโˆ’41๎‚€5040๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚+1๎‚€120๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3โˆ’1๎‚€288๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚4โˆ’1๎‚€480๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚5+1๎‚€720๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚6โˆ’1๎‚€5040๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚7๎‚ถ๐‘“๐‘›โˆ’2+โ„Ž2๎‚ต271๎‚€2016๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚โˆ’1๎‚€12๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3+1๎‚€18๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚4โˆ’1๎‚€480๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚5โˆ’1๎‚€180๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚6+1๎‚€1080๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚7๎‚ถ๐‘“๐‘›โˆ’1+โ„Ž2๎‚ต1111๎‚€2520๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚+12๎‚€๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚2โˆ’1๎‚€18๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚3โˆ’1๎‚€48๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚4+1๎‚€48๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚5+1๎‚€120๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚6โˆ’1๎‚€504๐‘ฅโˆ’๐‘ฅ๐‘›โ„Ž๎‚7๎‚ถ๐‘“๐‘›.(4.6)

On evaluating (4.6) at points ๐‘ฅ=๐‘ฅ๐‘›+1,๐‘ฅ๐‘›+2 and ๐‘ฅ๐‘›+3, we obtain the 3-point implicit block Linear Multistep methods ๐‘ฆ๐‘›+1=โˆ’๐‘ฆ๐‘›โˆ’1+2๐‘ฆ๐‘›+โ„Ž2๎‚€1๐‘“10๐‘›+1โˆ’1๐‘“240๐‘›+2๎‚+โ„Ž2๎‚€97๐‘“120๐‘›+1๐‘“10๐‘›โˆ’1โˆ’1๐‘“240๐‘›โˆ’2๎‚,๐‘ฆ๐‘›+2=โˆ’2๐‘ฆ๐‘›โˆ’1+3๐‘ฆ๐‘›+โ„Ž2๎‚€121๐‘“120๐‘›+1+11๐‘“120๐‘›+2โˆ’1๐‘“240๐‘›+3๎‚+โ„Ž2๎‚€103๐‘“60๐‘›+47๐‘“240๐‘›โˆ’1โˆ’1๐‘“120๐‘›โˆ’2๎‚,๐‘ฆ๐‘›+3=โˆ’3๐‘ฆ๐‘›โˆ’1+4๐‘ฆ๐‘›+โ„Ž2๎‚€29๐‘“15๐‘›+1+127๐‘“120๐‘›+2+1๐‘“15๐‘›+3๎‚+โ„Ž2๎‚€161๐‘“60๐‘›+4๐‘“15๐‘›โˆ’1โˆ’1๐‘“120๐‘›โˆ’2๎‚.(4.7)

On differentiating (4.6) and evaluating again at the same 3 discrete points of ๐‘ฅ, we obtain๐‘ฆโ€ฒ๐‘›+1=1โ„Ž๎€ท๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1๎€ธ๎‚€+โ„Ž589๐‘“1260๐‘›+1โˆ’389๐‘“10080๐‘›+2+1๐‘“252๐‘›+3๎‚๎‚€+โ„Ž5029๐‘“5040๐‘›+22๐‘“315๐‘›โˆ’1โˆ’1๐‘“2016๐‘›โˆ’2๎‚,๐‘ฆโ€ฒ๐‘›+2=1โ„Ž๎€ท๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1๎€ธ๎‚€+โ„Ž5933๐‘“5040๐‘›+1+407๐‘“1008๐‘›+2โˆ’149๐‘“10080๐‘›+3๎‚๎‚€+โ„Ž2063๐‘“2520๐‘›+1243๐‘“10080๐‘›โˆ’1โˆ’41๐‘“5040๐‘›โˆ’2๎‚,๐‘ฆโ€ฒ๐‘›+3=1โ„Ž๎€ท๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1๎€ธ๎‚€+โ„Ž157๐‘“252๐‘›+1+14059๐‘“10080๐‘›+2+397๐‘“1260๐‘›+3๎‚๎‚€+โ„Ž5813๐‘“5040๐‘›+1๐‘“315๐‘›โˆ’1+107๐‘“10080๐‘›โˆ’2๎‚.(4.8)

