Advances in Numerical Analysis

Advances in Numerical Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 513148 | 14 pages | https://doi.org/10.1155/2011/513148

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

Academic Editor: István Faragó
Received03 Feb 2011
Revised18 Apr 2011
Accepted26 May 2011
Published03 Aug 2011

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 ğ‘¦î…žî…ž=𝑓(𝑥,𝑦,ğ‘¦î…ž),𝑦(𝑥0)=ğ‘Ž,ğ‘¦î…ž(𝑥0)=𝑏. 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, ğ‘¦î…žî…žî€·=𝑓𝑥,𝑦,ğ‘¦î…žî€¸î€·ğ‘¥,𝑦0=ğ‘Ž,ğ‘¦î…žî€·ğ‘¥0=𝑏.(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 equationğ‘¦î…žî…žî€·=𝑓𝑥,𝑦,ğ‘¦î…žî€¸î€·ğ‘¥,𝑦0=ğ‘Ž,ğ‘¦î…žî€·ğ‘¥0=𝑏.(2.1)

The numerical solution to (2.1) can be obtained using a 𝑘-step explicit Linear Multistep Method (LMM) of the form𝑘𝑗=0𝛼𝑗𝑦𝑛+𝑗=ℎ2𝑘𝑗=0𝛽𝑗𝑓𝑛+𝑗,(2.2) where 𝑦𝑛+𝑗≈𝑦(𝑥𝑛+ğ‘—â„Ž), 𝑓𝑛+𝑗≡𝑓(𝑥𝑛+ğ‘—â„Ž,𝑦(𝑥𝑛+ğ‘—â„Ž),ğ‘¦î…ž(𝑥𝑛+ğ‘—â„Ž)), and 𝑥𝑛 is a discrete point at node point 𝑛. Where, 𝛼𝑗 and 𝛽𝑗 are parameters to be determined and usually 𝛽𝑘=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 𝑦′ 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 ğ‘¦î…žî…ž=𝑓(𝑥,𝑦), while 𝑦′ is set to zero. However, by the introduction a of continuous scheme, this is easily taken care of. Thus if 𝑦(𝑥) is a basis polynomial of the form𝑦(𝑥)=𝑝𝑗=0ğ‘Žğ‘—î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚ğ‘—.(2.3)

To derive an 𝑚 point block method, where 𝑚 is a positive integer, we set 𝑝=2(𝑚−1) for an explicit scheme or 𝑝=2𝑚+1 for an implicit scheme, interpolating (2.3) at points 𝑥𝑛+𝑗, 𝑗=0,1,2,…,𝑘, and collocating ğ‘¦î…žî…ž(𝑥) at points x𝑛+𝑗, 𝑗=0,1,2,…,𝑘, will result to a (𝑝+1) system of equation for arbitrary 𝑘, Okunuga and Ehigie [1], 𝑦𝑥𝑛+𝑗=𝑦𝑛+𝑗𝑓𝑥,𝑗=0,1,2,…,𝑘−1,𝑛+𝑗=𝑓𝑛+𝑗,𝑗=0,1,2,…,𝑘.(2.4)

The coefficients ğ‘Ž0,ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘ are obtained and substituted in (2.3) to obtain the Continuous Linear Multistep Scheme (CLMS) of the form 𝑌(𝑥)=𝑘−1𝑗=0𝛼𝑗(𝑥)𝑦𝑛+𝑗+ℎ2𝑘𝑗=0𝛽𝑗(𝑥)𝑓𝑛+𝑗.(2.5)

