ONE-STEP SECOND DERIVATIVE BLOCK INTRA-STEP POINTS FOR STIFF SYSTEMS OF INITIAL VALUE PROBLEMS

ABSTRACT

Modified class of single-step numerical schemes were proposed to improve the order of accuracy by imposing intra-step points in the formulation process of the proposed algorithms. The behaviour of modified numerical algorithm is of great concern when varieties  of  countable  off-grid  points  are  imposed  within  the  grid  points  in  the derivation process. In this study, we present a one-step second derivative modified algorithms for numerical solution of first order initial value problems of ODEs. The consistency, convergence and order of accuracy of the algorithms are improved by interpolating and collocating the power series polynomials at carefully selected intra- points generated by the Bhaskara cosine formula. The proposed methods are self- starting  and  applied  as  simultaneous  numerical  integrators  on  non-overlapping intervals. In order to further illustrate the effectiveness of the proposed algorithms, stiff systems of IVPs are considered and results obtained are compared with those from related schemes and other methods in the literature.

CHAPTER ONE

1.0       INTRODUCTION

1.1       Background to the Study

Numerical solutions for ordinary differential equations (ODEs) are very important in scientific computation,  as  they are widely used  for solution  of real  life problems. Analytic methods have been used in some literature to solve mathematical problems; but  some  of  the  problems  that  occur  in  real  life  cannot  be  solved  using  analytic methods. The importance of numerical methods is to find approximate solution to problems. This is usually achieved with steps, with each step improving the accuracy of the later, until enough accuracy of approximation is obtained. It is important to note that numerical methods cannot give an exact solution, therefore the errors involved are of great concern of study.

Many applications are modeled by systems of ordinary differential equations; these systems  exhibit  a  behaviour  known  as  stiffness.  Stiff  systems  are  considered challenging because explicit numerical methods designed for non-stiff problems are used with very small step sizes or do not converge at all. The knowledge of stiffness, occurring in differential equations came as a result of some pioneering works done by the two chemists, Curtiss and Hirschfelder (1952). Shampine and Gear (1971) expounded the characteristics of numerical methods used for solving problems with stiffness and discussed the different realistic goals when solving stiff problems which involve methods with strong stability properties for solving stiff systems. Models associated to these problems are first order system of ordinary differential equation of the form described in Burden and Faires (2011) as y1 y2 yn  f1  x, yx1, y2 ,…, yn f1  x, y1, y2 ,…, yn f1  x, y1, y2 ,…, yn (1.1) with  yn xn yn  , x x0 , xn0 0  0   where  f  satisfies the Lipschitz condition as given in Henrici (1962).

The common methods used to solve ODEs are categorized as single-step (multistage) methods such as Runge-Kutta methods and multistep (one-stage) methods such as Adams-Bashforth-Moulton methods (Avdelas and Simos, 1996). Stroller and Morrison (1958), Butcher (1965), Fang (2001), Watts and Shampine (1972) have fully studied implicit one-step methods.

However, problems frequently arise in which the magnitude of the derivative increases but the solution does not. In this situation, the error can grow so large that it dominates the calculations. Initial-value problems for which this is likely to occur are called stiff equations  and  are  quite  common,  particularly in  the study of vibrations,  chemical reactions, and electrical circuits. This class of differential equations usually have a term of the form ect in their exact solutions, where 𝑐 is a large positive constant for which explicit methods are unsuitable (Baraff, 1997). For this class of differential equations, implicit methods are recommended. In this research work, a class of block hybrid second derivative implicit single-step methods for solving ODEs will be constructed through  interpolation  and  collocation  techniques  –  Gragg and  Stetter (1964), Gear (1965), Kohfeld and Thompson (1967), Shampine and Watts (1969), Gladwell and Sayers (1976), Gupta (1978), Lambert (1991), Onumanyi et al. (1994), Akinfenwa et al. (2011), Mehdizadeh et al. (2012), Sahil et al. (2012) and Yakubu et al. (2017). The continuous representation generates a main discrete one-step second derivative block intra-step method (OSDBM) and additional methods which are combined and used as a block method to simultaneously produce approximations yn , yn1 at a block points xn , xn1,  h  xn1  xn , n 0,…, N1, on a partition a,b, where a,b is the interval of integration,  0,1 are the intra-step points, h is the constant step-size, n is a grid index and N  0 is the number of steps.

The method preserves the Rung-kutta traditional advantage of being self-starting and is more accurate since it is implemented as a block method. We note that block methods were first introduced by Milne (1953) for the purpose of obtaining starting values for predictor-corrector algorithms. However, Rosser (1967), developed Milne‟s idea into algorithms for general use. We emphasis that the continuous representation of our method is developed for general use, not only as a means of obtaining starting values for predictor-corrector algorithms; it generates a main discrete scheme and additional methods   which   are   combined   and   implemented   as   a   block   method   which simultaneously     generates     approximations     yn , yn j y xn , y  xn1 to     exact     solutions

1.2       Statement of the Research Problem

There exist certain classes of ordinary differential equations to which some numerical methods are not applicable. One of such classes is stiff system of ordinary differential equations   which   defies   explicit   methods   owing  to   its   property  of   containing components with different time scale which requires A-stability methods to be used. This places serious restriction on the choice of step-length to be used when applying some numerical methods in solving such problems. Explicit methods are said to be incapable for solving a stiff problem and as such, implicit methods (both single-step and multi-step) have been adopted over the years (Omar and Kuboye, 2015; Ndanusa and Tafida, 2016), most of which are used as predictor-corrector methods, requiring starting values. Implicit methods with large region of absolute stability are recommended and Improved Euler is one of such methods.

Several single-step methods and its modification have been formulated which have been very successful, but efforts are still being made towards having one which is of higher order.

In view of the above, some modified single-step methods incorporating second derivative with carefully selected off-step points are proposed, in which higher order is sought.

1.3       Aim and Objectives of the Study

This study aims at constructing a continuous formulation of one-step second derivative block  intra-step  methods  for  Stiff  System  of  ordinary  differential  equations.  The specific objectives are to;

1.   derive families of one-step method by incorporating in the derivation process, off- step points generated from the Bhaskara cosine approximation formula;

2.   compute order and error constant of the methods;

3.   establish the convergence and stability analysis of the methods;

4.   implement the methods for the solution of stiff systems of initial value problems of ODEs;

5.   establish the efficacy of the methods by comparing the results of the proposed methods with some existing methods found in the literatures.

1.4       Significance of the Study

This  research  shall  contribute  to  learning  in  the  aspect  of  numerical  methods  by deriving a class of one-step numerical schemes that are effective and suitable for the solution of stiff systems of ordinary differential equations with the believe that the proposed   methods   will   fulfill   some   required   states   of   numerical   conditions (convergence and stability).

1.5       Scope and Limitation of the Study

This research work focuses on the derivation and implementation of a class of one-step numerical schemes for the approximate solution of first order Stiff systems of initial value problems.  Higher order ODEs are not considered in this project.

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