The derivative formulae will be used to obtain the first derivative term in (2.1). Expressing the schemes (4.7) as block using our previous definition according to Fatunla [10], (4.7) becomesโŽ›โŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ฆ100010001๐‘›+1๐‘ฆ๐‘›+2๐‘ฆ๐‘›+3โŽžโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ฆ0โˆ’120โˆ’230โˆ’34๐‘›โˆ’2๐‘ฆ๐‘›โˆ’1๐‘ฆ๐‘›โŽžโŽŸโŽŸโŽŸโŽ +โ„Ž2โŽ›โŽœโŽœโŽœโŽœโŽœโŽ1โˆ’110024012112011โˆ’112024029151271120โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘“15๐‘›+1๐‘“๐‘›+2๐‘“๐‘›+3โŽžโŽŸโŽŸโŽŸโŽ +โ„Ž2โŽ›โŽœโŽœโŽœโŽœโŽœโŽโˆ’112401097โˆ’112012047240103โˆ’160412015161โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘“60๐‘›โˆ’2๐‘“๐‘›โˆ’1๐‘“๐‘›โŽžโŽŸโŽŸโŽŸโŽ .(4.9)

This scheme is also of the form (4.2).

5. Order, Consistency, Stability, and Convergence of the Methods

5.1. Order of the Methods

The methods (3.5) and (4.7) so derived are methods belonging to the class of LMM (2.2). So, if LMM (2.2) is a method associated with a linear difference operator,ล[]=๐‘ฆ(๐‘ฅ);โ„Ž๐‘˜๎“๐‘—=0๎€ท๐›ผ๐‘—๐‘ฆ(๐‘ฅ+๐‘—โ„Ž)โˆ’โ„Ž2๐›ฝ๐‘—๐‘ฆ๎…ž๎…ž๎€ธ,(๐‘ฅ+๐‘—โ„Ž)(5.1) where ๐‘ฆ(๐‘ฅ) is an arbitrary function continuously differentiable on the interval [๐‘Ž,๐‘]. The Taylors series expansion about the point ๐‘ฅ, ล[]๐‘ฆ(๐‘ฅ);โ„Ž=๐‘0๐‘ฆ(๐‘ฅ)+๐‘1โ„Ž๐‘ฆ๎…ž(๐‘ฅ)+๐‘2โ„Ž2๐‘ฆ๎…ž๎…ž(๐‘ฅ)+โ‹ฏ+๐‘๐‘žโ„Ž๐‘ž๐‘ฆ(๐‘ž)(๐‘ฅ),(5.2) where๐‘0=๐›ผ0+๐›ผ1+๐›ผ2+โ‹ฏ+๐›ผ๐‘˜,๐‘1=๐›ผ1+2๐›ผ2+โ‹ฏ+๐‘˜๐›ผ๐‘˜,๐‘2=1๎€ท๐›ผ2!1+22๐›ผ2+โ‹ฏ+๐‘˜2๐›ผ๐‘˜๎€ธโˆ’๎€ท๐›ฝ1+๐›ฝ2+โ‹ฏ+๐›ฝ๐‘˜๎€ธ,โ‹ฎ๐‘๐‘ž=1๎€ท๐›ผ๐‘ž!1+2๐‘ž๐›ผ2+โ‹ฏ+๐‘˜๐‘ž๐›ผ๐‘˜๎€ธโˆ’1๎€ท๐›ฝ(๐‘žโˆ’2)!1+2๐‘žโˆ’2๐›ฝ2+โ‹ฏ+๐‘˜๐‘žโˆ’2๐›ฝ๐‘˜๎€ธ,๐‘ž=3,4,โ€ฆ.(5.3)