This is evaluated for at 𝑥𝑛+𝑖, 𝑖=0,1,2,…𝑚 to obtain an 𝑚-point block method generally represented by Fatunla [10]. With the 𝑚-vector 𝑌𝑚 and 𝐹𝑚 specified as, 𝑌𝑛=𝑦𝑛+1,𝑦𝑛+2,𝑦𝑛+3,…,𝑦𝑛+𝑚𝑇,𝑌𝑛−1=𝑦𝑛,𝑦𝑛−1,𝑦𝑛−2,…,𝑦𝑛−𝑚+1𝑇,𝐹𝑛−1=𝑓𝑛,𝑓𝑛−1,𝑓𝑛−2,…,𝑓𝑛−𝑚+1𝑇,𝐹𝑛=𝑓𝑛+1,𝑓𝑛+2,𝑓𝑛+3,…,𝑓𝑛+𝑚𝑇.(2.6)

The 𝑟-block, 𝑚-point EBM will be generally represented as𝑌𝑛=𝑟𝑖=0𝐴(𝑖)𝑌𝑛−1+ℎ2𝑟𝑖=0𝐵(𝑖)𝐹𝑛−1,(2.7) where 𝐴(𝑖) and 𝐵(𝑖), 𝑖=0,1,2,…,𝑟, are 𝑚×𝑚 square matrix with elements ğ‘Žğ‘–ğ‘™ğ‘—, 𝑏𝑖𝑙𝑗 for 𝑙,𝑗=1,2,…,𝑚. The block scheme (2.7) is explicit if the coefficient Matrix 𝐵(0) is a null matrix.

3. Derivation of Explicit Block Methods

To derive a 1 block 3-point Explicit Block Method (EBM) that is, 𝑚=3, we set 𝑝=4. Let 𝑦(𝑥) be a basis function so that𝑦(𝑥)=𝑝𝑗=0ğ‘Žğ‘—î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚ğ‘—(3.1) while we interpolate (3.1) at point 𝑥=𝑥𝑛 and 𝑥𝑛−1 and collocate ğ‘¦î…žî…ž(𝑥) at 𝑥𝑛, 𝑥𝑛−1, and 𝑥𝑛−2 to obtain a system of equationsâŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽğ‘Ž100001−11−1100200002−612002−12480ğ‘Ž1ğ‘Ž2ğ‘Ž3ğ‘Ž4⎞⎟⎟⎟⎟⎟⎟⎟⎠=âŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽğ‘¦ğ‘›ğ‘¦ğ‘›âˆ’1ℎ2ğ‘“ğ‘›â„Ž2𝑓𝑛−1ℎ2𝑓𝑛−2⎞⎟⎟⎟⎟⎟⎟⎟⎠.(3.2) Solving the matrix equation above, we obtainğ‘Ž0=𝑦𝑛,ğ‘Ž1=𝑦𝑛−𝑦𝑛−1+7ℎ242𝑓𝑛+14ℎ2𝑓𝑛−1−1ℎ242𝑓𝑛−2,ğ‘Ž2=12ℎ2𝑓𝑛,ğ‘Ž3=14ℎ2𝑓𝑛−13ℎ2𝑓𝑛−1+1ℎ122𝑓𝑛−2,ğ‘Ž4=1ℎ242𝑓𝑛−1ℎ122𝑓𝑛−1+1ℎ242𝑓𝑛−2.(3.3)

Substituting the values ğ‘Ž0,ğ‘Ž1,ğ‘Ž2,ğ‘Ž3, and ğ‘Ž4 in (3.1), we obtain the CLMS𝑦(𝑥)=ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚ğ‘¦ğ‘›âˆ’1âˆ’î‚€î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚î‚ğ‘¦+1𝑛+ℎ2724ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚+12î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚2+14î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3+124ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚4𝑓𝑛+ℎ214î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚âˆ’13î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3−112ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚4𝑓𝑛−1+ℎ2−124ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚+124ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3+124ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚4𝑓𝑛−2.(3.4)

