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 𝑦=𝑓(𝑥,𝑦,𝑦),𝑦(𝑥0)=𝑎,𝑦(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, 𝑦=𝑓𝑥,𝑦,𝑦𝑥,𝑦0=𝑎,𝑦𝑥0=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 equation𝑦=𝑓𝑥,𝑦,𝑦𝑥,𝑦0=𝑎,𝑦𝑥0=𝑏.(2.1)

The numerical solution to (2.1) can be obtained using a 𝑘-step explicit Linear Multistep Method (LMM) of the form𝑘𝑗=0𝛼𝑗𝑦𝑛+𝑗=2𝑘𝑗=0𝛽𝑗𝑓𝑛+𝑗,(2.2) where 𝑦𝑛+𝑗𝑦(𝑥𝑛+𝑗), 𝑓𝑛+𝑗𝑓(𝑥𝑛+𝑗,𝑦(𝑥𝑛+𝑗),𝑦(𝑥𝑛+𝑗)), 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 𝑦=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)=𝑝𝑗=0𝑎𝑗𝑥𝑥𝑛𝑗.(2.3)

To derive an 𝑚 point block method, where 𝑚 is a positive integer, we set 𝑝=2(𝑚1) 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 𝑦(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], 𝑦𝑥𝑛+𝑗=𝑦𝑛+𝑗𝑓𝑥,𝑗=0,1,2,,𝑘1,𝑛+𝑗=𝑓𝑛+𝑗,𝑗=0,1,2,,𝑘.(2.4)

The coefficients 𝑎0,𝑎1,𝑎2,,𝑎𝑝 are obtained and substituted in (2.3) to obtain the Continuous Linear Multistep Scheme (CLMS) of the form 𝑌(𝑥)=𝑘1𝑗=0𝛼𝑗(𝑥)𝑦𝑛+𝑗+2𝑘𝑗=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+𝑚𝑇,𝑌𝑛1=𝑦𝑛,𝑦𝑛1,𝑦𝑛2,,𝑦𝑛𝑚+1𝑇,𝐹𝑛1=𝑓𝑛,𝑓𝑛1,𝑓𝑛2,,𝑓𝑛𝑚+1𝑇,𝐹𝑛=𝑓𝑛+1,𝑓𝑛+2,𝑓𝑛+3,,𝑓𝑛+𝑚𝑇.(2.6)

The 𝑟-block, 𝑚-point EBM will be generally represented as𝑌𝑛=𝑟𝑖=0𝐴(𝑖)𝑌𝑛1+2𝑟𝑖=0𝐵(𝑖)𝐹𝑛1,(2.7) where 𝐴(𝑖) and 𝐵(𝑖), 𝑖=0,1,2,,𝑟, are 𝑚×𝑚 square matrix with elements 𝑎𝑖𝑙𝑗, 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)=𝑝𝑗=0𝑎𝑗𝑥𝑥𝑛𝑗(3.1) while we interpolate (3.1) at point 𝑥=𝑥𝑛 and 𝑥𝑛1 and collocate 𝑦(𝑥) at 𝑥𝑛, 𝑥𝑛1, and 𝑥𝑛2 to obtain a system of equations𝑎10000111110020000261200212480𝑎1𝑎2𝑎3𝑎4=𝑦𝑛𝑦𝑛12𝑓𝑛2𝑓𝑛12𝑓𝑛2.(3.2) Solving the matrix equation above, we obtain𝑎0=𝑦𝑛,𝑎1=𝑦𝑛𝑦𝑛1+7242𝑓𝑛+142𝑓𝑛11242𝑓𝑛2,𝑎2=122𝑓𝑛,𝑎3=142𝑓𝑛132𝑓𝑛1+1122𝑓𝑛2,𝑎4=1242𝑓𝑛1122𝑓𝑛1+1242𝑓𝑛2.(3.3)