Definition 5.1. The method (2.2) is said to be of order ๐‘ if ๐‘0=๐‘1=๐‘2=โ‹ฏ=๐‘๐‘+1=0,๐‘๐‘+2โ‰ 0.(5.4)๐‘๐‘+2 is the error constant and ๐‘๐‘+2โ„Ž๐‘+2๐‘ฆ(๐‘+2)(๐‘ฅ๐‘›) is the truncation error at point ๐‘ฅ๐‘›.
Applying this definition to the individual methods (3.5) and (4.7) which make up the block explicit and implicit methods which is of the form (2.2), it is easily verified that each of the explicit methods (3.5) is of order ๐‘=(3,3,3)๐‘‡ with error constants [1/12,4/3,41/6]๐‘‡. Also applying this definitions on the implicit methods (4.7), the implicit method was of order ๐‘=(6,6,6)๐‘‡ with error constants [โˆ’439/4320,โˆ’3479/2880,โˆ’1393/180]๐‘‡.

Definition 5.2. A Linear Multistep Method of the form (2.2) is said to be consistent if the LMM is of order ๐‘โ‰ฅ1.

Since the methods derived in (3.5) and (4.7) are of order ๐‘โ‰ฅ1, therefore, the methods are consistent according to Definition 5.2.

5.2. 0-Stability of the Method

From literature, it is known that stability of a linear multistep method determines the manner in which the error is propagated as the numerical computation proceeds. Hence, it would be necessary to investigate the stability properties of the newly developed methods. In this paper, the 0-stability and the Region of Absolute Stability (RAS) of the methods are discussed.

Definition 5.3. The first characteristic polynomial, ๐œŒ(๐œ‰), associated with the linear multistep method (2.2), where it is the polynomial of degree ๐‘˜ whose coefficients are ๐›ผ๐‘— and the second characteristic polynomial ๐œŽ(๐œ‰) whose coefficients are ๐›ฝ๐‘—, is defined by ๐œŒ(๐œ‰)=๐‘˜๎“๐‘—=0๐›ผ๐‘—๐œ‰๐‘—๐œŽ,(5.5a)(๐œ‰)=๐‘˜๎“๐‘—=0๐›ฝ๐‘—๐œ‰๐‘—,(5.5b) where ๐œ‰โˆˆ๐ถ, ๐ถ is a set of complex numbers and a free variable. Stability is determined by the location of the roots of the characteristic polynomial.

Definition 5.4. The block method of form (2.7) and (4.2) is said to be 0-stable if the roots ๐œ‰๐‘—, ๐‘—=1(1)๐‘˜, of the first characteristic polynomial โˆ‘๐œŒ(๐œ‰)=det[๐‘˜๐‘–=0๐ด๐‘–๐œ‰๐‘˜โˆ’1]=0, ๐ด0=โˆ’๐ผ, satisfy |๐œ‰|โ‰ค1. If one of the roots is +1, we call the roots the principal roots of ๐œŒ(๐œ‰).

Definition 5.5. The Region of Absolute Stability (RAS) of methods of (2.7) and (4.2) is the set ๎€ฝโ„Ž๐‘…=2๐œ†โˆถforthatโ„Ž2๐œ†wheretherootsofthestabilitypolynomialareofabsolutelessthanone}.(5.5) However, in this paper, the boundary locus method will be used to plot and view the RAS. This is obtained using the first and second characteristic polynomials as ๐‘ง๐œŒ๎€ท๐‘’(๐œƒ)=๐‘–๐œƒ๎€ธ๐œŽ๎€ท๐‘’๐‘–๐œƒ๎€ธ.(5.6) Resolving this to real and imaginary parts and evaluating for values of ๐œƒโˆˆ(0,2๐œ‹) give the region of stability on a graph.