On evaluating (3.4) at points 𝑥=𝑥𝑛+𝑖,𝑖=1,2,3, we obtained the convergent explicit 3-point EBM as𝑦𝑛+1=−𝑦𝑛−1+2𝑦𝑛+ℎ213𝑓12𝑛−16𝑓𝑛−1+1𝑓12𝑛−2,𝑦𝑛+2=−2𝑦𝑛−1+3𝑦𝑛+ℎ2214𝑓𝑛−72𝑓𝑛−1+54𝑓𝑛−2,𝑦𝑛+3=−3𝑦𝑛−1+4𝑦𝑛+ℎ2312𝑓𝑛−15𝑓𝑛−1+112𝑓𝑛−2.(3.5)

Differentiating (3.4) and evaluating again at the same 3 discrete points of 𝑖, we obtain a block of first-order derivatives which can be used to determine the derivative term in the initial value problem (2.1).ğ‘¦î…žğ‘›+1=1â„Žî€·ğ‘¦ğ‘›âˆ’ğ‘¦ğ‘›âˆ’1+ℎ53𝑓24𝑛−13𝑓12𝑛−1+38𝑓𝑛−2,ğ‘¦î…žğ‘›+2=1â„Žî€·ğ‘¦ğ‘›âˆ’ğ‘¦ğ‘›âˆ’1+ℎ538𝑓𝑛−77𝑓12𝑛−1+55𝑓24𝑛−2,ğ‘¦î…žğ‘›+3=1â„Žî€·ğ‘¦ğ‘›âˆ’ğ‘¦ğ‘›âˆ’1+ℎ349𝑓24𝑛−714𝑓𝑛−1+161𝑓24𝑛−2.(3.6)

Expressing the schemes (3.5) as block using previous definition (2.7), we obtainâŽ›âŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘¦100010001𝑛+1𝑦𝑛+2𝑦𝑛+3⎞⎟⎟⎟⎠=âŽ›âŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘¦0−120−230−34𝑛−2𝑦𝑛−1ğ‘¦ğ‘›âŽžâŽŸâŽŸâŽŸâŽ +ℎ2⎛⎜⎜⎜⎜⎜⎝1−1126135124−72214112−15312âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘“ğ‘›âˆ’2𝑓𝑛−1ğ‘“ğ‘›âŽžâŽŸâŽŸâŽŸâŽ .(3.7)

Equation (3.7) is therefore said to be of the form (2.7). Thus (3.7) is represented notationally as 𝑌𝑛=𝐴(1)𝑌𝑛−1+ℎ2𝐵(1)𝐹𝑛−1.

4. Derivation of Implicit Block Methods

To derive a 1 block 3-point Implicit Block Method (IBM), we also define the following terms: 𝑌𝑛=𝑦𝑛+1,𝑦𝑛+2,𝑦𝑛+3𝑇,𝑌𝑛−1=𝑦𝑛,𝑦𝑛−1,𝑦𝑛−2𝑇,𝐹𝑛=𝑓𝑛+1,𝑓𝑛+2,𝑓𝑛+3𝑇,𝐹𝑛−1=𝑓𝑛,𝑓𝑛−1,𝑓𝑛−2𝑇.(4.1)

