UZOKA CHARLES  IFEANYI

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## PROJECT TOPIC AND MATERIAL ON MULTIDERIVATIVE LINEAR MULTISTEP METHODS FOR INITIAL VALUE PROBLEMS (IVPs) IN ORDINARY DIFFERENTIAL EQUATIONS (ODEs).

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## ABSTRACT

In this thesis, we constructed a four-step fourth derivative exponentially fitted integrator of order six and a six-step third derivative exponentially fitted integrator of order eight for the numerical integration of initial value problems in first order ordinary differential equations. The integrators which possess free parameters are based on predictor-corrector mode. The constructed formula of order six and eight are casted into an exponentially fitted formula. The stability analysis of the new methods was examined and the methods were implemented using Fortran program to solve some initial value problems in ordinary differential equations. Finally, the numerical results show that the new methods compete favourably with the existing methods in the literature.

Title page——————————————————————————————ii

Certification—————————————————————————————iii

Dedication—————————————————————————————–iv

Acknowledgement——————————————————————————–v

Table of content———————————————————————————–vi

List of tables————————————————————————————–viii

Abstract——————————————————————————————–ix

Chapter one

Introduction—————————————————————————————–1

• Background to the study——————————————————————1
• Ordinary Differential Equation———————————————————–2
• Initial value problems for first order Ordinary Differential Equation—————3
• Lipschitz Condition————————————————————————3

1.5      Numerical Methods for Solving Initial Value Problems in Ordinary Differential

Equations————————————————————————————4

• Justification———————————————————————————5
• Objective of the study———————————————————————7
• Research method—————————————————————————7

Chapter two

Literature review————————————————————————————9

2.1       Introduction———————————————————————————9

2.2       One-step methods————————————————————————–10

2.3       Multistep methods————————————————————————-12

2.4       Order and Error Constant—————————————————————–13

2.5       Multiderivative methods——————————————————————15

2.6       Explicit and Implicit methods————————————————————16

2.7       Taylor series method———————————————————————-18

2.8       Exponentially fitted multi derivative methods—————————————–18

Chapter three

Development of Integrator————————————————————————21

3.1       Introduction——————————————————————————–21

3.2       Derivation of the four step fourth derivative method of order six——————24

3.3       Derivation of the six step third derivative method of order eight——————-38

Chapter four

Stability analysis and numerical result———————————————————-58

4.1       Stability analysis————————————————————————–58

4.2       Stability analysis of four step fourth derivative method of order six————–59

4.3       Stability analysis of six step third derivative method of order eight—————81

4.4       Numerical examples and results——————————————————–101

Chapter five

Summary and conclusion————————————————————————-106

5.1       Summary———————————————————————————–106

5.2       Findings————————————————————————————106

5.3       Contribution to knowledge————————————————————–107

5.4       Conclusion———————————————————————————107

References——————————————————————————————-108

Appendix——————————————————————————————–111

List of Tables

Table 4.2.1      Table of values for stability function of predictor for four step fourth derivative method of order six——————————————————61

Table 4.2.2      Table of values for stability function of corrector for four step fourth derivative method of order six——————————————————63

Table 4.2.3      Table of values for free parameters for four step fourth derivative of order six—78

Table 4.2.4      Table of values for predictor-corrector integrator of order six—————–81

Table 4.3.1      Table of values for stability function of predictor for six step third derivative method of order eight—————————————————————-84

Table 4.3.2      Table of values for stability function of corrector for six step third derivative method of order eight—————————————————————-87

Table 4.3.3      Table of values for free parameters for six step third derivative method of order eight—————————————————————————–98

Table 4.3.4      Table of values for stability function of six step third derivative predictor-corrector integrator of order eight————————————————-101

Table (4.4.1a) Comparative analysis of result of problem 1 for four-step fourth derivative method of order six at x=1———————————————————-102

Table (4.4.1b) Comparative analysis of result on problem 1 for six-step third derivative method of order eight at x=1——————————————————————-103

Table (4.4.2a) Numerical result on problem 2 for four-step fourth derivative method of order six, at ———————————————————————————104

Table (4.4.2b) Numerical result on problem 2 for six-step third derivative method of order eight, at ———————————————————————————105

## CHAPTER ONE

INTRODUCTION

1.1       Background to the study

A differential equation is an equation involving a relation between an unknown function and one or more of its derivatives. Differential equations are among the most important mathematical tools used in producing models in the physical sciences, biological sciences, social and management sciences and engineering. They occur in connection with the mathematical description of problems that are encountered in various branches of science. Consequently, it constitutes a large and very important aspect of today’s mathematics.