The stability property of the 3-point EBM is determined by applying the scheme (3.7) to the test problem, ๐‘ฆ๎…ž๎…ž=๐œ†๐‘ฆ. By setting ๐‘ง=๐œ†โ„Ž2, the block scheme (3.7) becomes โŽ›โŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ฆ100010001๐‘›+1๐‘ฆ๐‘›+2๐‘ฆ๐‘›+3โŽžโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐‘ง๐‘ง12โˆ’1โˆ’62+13๐‘ง125๐‘ง4โˆ’2โˆ’7๐‘ง23+21๐‘ง4โˆ’11๐‘ง2โˆ’3โˆ’15๐‘ง4+31๐‘ง2โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ฆ๐‘›โˆ’2๐‘ฆ๐‘›โˆ’1๐‘ฆ๐‘›โŽžโŽŸโŽŸโŽŸโŽ .(5.7) This is of the form ๐‘Œ๐‘›=(๐ด(1)+๐‘ง๐ต(1))๐‘Œ๐‘›โˆ’1. The stability polynomial of this is given as๎€ท๎€ท๐ดdet๐œ‰โˆ’(1)+๐‘ง๐ต(1)๎€ธ๎€ธ=0.(5.8) Hence the stability polynomial of the 3-point EBM (3.7) is๐œ‰3+๐œ‰2๎‚€โˆ’145๎‚๎‚€12๐‘งโˆ’2+๐œ‰794๐‘ง2+376๎‚+๎‚€๐‘ง+1โˆ’๐‘ง3+54๐‘ง2โˆ’37๐‘ง๎‚12=0.(5.9) Substituting ๐‘ง=0 in (5.9), we obtain all the roots of the derived equation to be less than or equal to 1; hence it shows that the 3-point EBM generated is 0-stable.

Similarly, this is extended to the 3-point implicit block method (IBM) given in (4.9) and the stability polynomial obtained is ๎€ท๐œ‰๎€ทdet๐ผโˆ’๐‘ง๐ต(0)๎€ธโˆ’๎€ท๐ด(1)+๐‘ง๐ต(1)๎€ธ๎€ธ=0(5.10) which gives๐œ‰3๎‚€โˆ’59๐‘ง432003+11๐‘ง3602โˆ’31๎‚120๐‘ง+1+๐œ‰2๎‚€โˆ’11957๐‘ง216003โˆ’2153๐‘ง4802โˆ’679๎‚๎‚€80๐‘งโˆ’2+๐œ‰89๐‘ง43203+1๐‘ง152โˆ’14๎‚โˆ’๎‚€1๐‘ง+1๐‘ง216003โˆ’1๐‘ง14402+137๐‘ง๎‚240=0.(5.11) Substituting ๐‘ง=0 in (5.11), we obtain all the roots of the derived equation to be less than or equal to 1; hence it shows that the 3-point IBM generated is 0-stable.

Theorem 5.6. The LMM (2.2) is convergent iff it is consistent and 0-Stable.

The proof is given in Fatunla [11] and Lambert [12].

Since the consistency and 0-stability of the methods have been established, then the explicit block method (3.5) and the implicit block method (4.7) are convergent.

The Region of Absolute Stability (RAS) of the block methods in this paper are drawn based on the third scheme of the block. The RAS of the linear multistep methods in the EBM (3.5) is drawn with the Maple software and displayed in Figure 1 below while the RAS for the implicit block method (4.7) is displayed in Figure 2.

513148.fig.001
Figure 1: RAS for the explicit scheme.
513148.fig.002
Figure 2: RAS for implicit scheme.

It is observed that the RAS of the IBM is wider in range than the RAS of the EBM. This means that the implicit schemes will cope with Initial Value Problems better than the EBM in implementation with a higher step length.

6. Implementation of Schemes Generated

A Matlab code was developed for the implementation of the schemes in Sections 3 and 4 above. The code was designed so that it determines the initial points of the starting block methods with the analytical solution if it exists.