The 3- point IBM will be generally represented as𝑌𝑛=𝐴(1)𝑌𝑛−1+ℎ2𝐵(1)𝐹𝑛−1+𝐵(0)𝐹𝑛,(4.2) where 𝐴(1), 𝐵(0), and 𝐵(1) are 3×3 square matrix. Let 𝑦(𝑥) be a basis function so that 𝑦(𝑥)=𝑝𝑗=0ğ‘Žğ‘—î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚ğ‘—.(4.3) Setting 𝑝=7 for an implicit 3-point block scheme, we will interpolate (4.3) at points 𝑥𝑛 and 𝑥𝑛−1 and collocate ğ‘¦î…žî…ž(𝑥) at 6 points 𝑥𝑛+𝑖, 𝑖=−2,−1,0,1,2,3, to obtain a system of equations represented by the matrixâŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽğ‘Ž1−11−11−11−110000000002−1248−160480−1344002−612−2030−420020000000261220304200212481604801344002181085402430102060ğ‘Ž1ğ‘Ž2ğ‘Ž3ğ‘Ž4ğ‘Ž5ğ‘Ž6ğ‘Ž7⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=âŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽğ‘¦ğ‘›âˆ’1ğ‘¦ğ‘›â„Ž2𝑓𝑛−2ℎ2𝑓𝑛−1ℎ2ğ‘“ğ‘›â„Ž2𝑓𝑛+1ℎ2𝑓𝑛+2ℎ2𝑓𝑛+3⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(4.4) Solving the matrix equation, we obtain ğ‘Ž0=𝑦𝑛,ğ‘Ž1=𝑦𝑛+271ℎ20162𝑓𝑛−1+1111ℎ25202𝑓𝑛−𝑦𝑛−1−451ℎ50402𝑓𝑛+1+131ℎ50402𝑓𝑛+2−41ℎ50402𝑓𝑛−2,−37ℎ100802𝑓𝑛+3,ğ‘Ž2=12𝑓𝑛,ğ‘Ž31=−ℎ182𝑓𝑛−1ℎ242𝑓𝑛+2+1ℎ1202𝑓𝑛−2+16ℎ2𝑓𝑛+1−1ℎ122𝑓𝑛−1+1ℎ1802𝑓𝑛+3,ğ‘Ž45=−ℎ482𝑓𝑛−1ℎ2882𝑓𝑛+2−1ℎ2882𝑓𝑛−2+1ℎ182𝑓𝑛+1+1ℎ182𝑓𝑛−1,ğ‘Ž5=1ℎ482𝑓𝑛+7ℎ4802𝑓𝑛+2−1ℎ4802𝑓𝑛−2−7ℎ2402𝑓𝑛+1−1ℎ4802𝑓𝑛−1−1ℎ4802𝑓𝑛+3,ğ‘Ž6=1ℎ1202𝑓𝑛+1ℎ7202𝑓𝑛+2+1ℎ7202𝑓𝑛−2+1ℎ1802𝑓𝑛+1−1ℎ1802𝑓𝑛−1,ğ‘Ž71=−ℎ5042𝑓𝑛−1ℎ10082𝑓𝑛+2+1ℎ50402𝑓𝑛−2+1ℎ5042𝑓𝑛+1−1ℎ10802𝑓𝑛−1−1ℎ50402𝑓𝑛+3.(4.5)

Substituting the ğ‘Žğ‘–,𝑖=0,1,…,7 in (4.3), we obtain the CLMS,𝑦(𝑥)=âˆ’ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚ğ‘¦ğ‘›âˆ’1+î‚€î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚î‚ğ‘¦+1𝑛+ℎ2−4515040ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚+16î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3+118ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚4−7240ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚5−1180ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚6+1504ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚7𝑓𝑛+1+ℎ2−1315040ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚âˆ’124ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3−1288ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚4+7480ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚5+1720ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚6−11008ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚7𝑓𝑛+2+ℎ2−3710080ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚+1180ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3−1480ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚5+15040ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚7𝑓𝑛+3+ℎ2−415040ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚+1120ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3−1288ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚4−1480ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚5+1720ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚6−15040ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚7𝑓𝑛−2+ℎ22712016ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚âˆ’112ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3+118ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚4−1480ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚5−1180ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚6+11080ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚7𝑓𝑛−1+ℎ211112520ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚+12î‚€ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚2−118ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚3−148ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚4+148ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚5+1120ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚6−1504ğ‘¥âˆ’ğ‘¥ğ‘›â„Žî‚7𝑓𝑛.(4.6)

