DEVELOPMENT OF FALKNER-TYPE METHOD FOR NUMERICAL SOLUTION OF SECOND ORDER INITIAL VALUE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS

ABSTRACT

This research focuses on the formulation of block hybrid methods with power series as basic function through interpolation and collocation techniques for numerical solution of  second  order  initial  value  problems  in  ordinary differential  equations.  The  step number for the derived block hybrid method is k=2 with two off-step point and four off- step points. The basic properties of numerical methods were analyzed and findings revealed that the methods were consistent, zero-stable and convergent which makes them suitable for solving the class of problems considered such as linear and non-linear problems,  oscillatory  problems,  Dynamic  problem  and  Stiff  system.  The  results obtained from the proposed methods, show that the methods are of higher accuracy and have superiority over some existing methods considered in the literature.

CHAPTER ONE

1.0 INTRODUCTION

1.1  Background to the Study

The mathematical formulation of physical phenomena in science and engineering often leads to differential equations, which can be categorized as an ordinary differential equation (ODE) and a partial differential equation (PDE). This formulation will explain the behavior of the phenomenon in detail. The search for solutions of real-world problems requires solving ODEs and thus has been an important aspect of mathematical study. These mathematical models are represented in the form of first order or higher differential equation. The reliability of numerical approximation techniques in solving such problems has been proven by many researchers as the role of numerical methods in engineering problems solving has increased drastically in recent years.

Numerical analysis is the study of algorithms that use numerical approximation for problems of mathematical analysis. The numerical method for solving ordinary differential equations (ODEs) is the most powerful technique ever developed in continuous time dynamics; these are developed since most of the differential equations cannot be solved analytically (Chollum, 2004).

A differential equation, shortly DE is an equation involving a relationship between an unknown function and one or more of its derivatives.

Depending upon the domain of the functions involved we have ordinary differential equations, or shortly ODE, is when the unknown function depends on a single independent variable. Also, if it involves partial derivatives with respect to more, than one independent variable, then the differential equation is called a partial differential equation (PDE).

Our goal is to obtain a numerical solution for a second-order initial value problem

(I.V.P.) of the general form

Although it is possible to integrate a second-order I.V.P. by reducing it to a first order system and applying then one of the methods available for such systems, it seems more natural to provide numerical methods in order to integrate the problem directly.

The advantage of this procedure lies in the fact that they are able to exploit special information about ODEs, resulting in an increase in efficiency.

In order to get more accurate numerical solutions with less effort to solve the second- order differential equation, many scholars including Vigo-Aguiar and Ramos (2016), Mazzia and Nagy (2015), Mazzia et al. (2012), Sommeijer (1993), Brugnano and Trigiante (1998), Butcher and Hojjati (2005), Franco (2002), Mahmoud and Osman (2009), among others, developed different numerical methods to give the approximate solutions to (1.1) and higher-order ODEs without reducing it to a system of first-order ODEs. Some of the numerous numerical approaches presented by the aforementioned researchers included higher derivative multistep methods, Runge-Kutta methods, spline- collocation  methods,  and  Runge-Kutta-Nystrom  methods.  It  is  well-known  that  a Runge-Kutta-Nystrom method for solving (1.1) has a greater improvement as compared to standard Runge-Kutta methods. In case of a linear k-step method for first order ODEs, it becomes a 2k-step method for (1.1), thus increasing the computational work.

The need to improve on the aforementioned methods became important to researchers in this area. Different scholars such as Ramos and Singh (2016), Ramos and Rufai (2018), Ramos and Lorenzo (2010), have developed block methods for solving higher-order ODEs directly.In block methods, the approximations are simultaneously obtained at a number of 2 consecutive grid points in the interval of integration. These methods are less  costly  in  terms  of  number  of  function  evaluations  compared  to  the  reduction method and linear multistep methods.

A merit of the block methods over traditional predictor-corrector ones is that they give better approximate solutions when solving many problems of the form in (1.1) and higher-order ODEs directly.

This research is focused on development of Falkner-type method for numerical solution of second order initial value problems (IVPs) in ordinary differential equations.

1.2  Statement of the Research Problem

Several numerical methods cannot solve the second order problem directly without reducing to lower order equation. Also, these methods have been found to have major setbacks such as large computer storage memories because of too many auxiliary functions, wastage of computer time and a lot of human effort. Awoyemi (1992), the inability of the method to utilize additional information related to the specific ODEs and the lower order of accuracy of the methods used to solve the system of first order after it has been reduced compared to the increased dimension of the original problem.

Predictor-corrector method was also reported to have some major drawbacks due to the number of functions evaluated and the order of accuracy of corrector is higher than the predictor especially when there is need to interpolate and collocate at grid and off-grid point. This major setback of predictor-corrector methods are extensively addressed by Jator (2007). Also to overcome this setback of predictor-corrector methods researchers have proposed the block methods which gives solutions at each grid within the interval of integration without overlapping. They are implemented in a block-by-block fashion in other to reduce the burden of developing predictors (Jator 2007, Jator and Li 2009).

1.3  Aim and Objectives of the Study

The aim of this research work is to develop a Falkner-type method for the solution of the second order initial value problems (IVPs).

The following objectives are to be achieved

i.      To construct a block hybrid method for k=2 for the solution of second order initial value problems.

ii.      To  obtain  the  order  and  error  constant,  zero  stability,  consistency  and convergence of the method.

iii.     To apply the proposed method to solve initial value problems.

iv.      To compare the results with some existing methods found in the literature.

1.4 Justification of the Study

The development of Hybrid Falkner-Type method will enhance, enrich and strengthen the subject of numerical methods for solution of second order ODEs. The proposed methods show that, from a single continuous scheme, multiple finite difference methods can be obtained which would allow the new methods to be self-starting and it is useful in numerical solution at several points without starting values, hence, increases the speed of integration.

1.5  The Scope of the study

Second order initial value problems (IVPs) in ODEs were considered in this thesis.

1.6  Limitation of the Study

The research is limited to the following;

i.      Power  series  polynomial  was  considered  as  basis  function  because  of  its smoothness.

ii.      The research is limited to the formulation of Falkner-type method.

iii.     The research considers Second order initial value problems.

1.7 Falkner Type Method

The Falkner-type method is used to solve differential system of second order ordinary differential equations and its general form is as follows:

1.8  Maple Software

The software package has the ability to algebraically manipulate mathematical expressions and find symbolic solutions to certain problems such as those arising from ordinary and  partial  differential  equations.  In  this  work,  Maple (2015)  is  used  for simulation.

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