Thereafter it generated the values for ๐‘ฆ๐‘›+1, ๐‘ฆ๐‘›+2, and ๐‘ฆ๐‘›+3, using the block schemes directly for the explicit schemes and predictor-corrector technique for the implicit schemes using a fixed step size โ„Ž. So for ๐‘ฃ=10 corrections, the sequence of computation follows the ๐‘ƒ(๐ธ๐ถ)๐‘ฃ, where ๐‘ƒ, ๐ธ, and ๐ถ denote Predicting, Evaluating and Correcting as it is generally used in Predictor-Corrector modes for numerical computations with a desired accuracy Lambert [12].

7. Numerical Experiment

7.1. Experimental Problems

In this paper three standard problems are considered and our newly developed methods are used to solve these problems. The problems are presented below.

Problem 1. Consider the test problem for second-order ODE given by ๐‘ฆ๎…ž๎…ž=๐œ†๐‘ฆ,๐‘ฆ(0)=๐‘ฆ๎…ž(0)=1,with๐œ†=โˆ’1,0โ‰ค๐‘ฅโ‰ค1.(7.1)

This problem is known to have an analytical solution of ๐‘ฆ(๐‘ฅ)=cos๐‘ฅ+sin๐‘ฅ, and the results are presented in Table 1.

tab1
Table 1: Result of the test problem (1).

Problem 2. The Van der Pol equation which describes the Van der Pol oscillator is the second-order ODE ๐‘ฆ๎…ž๎…ž๎€ท=๐œ‡1โˆ’๐‘ฆ2๎€ธ๐‘ฆ๎…žโˆ’๐œ†๐‘ฆ,๐‘ฆ(0)=๐ด,๐‘ฆ๎…ž(0)=๐ต,0โ‰ค๐‘ฅโ‰ค1.(7.2) and it assumes some real positive numbers ๐œ‡ and ๐œ†. The problem was named after B. Van der Pol in 1926. This equation has attracted a lot of research attention both in nonlinear mechanics and control theory. This equation has no solution in terms of known tabulated transcendental function Fatunla [11]. To solve this directly using the schemes generated, we solve for ๐œ‡=10โˆ’4, 10โˆ’6, and 10โˆ’8 with ๐œ†=๐ด=๐ต=1. However, as ๐œ‡=0, (7.2) has the analytical solution ๐‘ฆ(๐‘ฅ)=cos(๐‘ฅ)+sin(๐‘ฅ).(7.3)
The results are presented using Maximum Error which is given in Table 2.

tab2
Table 2: Result of the Van der Pol problem (2).

Problem 3. The third problem is the second order ODE ๐‘ฆ๎…ž๎…ž=โˆ’101๐‘ฆโ€ฒโˆ’100๐‘ฆ,๐‘ฆ(0)=1,๐‘ฆ๎…ž(0)=0,0โ‰ค๐‘ฅโ‰ค1,(7.4) with exact solution 1๐‘ฆ=๎€ท99100๐‘’โˆ’๐‘ฅโˆ’๐‘’โˆ’100๐‘ฅ๎€ธ.(7.5)
The results obtained by using the 3-point EBM and IBM are presented in Table 3.

tab3
Table 3: Result of problem (3).
7.2. Numerical Results

The Numerical results for the solution of the problems illustrated in the previous subsection will be presented in form of the Maximum Error.

It would be observed that in Problems 2, the explicit methods compare favourably with the implicit scheme but the accuracy of the methods increases as ๐œ‡ decreases. Whereas in Problems 1 and 3 the explicit block methods produce a poorer result compared to the implicit method. However, in Problem 3 the explicit method failed for a step length โ„Ž=0.01, 0.005, and 0.0025, while the implicit method shows its superiority by producing results.

8. Conclusion

We have been able to derive some 3-point Implicit and Explicit Block Methods via collocation multistep technique. This block schemes derived in this paper have been represented in form of (2.7), which is a representation given by Fatunla [10]. This representation of the schemes generated as a single block methods will yield 3 points on implementation. The Order, Stability, Consistency, and Convergence of these schemes were established as stated. These derived methods were implemented on standard mechanical problems and their results were found to be sufficiently accurate for various values of step length.

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