On evaluating (4.6) at points 𝑥=𝑥𝑛+1,𝑥𝑛+2 and 𝑥𝑛+3, we obtain the 3-point implicit block Linear Multistep methods 𝑦𝑛+1=−𝑦𝑛−1+2𝑦𝑛+ℎ21𝑓10𝑛+1−1𝑓240𝑛+2+ℎ297𝑓120𝑛+1𝑓10𝑛−1−1𝑓240𝑛−2,𝑦𝑛+2=−2𝑦𝑛−1+3𝑦𝑛+ℎ2121𝑓120𝑛+1+11𝑓120𝑛+2−1𝑓240𝑛+3+ℎ2103𝑓60𝑛+47𝑓240𝑛−1−1𝑓120𝑛−2,𝑦𝑛+3=−3𝑦𝑛−1+4𝑦𝑛+ℎ229𝑓15𝑛+1+127𝑓120𝑛+2+1𝑓15𝑛+3+ℎ2161𝑓60𝑛+4𝑓15𝑛−1−1𝑓120𝑛−2.(4.7)

On differentiating (4.6) and evaluating again at the same 3 discrete points of 𝑥, we obtain𝑦′𝑛+1=1â„Žî€·ğ‘¦ğ‘›âˆ’ğ‘¦ğ‘›âˆ’1+ℎ589𝑓1260𝑛+1−389𝑓10080𝑛+2+1𝑓252𝑛+3+ℎ5029𝑓5040𝑛+22𝑓315𝑛−1−1𝑓2016𝑛−2,𝑦′𝑛+2=1â„Žî€·ğ‘¦ğ‘›âˆ’ğ‘¦ğ‘›âˆ’1+ℎ5933𝑓5040𝑛+1+407𝑓1008𝑛+2−149𝑓10080𝑛+3+ℎ2063𝑓2520𝑛+1243𝑓10080𝑛−1−41𝑓5040𝑛−2,𝑦′𝑛+3=1â„Žî€·ğ‘¦ğ‘›âˆ’ğ‘¦ğ‘›âˆ’1+ℎ157𝑓252𝑛+1+14059𝑓10080𝑛+2+397𝑓1260𝑛+3+ℎ5813𝑓5040𝑛+1𝑓315𝑛−1+107𝑓10080𝑛−2.(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) becomesâŽ›âŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘¦100010001𝑛+1𝑦𝑛+2𝑦𝑛+3⎞⎟⎟⎟⎠=âŽ›âŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘¦0−120−230−34𝑛−2𝑦𝑛−1ğ‘¦ğ‘›âŽžâŽŸâŽŸâŽŸâŽ +ℎ2⎛⎜⎜⎜⎜⎜⎝1−110024012112011−112024029151271120âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘“15𝑛+1𝑓𝑛+2𝑓𝑛+3⎞⎟⎟⎟⎠+ℎ2⎛⎜⎜⎜⎜⎜⎝−112401097−112012047240103−160412015161âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘“60𝑛−2𝑓𝑛−1ğ‘“ğ‘›âŽžâŽŸâŽŸâŽŸâŽ .(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,Ł[]=𝑦(𝑥);â„Žğ‘˜î“ğ‘—=0𝛼𝑗𝑦(𝑥+ğ‘—â„Ž)−ℎ2ğ›½ğ‘—ğ‘¦î…žî…žî€¸,(𝑥+ğ‘—â„Ž)(5.1) where 𝑦(𝑥) is an arbitrary function continuously differentiable on the interval [ğ‘Ž,𝑏]. The Taylors series expansion about the point 𝑥, Ł[]𝑦(𝑥);ℎ=𝑐0𝑦(𝑥)+𝑐1â„Žğ‘¦î…ž(𝑥)+𝑐2ℎ2ğ‘¦î…žî…ž(𝑥)+⋯+ğ‘ğ‘žâ„Žğ‘žğ‘¦(ğ‘ž)(𝑥),(5.2) where𝑐0=𝛼0+𝛼1+𝛼2+⋯+𝛼𝑘,𝑐1=𝛼1+2𝛼2+⋯+𝑘𝛼𝑘,𝑐2=1𝛼2!1+22𝛼2+⋯+𝑘2𝛼𝑘−𝛽1+𝛽2+⋯+𝛽𝑘,â‹®ğ‘ğ‘ž=1î€·ğ›¼ğ‘ž!1+2ğ‘žğ›¼2+⋯+ğ‘˜ğ‘žğ›¼ğ‘˜î€¸âˆ’1𝛽(ğ‘žâˆ’2)!1+2ğ‘žâˆ’2𝛽2+⋯+ğ‘˜ğ‘žâˆ’2𝛽𝑘,ğ‘ž=3,4,….(5.3)