Differential equation is a process by which solutions can be sort to some real life problems. These problems can either be solved by the use of analytical techniques or by numerical methods. Since most ordinary differential equations are not analytically solvable, numerical methods are often better option. Many methods have been proposed and used by different authors with the aim of providing accurate solutions to the various types of differential equations. Differential equation is divided into two parts, ordinary differential equation and partial differential equations; here our work is centred on proposing a technique that can solve problems in ordinary differential equations, although many of such methods already exist. Our focus here is on numerical solutions to ordinary differential equations with particular emphasis on the use of linear multistep methods.

Stiff differential systems including the building energy simulation problems, are difficult and costly to compute. Standard explicit solvers are compact, and time stepping with them is cheap, but many active increments are required. Implicit solvers offer stability for any time increment at the cost of a lot of computation per step. What is needed is a method that can take a long time cheaply. Exponential fitting methods offer this option. Abhulimen (2006).

The rational behind the development of this kind of numerical integrator is that exponentially fitted formulae possess a large region of absolute stability when compared to conventional ones, Hochbruck, Lubich, Selhfer (1998).

In the last decades, several authors such as Enright (1974), Enright and Pryce (1983), Brown (1977), Cash (1981), Jackson and Kenue (1974) Voss (1988), Okunuga (1994), Abhulimen and Okunuga (2008), and Abhulimen and Omeike (2011) developed second derivative integrators for the numerical solutions of stiff differential equations. These integrators however were found to be A-stable, particularly for stiff problems whose solutions have exponential functions.

1.2       Ordinary differential equations (ODEs):

Many problems in science and other areas involving rate of change usually resolve into ODEs. The most general form which ODEs may assume is giving by;

1.2.1)

where  is the independent variable,  is the dependent variable,

So that

more compactly we represent (1.2.1) in vector form as;

where  so that  and  denotes transpose.

1.3       Initial Value Problems (IVPs) for First Order Ordinary Differential Equation

The first order differential equation  may possess an infinite number of solutions. For example the function  is, for any value of the constant , a solution of the differential equation , where , is a given constant. We can pick out any particular solution by prescribing an initial value condition, . For the above example, the particular solution satisfying this initial condition is easily found to be  we say that the differential equation together with an initial condition constitutes an initial value problem,

1.4       Lipschitz Condition

Theorem 1.1: let  be a real function and continuous for all points  in the region  defined by , containing initial values  where  are finite. Let there exist a constant  such that for any  and for any pairs  for which  are both in

then for any giving number . The initial value problem (1.3.1) has a unique solution .   is called the Lipschitz constant. This condition maybe thought of as being intermediate between differentiability and continuity, such that:

is continuously differentiable w.r.t.  for all  in .

satisfies a lipschitz condition w.r.t.  for all  in .

is continuous w.r.t.  for all  in .

In particular, if  possesses a continuous derivative with w.r.t.  for all  in , then by mean value theorem,

where  is a point in the interior of interval whose end-point are  and  and  and  are both in . Then the lipschitz constant  of the system may be taken to be

(1.4.3)

1.5      Numerical Methods for Solving Initial Value Problems in Ordinary Differential Equations

Numerical methods are methods used for solving Ordinary Differential Equations. According to Shepley (1989), numerical methods are employed in the solution of the differential equation  with the initial-condition  to obtain approximate solution at various selected values of x with the aim of having exact solution. To do this we set  as the exact solution of the problem, and let  denote a small positive increment in x. Let  and consider

( . A numerical method will use the differential equation and the condition to successively approximate these exact values ( . Let  be the approximations to  respectively, so that finding  and finding an approximation to  mean the same thing. In finding the approximation   we proceed in the following way:

First, we find  using the method of interest to solve the differential equation   with the initial value . Then  is estimated using the estimate ,  is estimated using the estimate , and so on, so that in general,  is estimated using the estimate . A method which proceeds in this manner is called a one-step method. On the other hand, in finding   some methods actually use several of the preceding approximations   to estimate the differential equation   with the given initial condition . Such methods cannot find   from with the initial condition . Hence such methods are called multi-step method. To use a multi-step method, the first few   must be found by a starting method, until a sufficient number of them are on hand to begin using the continuing method. Most of our attention in this sense will be devoted to starting methods. Shepley continues by saying that given an approximation  to , the absolute error, or simply error is defined as ; the error measures how far away the approximation  is from the exact value . Naturally we hope that any given numerical method will keep the error small, that is, the method should have some level of accuracy.

