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Title Page                                                                                                        ii

Certification                                                                                                    iii

Dedication                                                                                                      iv

Acknowledgment                                                                                                       v

Table of Content                                                                                                         vi

Abstract                                                                                                                      ix



1.1 Introduction                                                                                                          1

1.2 Differential Equation                                                                                            2

1.2.1 Types of Differential Equation                                                              3

1.2.2 Ordinary Differential Equations                                                                        3

1.2.3 Partial Differential Equation                                                                              3

1.3 Initial Value Problems                                                                              4

1.4 Objective of the Study                                                                                         4

1.5 Method of Research                                                                                             4




  • Introduction 6
  • Existing Numerical Methods 7

2.2a Definitions Associated with one step Methods                                                  7

2.3 Runge Kutta Methods                                                                                          9

2.3a Classes of Runge Kutta Methods                                                                       9

2.4 Linear Multistep Method                                                                          10

2.4b Linear Multistep family                                                                          12

2.4c Adams Methods                                                                                      12

2.4d Backward Differentiation Formula (BDF)                                                         13

2.5 The Hybrid Method                                                                                              14

2.6a Parallel Block Method                                                                            17

2.6b Parallel Block Implicit Method                                                               19

2.6c Parallel Block Predictor- Corrector (PBPC) Method                                          21

2.6d Parallel Block Methods For Stiff Equation                                                        23

2.6e Block Runge Kutta Method                                                                               25



Development of the Method

3.1 Introduction                                                                                                          27




4.0 Preamble                                                                                                   51

4.1 The Basic Properties of the Method                                                                     51

4.5 Consistency                                                                                                          52

4.6 Convergence:                                                                                                        52

4.7 Zero Stability                                                                                                        53

4.8 Region of Absolute Stability                                                                                57



Implementation and Numerical Results

5.0 Introduction                                                                      59

5.1 Algorithm                                                                                   59

5.2 Numerical Experiment and Results                                             59


A self starting six step ten order block method with three off-grids points have been derived for solving Ordinary Differential Equations (ODE) using interpolation and collocation procedures. Multiple numerical integrators were arrived at. These integrators are combined into a single block matrix equations. Our method was found to be zero stable, consistent  and convergent. The numerical examples considered showed that our method gave better accuracy than some existing methods. The stability analysis shows that our method is A-stable and of order 10.




The development of numerical integration formula for stiff problems has attracted great attention in the past. It is also important to note that mathematical model of physical situation, in kinetic chemical controls, and electric theory often leads to stiff Ordinary Differential Equations.

Stiffness in common terms refers to a difficult and severe situation than usual. One of the basic problems that are in the solution of stiff problem is that of numerical stability.

In this thesis, emphasis will be on the solution of initial value problem of Ordinary Differential Equation.


We seek a solution on the range , where a and b are finite, and we assume that f satisfies the continuous which guarantee that the problem have a unique continuous differentiable solution which we shall denote by

Various researchers have discussed and published literature in the solution of ordinary differential equation of (1.1) amongst other are Lie and Norsett (1980), Serisena (1989). These researchers have developed various numerical methods to solve such initial value problems.

In the course of this work, we shall examine various existing methods which will act as motivation in developing our scheme.


1.2 Differential Equation

Differential equation is a mathematical equation which relates a function with it derivatives. The function usually represents physical quantities, while the derivatives represent their rate of change and the equation defines a relationship between the two. Because such relationships are extremely common, differential equation plays a pivotal role in many disciplines including Engineering.

In Mathematics, differential equations are studied from several directions mostly concerned with their solution, the set of functions that satisfy the equation.

Differential equation first came into existence with the invention of Calculus by Newton and the three kinds of differential equation according to Newton are


He solved these given equations and others using infinite series and discus the non uniqueness of solution.

Other examples of differential equation are Bernouli (1965), differential equations, which have the form , Euler-Lagrange equation (1750), which he uses in the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.




Differential Equations can be divided into various types. Apart from the description of the properties of the equation itself. These classes of differential equations can help inform the choice of approach to solutions.


This is an equation containing a function of one independent variable and its derivatives. Linear differential equation can be added and multiplied by coefficients Ordinary Differential Equations that lack additive solution are known as non linear, and solving them is more intricate.

Ordinary Differential Equations can be used to describe a wide variety of phenomenon such as sound, heat, electrostatics e.t.c.


A Partial Differential Equation is a differential equation that contains multi-derivative function and their partial derivatives. They are used to formulate problem involving function of several variable, and are either solved by hand or used to create a relevant computer model.


Initial value problem is an Ordinary Differential Equation together with a specified value called the initial condition of the unknown function at a given point in the domain of the solution.




A linear multistep method is a computational methods for determining the numerical solution of initial value problems of ordinary differential equations which form a linear relation between .

The general formula is given as


Where  is the numerical solution of the initial value problems


The overall aim of the study is to modify polynomial basis function

The specific objectives are as follows:

  1. To derive an order ten continuous hybrid block method.
  2. To investigate the stability properties, characteristics, consistency and the nature of convergence of the integrator constructed.
  • To implement the integrators and compare it with existing methods.
  1. Apply MAPLE programming for the implementation of the integrator and compare their performance with some established results.



In carrying out our research satisfactorily, we intend to:

  1. Adopt and modify polynomial basis function of the form
  2. Use interpolation and collocation procedure to choose interpolating point S
  • Determine coefficients by taking the derivative of , given as and substituting into .
  1. Collocate at various point where
  2. Investigating and analyse the stability of our derived method to ascertain if it satisfies the A-stable condition.
  3. Experimenting the derived formulas by using aropriate MAPLE algorithm to solve some IVPs.
  • Compare result of our formulas with that of some existing formulas with the aim of ascertaining level of accuracy.




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