Definition 5.1. The method (2.2) is said to be of order 𝑝 if 𝑐0=𝑐1=𝑐2=⋯=𝑐𝑝+1=0,𝑐𝑝+2≠0.(5.4)𝑐𝑝+2 is the error constant and 𝑐𝑝+2â„Žğ‘+2𝑦(𝑝+2)(𝑥𝑛) is the truncation error at point 𝑥𝑛.
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 𝑝=(3,3,3)𝑇 with error constants [1/12,4/3,41/6]𝑇. Also applying this definitions on the implicit methods (4.7), the implicit method was of order 𝑝=(6,6,6)𝑇 with error constants [−439/4320,−3479/2880,−1393/180]𝑇.

Definition 5.2. A Linear Multistep Method of the form (2.2) is said to be consistent if the LMM is of order 𝑝≥1.

Since the methods derived in (3.5) and (4.7) are of order 𝑝≥1, 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, 𝜌(𝜉), associated with the linear multistep method (2.2), where it is the polynomial of degree 𝑘 whose coefficients are 𝛼𝑗 and the second characteristic polynomial ğœŽ(𝜉) whose coefficients are 𝛽𝑗, is defined by 𝜌(𝜉)=𝑘𝑗=0ğ›¼ğ‘—ğœ‰ğ‘—ğœŽ,(5.5a)(𝜉)=𝑘𝑗=0𝛽𝑗𝜉𝑗,(5.5b) where 𝜉∈𝐶, 𝐶 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 𝜉𝑗, 𝑗=1(1)𝑘, of the first characteristic polynomial ∑𝜌(𝜉)=det[𝑘𝑖=0𝐴𝑖𝜉𝑘−1]=0, 𝐴0=−𝐼, satisfy |𝜉|≤1. If one of the roots is +1, we call the roots the principal roots of 𝜌(𝜉).

Definition 5.5. The Region of Absolute Stability (RAS) of methods of (2.7) and (4.2) is the set î€½â„Žğ‘…=2𝜆∶forthatℎ2𝜆wheretherootsofthestabilitypolynomialareofabsolutelessthanone}.(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 𝑧𝜌𝑒(𝜃)=ğ‘–ğœƒî€¸ğœŽî€·ğ‘’ğ‘–ğœƒî€¸.(5.6) Resolving this to real and imaginary parts and evaluating for values of 𝜃∈(0,2𝜋) 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, ğ‘¦î…žî…ž=𝜆𝑦. By setting 𝑧=ğœ†â„Ž2, the block scheme (3.7) becomes âŽ›âŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘¦100010001𝑛+1𝑦𝑛+2𝑦𝑛+3⎞⎟⎟⎟⎠=âŽ›âŽœâŽœâŽœâŽœâŽœâŽğ‘§ğ‘§12−1−62+13𝑧125𝑧4−2−7𝑧23+21𝑧4−11𝑧2−3−15𝑧4+31𝑧2âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽğ‘¦ğ‘›âˆ’2𝑦𝑛−1ğ‘¦ğ‘›âŽžâŽŸâŽŸâŽŸâŽ .(5.7) This is of the form 𝑌𝑛=(𝐴(1)+𝑧𝐵(1))𝑌𝑛−1. The stability polynomial of this is given as𝐴det𝜉−(1)+𝑧𝐵(1)=0.(5.8) Hence the stability polynomial of the 3-point EBM (3.7) is𝜉3+𝜉2−14512𝑧−2+𝜉794𝑧2+376+𝑧+1−𝑧3+54𝑧2−37𝑧12=0.(5.9) Substituting 𝑧=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 𝜉det𝐼−𝑧𝐵(0)−𝐴(1)+𝑧𝐵(1)=0(5.10) which gives𝜉3−59𝑧432003+11𝑧3602−31120𝑧+1+𝜉2−11957𝑧216003−2153𝑧4802−67980𝑧−2+𝜉89𝑧43203+1𝑧152−14−1𝑧+1𝑧216003−1𝑧14402+137𝑧240=0.(5.11) Substituting 𝑧=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.

