ABSTRACT The aim of this research is to investigate numerically the problem on solving Delay Differential Equations (DDEs) using Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method. This research also discussed the problem on solving Ordinary Differential Equations (ODEs) using the same method. In this research, we also developed the algorithm of Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method and the algorithm of Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method incorporated with Hermite Interpolation. Finally, we apply this method to a real life problem and we choose Food-Limited Model. In this research, we use Mathematica 7 software and Microsoft Excel to conduct the calculation.
ABSTRAK Matlamat utama kajian ini ialah untuk mengkaji secara keadah berangka masalah yang melibatkan Persamaan Pembeza Lengah dengan menggunakan Kaedah Pseudo-Runge Kutta tahap kedua peringkat keempat. Kajian ini turut membincangkan penyelesaian kepada Persamaan Pembeza Biasa dengan menggunakan kaedah yang sama. Di dalam kajian ini, algoritma berkaitan dengan Kaedah Nakashima Pseudo-Runge Kutta tahap kedua peringkat keempat dan algoritma bagi kaedah ini dengan menyertakan interpolasi Hermite turut dirumuskan. Akhir sekali, kaedah ini diaplikasikan di dalam masalah kehidupan seharian dan kami memilih Model Had Makanan. Di dalam kajian ini, perisian Mathematica 7 dan Microsoft Excel digunakan dalam membuat pengiraan.
TABLES OF CONTENTS CHAPTER TITLE PAGE STUDENT S DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF ABBREVIATIONS LIST OF SYMBOLS LIST OF APPENDICES ii iii iv v vi vii x xii xiii xiv xv 1 INTRODUCTION 1 1.1 Background of the problem 1 1.2 Statement of the problem 2 1.3 Objectives of the study 3 1.4 Scope of the study 3 1.5 Significant of the study 4
1.6 Organization of the study 4 2 LITERATURE REVIEW 6 2.1 Pseudo-Runge-Kutta (PRK) Methods 6 2.2 Delay Differential Equations (DDEs) 8 3 SOLVING ORDINARY DIFFERENTIAL 14 EQUATIONS USING PSEUDO-RUNGE- KUTTA METHOD 3.1 Introduction 14 3.2 Ordinary Differential Equations (ODEs) 16 3.3 Nakashima s 2 Stages 4 th Order Pseudo- 17 Runge-Kutta Method 3.3.1 Conclusion for Example 3.1 and 35 Example 3.2 3.3.2 Conclusion for Example 3.3 and 51 Example 3.4 4 SOLVING DELAY DIFFERENTIAL 54 EQUATIONS USING PSEUDO-RUNGE- KUTTA METHOD 4.1 Introduction 54 4.2 Delay Differential Equations (DDEs) 55 4.3 Hermite Interpolation 56 4.4 Nakashima s 2 Stages 4 th Order Pseudo- 58 Runge-Kutta Method incorporates with Hermite Interpolation 4.4.1 Conclusion for Example 4.1 and 88 Example 4.2
5 APPLICATION 91 5.1 Introduction 91 5.2 Food-Limited Population Model 92 5.3 Discussion 99 6 CONCLUSION AND SUGGESTIONS 101 6.1 Conclusion 101 6.2 Suggestions 103 REFERENCES 104 APPENDICES 107
CHAPTER 1 INTRODUCTION 1.1 Background of the Problem In Numerical Analysis, the Runge-Kutta (RK) Methods are an important family of implicit and explicit iterative methods for the approximation of the solutions of Ordinary Differential Equations (ODEs). Runge-Kutta (RK) Methods were developed by German Mathematicians C. Runge and M.W. Kutta around 1900. Pseudo-Runge-Kutta (PRK) Method is also one of the Runge-Kutta (RK) Methods family. In this research, 2 Stages 4 th Order Pseudo-Runge-Kutta Method, which is a type of Pseudo-Runge-Kutta (PRK) Methods will be discussed. According to F. Constabile, Pseudo-Runge-Kutta (PRK) Methods require fewer evaluations than Runge- Kutta (RK) Method of the same order. Fourth Order Runge-Kutta (RK4) Method will be used to solve Initial Value Problems (IVPs) in order to get the first term.
The research begins on viewing the history of Pseudo-Runge-Kutta (PRK) Method and Delay Differential Equations (DDEs). Then, we will rederive Pseudo- Runge-Kutta (PRK) Method according to Nakashima (1982) using Wolfram Mathematica 7. We will also discuss on some examples of Ordinary Differential Equations (ODEs) and Delay Differential Equations (DDEs) using Pseudo-Runge-Kutta (PRK) Method. Finally, this method will be applied to solve real life problem. the problem on Food-Limited Model will be discussed in this research. 1.2 Statement of the Problem We wishes to study the paper by Nakashima (1982) with an attempt to rederive Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method and used to solve Ordinary Differential Equations (ODEs). Besides that, this research will also derive Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method incorporates with Hermite Interpolation and used to solve Delay Differential Equations (DDEs). In this research, the Wolfram Mathematica 7 and Microsoft Office Excel 2007 will be used in order to derive and to solve the problems.
1.3 Objectives of the Study The main objectives of this research are: a. To rederive Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method according to Nakashima (1982). b. To derive Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method incorporate with Hermite Interpolation. c. To solve the problems of Ordinary Differential Equations (ODEs) and Delay Differential Equations (DDEs) using Nakashima s 2 Stages 4 th Order Pseudo- Runge-Kutta Method and analyzed it. d. To apply Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method on solving the real life problem of Delay Differential Equations (DDEs) and analyzed it. 1.4 Scope of the Study This research is focused on Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method to solve Ordinary Differential Equations (ODEs) and this method is incorporated with Hermite Interpolation to solve Delay Differential Equations (DDEs). The scope is limited to single delay of Delay Differential Equations (DDEs).
1.5 Significant of the Study This research will give the alternative method to solve Ordinary Differential Equations (ODEs) and Delay Differential Equations (DDEs). This research will leads to further study on the Pseudo-Runge-Kutta Method with different order and on applying Pseudo-Runge-Kutta method to solve the related problem to Delay Differential Equations (DDEs). Besides that, the research will also enhance understanding on Delay Differential Equations (DDEs). 1.6 Organization of the Study In chapter one, we discuss the background of the study, statement of the problem and objectives for this study. Besides that, we also state the scope of the study and the significant of the study. Lastly, we will also discuss on the organization of the study. In chapter two, some literature review on Pseudo-Runge-Kutta Method will be discussed. In this chapter, the literature review of Delay Differential Equations (DDEs) on the methods that been used to solve Delay Differential Equations (DDEs) will be discussed. In chapter three, we will discuss on Pseudo-Runge-Kutta Method in order to solve Ordinary Differential equations (ODEs). In this chapter, the research will rederive Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method. Then some example of
using Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method on solving Ordinary Differential Equationa (ODEs). In Chapter four, we will focus on Nakashima s 2 Stages 4 th Order Pseudo- Runge-Kutta Method incorporate with Hermite Interpolation. This method will be derived and used to solve problem on Delay differential Equations (DDEs). In chapter five, we will discuss on the applications of using Nakashima a 2 Stages 4 th Order Pseudo-Runge-Kutta Method incorporate with Hermite Interpolation. The application base on Food-Limited Model will be discussed in this chapter. Lastly, in chapter six, we will discuss on the conclusion for this study. Besides that, we will also give some suggestions for the further study in order to improve the study on Pseudo-Runge-Kutta (PRK) Methods and Delay Differential Equations (DDEs).