Substituting the values 𝑎0,𝑎1,𝑎2,𝑎3, and 𝑎4 in (3.1), we obtain the CLMS𝑦(𝑥)=𝑥𝑥𝑛𝑦𝑛1𝑥𝑥𝑛𝑦+1𝑛+2724𝑥𝑥𝑛+12𝑥𝑥𝑛2+14𝑥𝑥𝑛3+124𝑥𝑥𝑛4𝑓𝑛+214𝑥𝑥𝑛13𝑥𝑥𝑛3112𝑥𝑥𝑛4𝑓𝑛1+2124𝑥𝑥𝑛+124𝑥𝑥𝑛3+124𝑥𝑥𝑛4𝑓𝑛2.(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=𝑦𝑛1+2𝑦𝑛+213𝑓12𝑛16𝑓𝑛1+1𝑓12𝑛2,𝑦𝑛+2=2𝑦𝑛1+3𝑦𝑛+2214𝑓𝑛72𝑓𝑛1+54𝑓𝑛2,𝑦𝑛+3=3𝑦𝑛1+4𝑦𝑛+2312𝑓𝑛15𝑓𝑛1+112𝑓𝑛2.(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).𝑦𝑛+1=1𝑦𝑛𝑦𝑛1+53𝑓24𝑛13𝑓12𝑛1+38𝑓𝑛2,𝑦𝑛+2=1𝑦𝑛𝑦𝑛1+538𝑓𝑛77𝑓12𝑛1+55𝑓24𝑛2,𝑦𝑛+3=1𝑦𝑛𝑦𝑛1+349𝑓24𝑛714𝑓𝑛1+161𝑓24𝑛2.(3.6)

Expressing the schemes (3.5) as block using previous definition (2.7), we obtain𝑦100010001𝑛+1𝑦𝑛+2𝑦𝑛+3=𝑦012023034𝑛2𝑦𝑛1𝑦𝑛+2111261351247221411215312𝑓𝑛2𝑓𝑛1𝑓𝑛.(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)𝑌𝑛1+2𝐵(1)𝐹𝑛1.

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+3𝑇,𝑌𝑛1=𝑦𝑛,𝑦𝑛1,𝑦𝑛2𝑇,𝐹𝑛=𝑓𝑛+1,𝑓𝑛+2,𝑓𝑛+3𝑇,𝐹𝑛1=𝑓𝑛,𝑓𝑛1,𝑓𝑛2𝑇.(4.1)

The 3- point IBM will be generally represented as𝑌𝑛=𝐴(1)𝑌𝑛1+2𝐵(1)𝐹𝑛1+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)=𝑝𝑗=0𝑎𝑗𝑥𝑥𝑛𝑗.(4.3) Setting 𝑝=7 for an implicit 3-point block scheme, we will interpolate (4.3) at points 𝑥𝑛 and 𝑥𝑛1 and collocate 𝑦(𝑥) at 6 points 𝑥𝑛+𝑖, 𝑖=2,1,0,1,2,3, to obtain a system of equations represented by the matrix𝑎1111111110000000002124816048013440026122030420020000000261220304200212481604801344002181085402430102060𝑎1𝑎2𝑎3𝑎4𝑎5𝑎6𝑎7=𝑦𝑛1𝑦𝑛2𝑓𝑛22𝑓𝑛12𝑓𝑛2𝑓𝑛+12𝑓𝑛+22𝑓𝑛+3.(4.4) Solving the matrix equation, we obtain 𝑎0=𝑦𝑛,𝑎1=𝑦𝑛+27120162𝑓𝑛1+111125202𝑓𝑛𝑦𝑛145150402𝑓𝑛+1+13150402𝑓𝑛+24150402𝑓𝑛2,37100802𝑓𝑛+3,𝑎2=12𝑓𝑛,𝑎31=182𝑓𝑛1242𝑓𝑛+2+11202𝑓𝑛2+162𝑓𝑛+11122𝑓𝑛1+11802𝑓𝑛+3,𝑎45=482𝑓𝑛12882𝑓𝑛+212882𝑓𝑛2+1182𝑓𝑛+1+1182𝑓𝑛1,𝑎5=1482𝑓𝑛+74802𝑓𝑛+214802𝑓𝑛272402𝑓𝑛+114802𝑓𝑛114802𝑓𝑛+3,𝑎6=11202𝑓𝑛+17202𝑓𝑛+2+17202𝑓𝑛2+11802𝑓𝑛+111802𝑓𝑛1,𝑎71=5042𝑓𝑛110082𝑓𝑛+2+150402𝑓𝑛2+15042𝑓𝑛+1110802𝑓𝑛1150402𝑓𝑛+3.(4.5)