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 𝑦𝑛+1, 𝑦𝑛+2, and 𝑦𝑛+3, using the block schemes directly for the explicit schemes and predictor-corrector technique for the implicit schemes using a fixed step size ℎ. So for 𝑣=10 corrections, the sequence of computation follows the 𝑃(𝐸𝐶)𝑣, where 𝑃, 𝐸, and 𝐶 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 ğ‘¦î…žî…ž=𝜆𝑦,𝑦(0)=ğ‘¦î…ž(0)=1,with𝜆=−1,0≤𝑥≤1.(7.1)

This problem is known to have an analytical solution of 𝑦(𝑥)=cos𝑥+sin𝑥, and the results are presented in Table 1.


ℎ Explicit max. error Implicit max. error

0.01 5.12 𝐸 − 0 7 6.55 𝐸 − 1 4
0.005 6.30 𝐸 − 0 8 2.12 𝐸 − 1 3
0.0025 7.81 𝐸 − 0 9 1.78 𝐸 − 1 3
0.001 4.95 𝐸 − 1 0 9.43 𝐸 − 1 3
0.0005 5.52 𝐸 − 1 1 8.00 𝐸 − 1 2
0.00025 1.15 𝐸 − 1 1 1.77 𝐸 − 1 1

Problem 2. The Van der Pol equation which describes the Van der Pol oscillator is the second-order ODE ğ‘¦î…žî…žî€·=𝜇1−𝑦2î€¸ğ‘¦î…žâˆ’ğœ†ğ‘¦,𝑦(0)=𝐴,ğ‘¦î…ž(0)=𝐵,0≤𝑥≤1.(7.2) and it assumes some real positive numbers 𝜇 and 𝜆. 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 𝜇=10−4, 10−6, and 10−8 with 𝜆=𝐴=𝐵=1. However, as 𝜇=0, (7.2) has the analytical solution 𝑦(𝑥)=cos(𝑥)+sin(𝑥).(7.3)
The results are presented using Maximum Error which is given in Table 2.


𝜇 = 1 0 − 4 𝜇 = 1 0 − 6 𝜇 = 1 0 − 8
ℎ Explicit Implicit Explicit Implicit Explicit Implicit
max. errormax. errormax. errormax. errormax. errormax. error

0.01 1.15 𝐸 − 0 5 1.10 𝐸 − 0 5 6.22 𝐸 − 0 7 1.10 𝐸 − 0 7 5.13 𝐸 − 0 7 1.10 𝐸 − 0 9
0.005 1.10 𝐸 − 0 5 1.10 𝐸 − 0 5 1.72 𝐸 − 0 7 1.10 𝐸 − 0 7 6.48 𝐸 − 0 8 1.10 𝐸 − 0 9
0.0025 1.10 𝐸 − 0 5 1.10 𝐸 − 0 5 1.17 𝐸 − 0 7 1.10 𝐸 − 0 7 8.90 𝐸 − 0 9 1.10 𝐸 − 0 9
0.001 1.10 𝐸 − 0 5 1.10 𝐸 − 0 5 1.10 𝐸 − 0 7 1.10 𝐸 − 0 7 1.59 𝐸 − 0 9 1.10 𝐸 − 0 9
0.0005 1.10 𝐸 − 0 5 1.10 𝐸 − 0 5 1.10 𝐸 − 0 7 1.10 𝐸 − 0 7 1.16 𝐸 − 0 9 1.10 𝐸 − 0 9
0.00025 1.10 𝐸 − 0 5 1.10 𝐸 − 0 5 1.10 𝐸 − 0 7 1.10 𝐸 − 0 7 1.07 𝐸 − 0 9 1.10 𝐸 − 0 9