1.5       JUSTIFICATION

Numerical method is a part of numerical analysis which studies the methods for finding numerical approximations to the solution of ordinary differential equations. Some of the existing methods for solving IVPs are the one-step methods e.g Runge-kutta, milne method, the linear multistep methods, the hybrid methods.

Of all numerical methods for the numerical solution of this initial value problem the easiest to implement is the Euler’s rule. Lambert (1973).

According to Sheply (1989), numerical methods are employed in the solution of the differential equation  with the initial condition  to obtain approximate solution at various selected values of .

A good and potential numerical method for the solution of initial value problems in ordinary differential equations must possess good accuracy and some reasonable wide region of absolute stability, Dahlquist (1973).

Stability is the property of a numerical method to keep the errors bounded as the computation advances Abhulimen (2009). A-stability is one of the important stability requirements for a linear multistep method, Enright (1974). Dahlquist proved that the order of an A-stable linear multistep method is  which linear multistep method is limited by the requirement of A-stability. This has made researchers to find other classes of numerical methods for higher order to solve differential equations.

Liniger and Willoughby (1970) developed the concept of exponential fittings in the course of higher order A-stable numerical methods, which allows free parameters that are chosen to fit some exponential integration functions that satisfies integration formula exactly. Recently exponential integrators have become an active area of research.

Before designing our integrators we considered many methods and were motivated by the striking proposals made by the following authors:

Jackson and Kenue (1974), Cash (1981), Okunuga (1992), Hochbuck, Lubish et al (1998), Abhulimen and Otunta (2009), Okunuga (1999), Abhulimen and Otunta (2006), Vigo and Martin (2006, 2007), Abhulimen (2006, 2014), Abhulimen and Okunuga (2008).

Most of these developed formulae allow free parameters and act as a motivation in deriving a 4-step 4th derivative method of order 6 and 6-step 3rd derivative method of order 8 with one free parameter.

It is important to note that for systems for which exponential fitting is appropriate, it is usually found that exponential fitted integration formulas are substantially more efficient than conventional ones. Abhulimen and Omeike (2011). The exponential fitting method also offers favourably properties in the integration of differential equations whose Jacobian has large imaginary eigenvalues (Hochbruck et al.1998).

1.7       OBJECTIVES OF THE STUDY

The overall aim of the study is to derive a four-step fourth derivative exponentially fitted integrator of order six and a six-step third derivative exponentially fitted integrator of order eight which are A-stable for all choices of the fitting parameters.

The specific objectives of the study are as follows:

1. To derive exponentially fitted integrators with one free parameter.
2. To investigate the stability properties, whether they are A-stable of higher order.
3. To implement the integrator using Fortran package (plato 4.51) and compare with existing ones.

1.8       RESEARCH METHOD

In the spirit of Cash (1981) and Abhulimen (2014), the general form taken in the derivation of our new integrator is given as:

where equation (1.7) and (1.8) are use as predictor and corrector respectively, and  are positive integers,  is a step length. The coefficients  and  are real constants.  are approximate to , ,

When deriving exponentially fitted methods, the approach is to allow both (1.8.1) and (1.8.2) to possess free parameters which allow it to be fitted automatically to exponential functions. Abhulimen and Omeike (2011).

1. Removing the arbitrary constant by assuming throughout that and using Taylor’s series expansion to develop and generate parameters for 4 and 6 step fourth and third derivative integrator of order 6 and 8 respectively.
2. Fitting the derived formula exponentially by introducing a scalar test equation, in which the values of the free parameters are solved for. Then using the procedure of Liniger (1970) as introduced by Abhulimen (2011, 2014).
3. Investigating and analyzing the stability of our derived methods (formula), to ascertain if it satisfies the A-stable condition. In doing these, the range of values of the free parameters in the open half plane will be required.
4. Experimenting the derived formulae by using appropriate FORTRAN algorithm (plato 4.51) to solve some IVPs.
5. Comparing the results of our formulae with that of some existing formulae with the aim of ascertaining the level of accuracy.INTRODUCTION

1.1       Background to the study

A differential equation is an equation involving a relation between an unknown function and one or more of its derivatives. Differential equations are among the most important mathematical tools used in producing models in the physical sciences, biological sciences, social and management sciences and engineering. They occur in connection with the mathematical description of problems that are encountered in various branches of science. Consequently, it constitutes a large and very important aspect of today’s mathematics.

Differential equation is a process by which solutions can be sort to some real life problems. These problems can either be solved by the use of analytical techniques or by numerical methods. Since most ordinary differential equations are not analytically solvable, numerical methods are often better option. Many methods have been proposed and used by different authors with the aim of providing accurate solutions to the various types of differential equations. Differential equation is divided into two parts, ordinary differential equation and partial differential equations; here our work is centred on proposing a technique that can solve problems in ordinary differential equations, although many of such methods already exist. Our focus here is on numerical solutions to ordinary differential equations with particular emphasis on the use of linear multistep methods.

Stiff differential systems including the building energy simulation problems, are difficult and costly to compute. Standard explicit solvers are compact, and time stepping with them is cheap, but many active increments are required. Implicit solvers offer stability for any time increment at the cost of a lot of computation per step. What is needed is a method that can take a long time cheaply. Exponential fitting methods offer this option. Abhulimen (2006).

The rational behind the development of this kind of numerical integrator is that exponentially fitted formulae possess a large region of absolute stability when compared to conventional ones, Hochbruck, Lubich, Selhfer (1998).

In the last decades, several authors such as Enright (1974), Enright and Pryce (1983), Brown (1977), Cash (1981), Jackson and Kenue (1974) Voss (1988), Okunuga (1994), Abhulimen and Okunuga (2008), and Abhulimen and Omeike (2011) developed second derivative integrators for the numerical solutions of stiff differential equations. These integrators however were found to be A-stable, particularly for stiff problems whose solutions have exponential functions.

1.2       Ordinary differential equations (ODEs):

Many problems in science and other areas involving rate of change usually resolve into ODEs. The most general form which ODEs may assume is giving by;

1.2.1)

where  is the independent variable,  is the dependent variable,

So that

more compactly we represent (1.2.1) in vector form as;

where  so that  and  denotes transpose.

1.3       Initial Value Problems (IVPs) for First Order Ordinary Differential Equation

The first order differential equation  may possess an infinite number of solutions. For example the function  is, for any value of the constant , a solution of the differential equation , where , is a given constant. We can pick out any particular solution by prescribing an initial value condition, . For the above example, the particular solution satisfying this initial condition is easily found to be  we say that the differential equation together with an initial condition constitutes an initial value problem,

1.4       Lipschitz Condition

Theorem 1.1: let  be a real function and continuous for all points  in the region  defined by , containing initial values  where  are finite. Let there exist a constant  such that for any  and for any pairs  for which  are both in

then for any giving number . The initial value problem (1.3.1) has a unique solution .   is called the Lipschitz constant. This condition maybe thought of as being intermediate between differentiability and continuity, such that:

is continuously differentiable w.r.t.  for all  in .

satisfies a lipschitz condition w.r.t.  for all  in .

is continuous w.r.t.  for all  in .

In particular, if  possesses a continuous derivative with w.r.t.  for all  in , then by mean value theorem,

where  is a point in the interior of interval whose end-point are  and  and  and  are both in . Then the lipschitz constant  of the system may be taken to be

(1.4.3)

1.5      Numerical Methods for Solving Initial Value Problems in Ordinary Differential Equations

Numerical methods are methods used for solving Ordinary Differential Equations. According to Shepley (1989), numerical methods are employed in the solution of the differential equation  with the initial-condition  to obtain approximate solution at various selected values of x with the aim of having exact solution. To do this we set  as the exact solution of the problem, and let  denote a small positive increment in x. Let  and consider

( . A numerical method will use the differential equation and the condition to successively approximate these exact values ( . Let  be the approximations to  respectively, so that finding  and finding an approximation to  mean the same thing. In finding the approximation   we proceed in the following way:

First, we find  using the method of interest to solve the differential equation   with the initial value . Then  is estimated using the estimate ,  is estimated using the estimate , and so on, so that in general,  is estimated using the estimate . A method which proceeds in this manner is called a one-step method. On the other hand, in finding   some methods actually use several of the preceding approximations   to estimate the differential equation   with the given initial condition . Such methods cannot find   from with the initial condition . Hence such methods are called multi-step method. To use a multi-step method, the first few   must be found by a starting method, until a sufficient number of them are on hand to begin using the continuing method. Most of our attention in this sense will be devoted to starting methods. Shepley continues by saying that given an approximation  to , the absolute error, or simply error is defined as ; the error measures how far away the approximation  is from the exact value . Naturally we hope that any given numerical method will keep the error small, that is, the method should have some level of accuracy.

1.5       JUSTIFICATION

Numerical method is a part of numerical analysis which studies the methods for finding numerical approximations to the solution of ordinary differential equations. Some of the existing methods for solving IVPs are the one-step methods e.g Runge-kutta, milne method, the linear multistep methods, the hybrid methods.

Of all numerical methods for the numerical solution of this initial value problem the easiest to implement is the Euler’s rule. Lambert (1973).

According to Sheply (1989), numerical methods are employed in the solution of the differential equation  with the initial condition  to obtain approximate solution at various selected values of .

A good and potential numerical method for the solution of initial value problems in ordinary differential equations must possess good accuracy and some reasonable wide region of absolute stability, Dahlquist (1973).

Stability is the property of a numerical method to keep the errors bounded as the computation advances Abhulimen (2009). A-stability is one of the important stability requirements for a linear multistep method, Enright (1974). Dahlquist proved that the order of an A-stable linear multistep method is  which linear multistep method is limited by the requirement of A-stability. This has made researchers to find other classes of numerical methods for higher order to solve differential equations.

Liniger and Willoughby (1970) developed the concept of exponential fittings in the course of higher order A-stable numerical methods, which allows free parameters that are chosen to fit some exponential integration functions that satisfies integration formula exactly. Recently exponential integrators have become an active area of research.

Before designing our integrators we considered many methods and were motivated by the striking proposals made by the following authors:

Jackson and Kenue (1974), Cash (1981), Okunuga (1992), Hochbuck, Lubish et al (1998), Abhulimen and Otunta (2009), Okunuga (1999), Abhulimen and Otunta (2006), Vigo and Martin (2006, 2007), Abhulimen (2006, 2014), Abhulimen and Okunuga (2008).

Most of these developed formulae allow free parameters and act as a motivation in deriving a 4-step 4th derivative method of order 6 and 6-step 3rd derivative method of order 8 with one free parameter.

It is important to note that for systems for which exponential fitting is appropriate, it is usually found that exponential fitted integration formulas are substantially more efficient than conventional ones. Abhulimen and Omeike (2011). The exponential fitting method also offers favourably properties in the integration of differential equations whose Jacobian has large imaginary eigenvalues (Hochbruck et al.1998).

1.7       OBJECTIVES OF THE STUDY

The overall aim of the study is to derive a four-step fourth derivative exponentially fitted integrator of order six and a six-step third derivative exponentially fitted integrator of order eight which are A-stable for all choices of the fitting parameters.

The specific objectives of the study are as follows:

1. To derive exponentially fitted integrators with one free parameter.
2. To investigate the stability properties, whether they are A-stable of higher order.
3. To implement the integrator using Fortran package (plato 4.51) and compare with existing ones.

1.8       RESEARCH METHOD

In the spirit of Cash (1981) and Abhulimen (2014), the general form taken in the derivation of our new integrator is given as:

where equation (1.7) and (1.8) are use as predictor and corrector respectively, and  are positive integers,  is a step length. The coefficients  and  are real constants.  are approximate to , ,

When deriving exponentially fitted methods, the approach is to allow both (1.8.1) and (1.8.2) to possess free parameters which allow it to be fitted automatically to exponential functions. Abhulimen and Omeike (2011).

1. Removing the arbitrary constant by assuming throughout that and using Taylor’s series expansion to develop and generate parameters for 4 and 6 step fourth and third derivative integrator of order 6 and 8 respectively.
2. Fitting the derived formula exponentially by introducing a scalar test equation, in which the values of the free parameters are solved for. Then using the procedure of Liniger (1970) as introduced by Abhulimen (2011, 2014).
3. Investigating and analyzing the stability of our derived methods (formula), to ascertain if it satisfies the A-stable condition. In doing these, the range of values of the free parameters in the open half plane will be required.
4. Experimenting the derived formulae by using appropriate FORTRAN algorithm (plato 4.51) to solve some IVPs.
5. Comparing the results of our formulae with that of some existing formulae with the aim of ascertaining the level of accuracy.

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