Substituting the 𝑎𝑖,𝑖=0,1,,7 in (4.3), we obtain the CLMS,𝑦(𝑥)=𝑥𝑥𝑛𝑦𝑛1+𝑥𝑥𝑛𝑦+1𝑛+24515040𝑥𝑥𝑛+16𝑥𝑥𝑛3+118𝑥𝑥𝑛47240𝑥𝑥𝑛51180𝑥𝑥𝑛6+1504𝑥𝑥𝑛7𝑓𝑛+1+21315040𝑥𝑥𝑛124𝑥𝑥𝑛31288𝑥𝑥𝑛4+7480𝑥𝑥𝑛5+1720𝑥𝑥𝑛611008𝑥𝑥𝑛7𝑓𝑛+2+23710080𝑥𝑥𝑛+1180𝑥𝑥𝑛31480𝑥𝑥𝑛5+15040𝑥𝑥𝑛7𝑓𝑛+3+2415040𝑥𝑥𝑛+1120𝑥𝑥𝑛31288𝑥𝑥𝑛41480𝑥𝑥𝑛5+1720𝑥𝑥𝑛615040𝑥𝑥𝑛7𝑓𝑛2+22712016𝑥𝑥𝑛112𝑥𝑥𝑛3+118𝑥𝑥𝑛41480𝑥𝑥𝑛51180𝑥𝑥𝑛6+11080𝑥𝑥𝑛7𝑓𝑛1+211112520𝑥𝑥𝑛+12𝑥𝑥𝑛2118𝑥𝑥𝑛3148𝑥𝑥𝑛4+148𝑥𝑥𝑛5+1120𝑥𝑥𝑛61504𝑥𝑥𝑛7𝑓𝑛.(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=𝑦𝑛1+2𝑦𝑛+21𝑓10𝑛+11𝑓240𝑛+2+297𝑓120𝑛+1𝑓10𝑛11𝑓240𝑛2,𝑦𝑛+2=2𝑦𝑛1+3𝑦𝑛+2121𝑓120𝑛+1+11𝑓120𝑛+21𝑓240𝑛+3+2103𝑓60𝑛+47𝑓240𝑛11𝑓120𝑛2,𝑦𝑛+3=3𝑦𝑛1+4𝑦𝑛+229𝑓15𝑛+1+127𝑓120𝑛+2+1𝑓15𝑛+3+2161𝑓60𝑛+4𝑓15𝑛11𝑓120𝑛2.(4.7)

On differentiating (4.6) and evaluating again at the same 3 discrete points of 𝑥, we obtain𝑦𝑛+1=1𝑦𝑛𝑦𝑛1+589𝑓1260𝑛+1389𝑓10080𝑛+2+1𝑓252𝑛+3+5029𝑓5040𝑛+22𝑓315𝑛11𝑓2016𝑛2,𝑦𝑛+2=1𝑦𝑛𝑦𝑛1+5933𝑓5040𝑛+1+407𝑓1008𝑛+2149𝑓10080𝑛+3+2063𝑓2520𝑛+1243𝑓10080𝑛141𝑓5040𝑛2,𝑦𝑛+3=1𝑦𝑛𝑦𝑛1+157𝑓252𝑛+1+14059𝑓10080𝑛+2+397𝑓1260𝑛+3+5813𝑓5040𝑛+1𝑓315𝑛1+107𝑓10080𝑛2.(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) becomes𝑦100010001𝑛+1𝑦𝑛+2𝑦𝑛+3=𝑦012023034𝑛2𝑦𝑛1𝑦𝑛+21110024012112011112024029151271120𝑓15𝑛+1𝑓𝑛+2𝑓𝑛+3+2112401097112012047240103160412015161𝑓60𝑛2𝑓𝑛1𝑓𝑛.(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);𝑘𝑗=0𝛼𝑗𝑦(𝑥+𝑗)2𝛽𝑗𝑦,(𝑥+𝑗)(5.1) where 𝑦(𝑥) is an arbitrary function continuously differentiable on the interval [𝑎,𝑏]. The Taylors series expansion about the point 𝑥, Ł[]𝑦(𝑥);=𝑐0𝑦(𝑥)+𝑐1𝑦(𝑥)+𝑐22𝑦(𝑥)++𝑐𝑞𝑞𝑦(𝑞)(𝑥),(5.2) where𝑐0=𝛼0+𝛼1+𝛼2++𝛼𝑘,𝑐1=𝛼1+2𝛼2++𝑘𝛼𝑘,𝑐2=1𝛼2!1+22𝛼2++𝑘2𝛼𝑘𝛽1+𝛽2++𝛽𝑘,𝑐𝑞=1𝛼𝑞!1+2𝑞𝛼2++𝑘𝑞𝛼𝑘1𝛽(𝑞2)!1+2𝑞2𝛽2++𝑘𝑞2𝛽𝑘,𝑞=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+20.(5.4)𝑐𝑝+2 is the error constant and 𝑐𝑝+2𝑝+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 𝑝1.

Since the methods derived in (3.5) and (4.7) are of order 𝑝1, 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 𝜎(𝜉) whose coefficients are 𝛽𝑗, is defined by 𝜌(𝜉)=𝑘𝑗=0𝛼𝑗𝜉𝑗𝜎,(5.5a)(𝜉)=𝑘𝑗=0𝛽𝑗𝜉𝑗,(5.5b) where 𝜉𝐶, 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=0𝐴𝑖𝜉𝑘1]=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 𝑅=2𝜆forthat2𝜆wheretherootsofthestabilitypolynomialareofabsolutelessthanone}.(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 𝑧𝜌𝑒(𝜃)=𝑖𝜃𝜎𝑒𝑖𝜃.(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, 𝑦=𝜆𝑦. By setting 𝑧=𝜆2, the block scheme (3.7) becomes 𝑦100010001𝑛+1𝑦𝑛+2𝑦𝑛+3=𝑧𝑧12162+13𝑧125𝑧427𝑧23+21𝑧411𝑧2315𝑧4+31𝑧2𝑦𝑛2𝑦𝑛1𝑦𝑛.(5.7) This is of the form 𝑌𝑛=(𝐴(1)+𝑧𝐵(1))𝑌𝑛1. The stability polynomial of this is given as𝐴det𝜉(1)+𝑧𝐵(1)=0.(5.8) Hence the stability polynomial of the 3-point EBM (3.7) is𝜉3+𝜉214512𝑧2+𝜉794𝑧2+376+𝑧+1𝑧3+54𝑧237𝑧12=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 𝜉det𝐼𝑧𝐵(0)𝐴(1)+𝑧𝐵(1)=0(5.10) which gives𝜉359𝑧432003+11𝑧360231120𝑧+1+𝜉211957𝑧2160032153𝑧480267980𝑧2+𝜉89𝑧43203+1𝑧152141𝑧+1𝑧2160031𝑧14402+137𝑧240=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 . 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 𝑦=𝜆𝑦,𝑦(0)=𝑦(0)=1,with𝜆=1,0𝑥1.(7.1)

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

Problem 2. The Van der Pol equation which describes the Van der Pol oscillator is the second-order ODE 𝑦=𝜇1𝑦2𝑦𝜆𝑦,𝑦(0)=𝐴,𝑦(0)=𝐵,0𝑥1.(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=104, 106, and 108 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.

Problem 3. The third problem is the second order ODE 𝑦=101𝑦100𝑦,𝑦(0)=1,𝑦(0)=0,0𝑥1,(7.4) with exact solution 1𝑦=99100𝑒𝑥𝑒100𝑥.(7.5)
The results obtained by using the 3-point EBM and IBM are presented in Table 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 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 =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.