Problem 3. The third problem is the second order ODE ğ‘¦î…žî…ž=−101𝑦′−100𝑦,𝑦(0)=1,ğ‘¦î…ž(0)=0,0≤𝑥≤1,(7.4) with exact solution 1𝑦=99100𝑒−𝑥−𝑒−100𝑥.(7.5)
The results obtained by using the 3-point EBM and IBM are presented in Table 3.


ℎ Explicit max. errorImplicit max. error

0.01 Failed 6.90 𝐸 − 0 6
0.005 Failed 7.05 𝐸 − 0 9
0.0025 Failed 4.34 𝐸 − 1 0
0.001 3.40 𝐸 − 0 6 2.79 𝐸 − 1 2
0.0005 6.17 𝐸 − 0 7 1.10 𝐸 − 1 3
0.00025 9.17 𝐸 − 0 8 5.07 𝐸 − 1 4

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 𝜇 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 ℎ=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.

References

  1. S. A. Okunuga and J. Ehigie, “A new derivation of continuous collocation multistep methods using power series as basis function,” Journal of Modern Mathematics and Statistics, vol. 3, no. 2, pp. 43–50, 2009. View at: Google Scholar
  2. P. Onumanyi, D. O. Awoyemi, S. N. Jator, and U. W. Sirisena, “New linear multistep methods with continuous coefficients for first order IVPs,” Journal of the Nigerian Mathematical Society, vol. 13, pp. 37–51, 1994. View at: Google Scholar
  3. R. L. Brown, “Some characteristics of implicit multistep multi-derivative integration formulas,” SIAM Journal on Numerical Analysis, vol. 14, no. 6, pp. 982–993, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. P. Onumanyi, J. Fatokun, and B. O. Adejo, “Accurate numerical differentiation by continuous integrators for ordinary differential equations,” Journal of the Nigerian Mathematical Society, vol. 27, pp. 69–90, 2008. View at: Google Scholar | Zentralblatt MATH
  5. F. Ismail, L. K. Yap, and O. Mohamad, “Explicit and implicit 3-point block methods for solving special second order ordinary differential equations directly,” International Journal of Mathematical Analysis, vol. 3, no. 5, pp. 239–254, 2009. View at: Google Scholar | Zentralblatt MATH
  6. J. O. Ehigie, S. A. Okunuga, and A. B. Sofoluwe, “On generalized 2-step continuous linear multistep method of hybrid type for the integration of second order ordinary differential equations,” Archives of Applied Science Research, vol. 2, no. 6, pp. 362–372, 2010. View at: Google Scholar
  7. W. E. Milne, Numerical Solution of Differential Equations, John Wiley & Sons, New York, NY, USA, 1953.
  8. J. B. Rosser, “A Runge-Kutta for all seasons,” SIAM Review, vol. 9, pp. 417–452, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. L. F. Shampine and H. A. Watts, “Block implicit one-step methods,” Mathematics of Computation, vol. 23, pp. 731–740, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. S. O. Fatunla, “Parallel methods for second order ODEs,” in Monogragh- Computational Ordinary Differential Equations, S. O. Fatunla, Ed., pp. 87–99, University Press Plc, Ibadan, Nigeria, 1992. View at: Google Scholar
  11. S. O. Fatunla, Numerical Methods for Fatunla Initial Value Problems in Ordinary Differential Equations, Academic Press, New York, NY, USA, 1988.
  12. J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1973.

Copyright © 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.

1123 Views | 648 Downloads | 3 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder