UNIVERSITI PUTRA MALAYSIA BLOCK MULTISTEP METHODS FOR SOLVING FIRST ORDER RETARDED AND NEUTRAL DELAY DIFFERENTIAL EQUATIONS NURUL HUDA BT ABDUL AZIZ IPM 2015 17
BLOCK MULTISTEP METHODS FOR SOLVING FIRST ORDER RETARDED AND NEUTRAL DELAY DIFFERENTIAL EQUATIONS By NURUL HUDA BT ABDUL AZIZ Thesis Submitted to the School of Graduate Studies, Universiti Putra Malaysia, in Fulfilment of the Requirements for the Degree of Doctor of Philosophy December 2015
COPYRIGHT All material contained within the thesis, including without limitation text, logos, icons, photographs and all other artwork, is copyright material of Universiti Putra Malaysia unless otherwise stated. Use may be made of any material contained within the thesis for non-commercial purposes from the copyright holder. Commercial use of material may only be made with the express, prior, written permission of Universiti Putra Malaysia. Copyright Universiti Putra Malaysia
DEDICATIONS My deepest wish to my lovely husband for his great support, understanding and being a strength for my PhD journey. To my mother, father and siblings, thank you so much for all the Du a that always encourage and support me. Thank you so much.
Abstract of thesis presented to the Senate of Universiti Putra Malaysia in fulfilment of the requirement for the degree of Doctor of Philosophy BLOCK MULTISTEP METHODS FOR SOLVING FIRST ORDER RETARDED AND NEUTRAL DELAY DIFFERENTIAL EQUATIONS By NURUL HUDA BT ABDUL AZIZ December 2015 Chairman: Zanariah Abdul Majid, PhD Faculty : Institute for Mathematical Research This thesis investigates the numerical solutions for solving first order retarded and neutral delay differential equations and its analysis in block multistep methods. The investigation begins by solving the retarded delay differential equations (RDDE) using 1-point multistep method and it then extended to the 2-point and 3-point block multistep methods. There are two strategies that have been implemented in the numerical solutions which are the variable step size and the variable step size and variable order. In the developed algorithm, a Newton divided difference interpolation has been used in approximating a non-vanishing delay, whilst a new approach that is based on an adaptation of predictor-corrector scheme has been used to handle the vanishing delay problems. Special attention according to the numerical treatment of discontinuity in the delay solution is also presented where the techniques developed is from the adaptation of detect, locate and treat strategy. The results of these strategies are shown that it can be applied in the block multistep methods and able to solve even a type of state-dependent delays. The numerical solution of RDDE is then extended for solving the neutral delay differential equations (NDDE) of non-discontinuity and discontinuity cases. A new subroutine function has been added in the developed algorithm with the capabilities to approximate the neutral delay term using the interpolation of function evaluation. The interpolation accuracy that has been obtained from this approach is reliable and has an advantage in the less computational work. i
The analysis of all numerical methods including order, error constant, consistency, zero-stability and convergence properties are also have been presented. The proposed methods have shown to have a convergence when the numerical solution approaches to the exact solution as the step size h tends to zero. From the stability properties, it has been determined that the block multistep methods of RDDE have P-stability and Q-stability regions, whilst for the NDDE has NP-stability regions as the regions shrink when the step size ratios decreased. In conclusion, the performance of the proposed methods and the developed approaches are reliable and suitable for solving both retarded and neutral delay differential equations. Some advantages in terms of total number of steps and function calls of these proposed methods also have been identified when the comparison of the numerical results are made with the existing methods. ii
Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagai memenuhi keperluan untuk ijazah Doktor Falsafah KAEDAH BLOK MULTILANGKAH BAGI PENYELESAIAN PERINGKAT PERTAMA PERSAMAAN PEMBEZAAN TUNDA LEWAT DAN NEUTRAL Oleh NURUL HUDA BT ABDUL AZIZ Disember 2015 Pengerusi: Zanariah Abdul Majid, PhD Fakulti : Institut Penyelidikan Matematik Tesis ini mengkaji penyelesaian berangka bagi menyelesaikan persamaan pembezaan tunda lewat dan neutral peringkat pertama dan analisisnya dalam kaedah blok multilangkah. Kajian ini dimulakan dengan menyelesaikan persamaan pembezaan tunda lewat (PPTL) menggunakan kaedah multilangkah 1-titik dan kemudiannya diperluaskan kepada kaedah blok multilangkah 2-titik dan 3-titik. Terdapat dua strategi yang telah dilaksanakan dalam penyelesaian berangka ini iaitu saiz langkah berubah, dan saiz langkah dan peringkat berubah. Dalam algoritma yang dibangunkan, interpolasi perbezaan terbahagi Newton telah digunakan dalam menganggarkan tunda tidak terhapus, manakala satu pendekatan baru yang berasaskan kepada adaptasi skim peramal-pembetul telah digunakan untuk menangani masalah tunda terhapus. Pemerhatian khusus terhadap rawatan berangka ketidakselanjaran juga dipersembahkan di mana teknik yang dibangunkan adalah daripada penyesuaian strategi mengesan, menempat dan merawat. Hasil daripada strategi ini menunjukkan bahawa ianya boleh digunakan dalam kaedah blok multilangkah dan mampu menyelesaikan walaupun tunda jenis yang bergantung kepada keadaan. Penyelesaian berangka bagi PPTL kemudiannya diperluaskan untuk menyelesaikan persamaan pembezaan tunda neutral (PPTN) bagi kes keselanjaran dan ketidakselanjaran. Satu fungsi subrutin baru telah ditambah dalam algoritma yang dibangunkan dengan keupayaan untuk menganggar istilah tunda neutral dengan menggunakan interpolasi daripada penilaian fungsi. Ketepatan interpolasi yang diperoleh daripada pendekatan ini boleh dipercayai dan mempunyai faedah dalam mengurangkan kerja komputasi. iii
Analisis bagi semua kaedah termasuk ciri-ciri peringkat, pemalar ralat, konsistensi, kestabilan-sifar dan penumpuan juga turut dipersembahkan. Kaedah-kaedah yang dicadangkan telah menunjukkan bahawa ia mempunyai penumpuan apabila penyelesaian berangka menghampiri penyelesaian sebenar bagi saiz langkah h yang cenderung kepada sifar. Daripada ciri-ciri kestabilan, ia telah ditentukan bahawa kaedah blok multilangkah bagi PPTL mempunyai kawasan kestabilan-p dan kestabilan-q, manakala bagi PPTN ia mempunyai kawasan kestabilan-np iaitu kawasan kestabilan mengecil apabila nisbah saiz langkah berkurang. Kesimpulannya, prestasi bagi kaedah-kaedah yang dicadangkan dan pendekatan yang dibangunkan adalah boleh dipercayai dan sesuai bagi menyelesaikan kedua-dua persamaan pembezaan tunda jenis lewat dan neutral. Beberapa kelebihan darisegi jumlah bilangan langkah dan penilaian fungsi daripada kaedah-kaedah yang dicadangkan juga dikenal pasti apabila perbandingan keputusan berangka dibuat dengan kaedah yang sedia ada. iv
ACKNOWLEDGEMENTS In the name of Allah, the most Gracious and the most Merciful. Alhamdulillah, all praise to Allah for the strengths and His blessing in completing this thesis. Special appreciation goes to my supervisor, Prof. Dr. Zanariah Bt Abdul Majid, for her supervision and constant support. Her invaluable help of constructive comments and suggestions throughout the experimental and thesis work have contributed to the success of this research. Not forgotten, my appreciation to my co-supervisor, Prof. Dato Dr. Mohamed Bin Suleiman and Prof. Dr. Fudziah Bt Ismail for their support and knowledge regarding this topic. My special thanks goes to my dear husband, Ahmad Shahrimi Bin Ismail for his deeply love, support, understanding and caring. His great patience at all time has made this journey become wonderful and full of inspiration. The last but not least, my deepest gratitute and appreciation goes to my parents Abdul Aziz Bin Mohd Akhir and Rosni Bt Saad for their continuous support, love and prayers when I was up and down in every aspect of my life. To my mom, thank you so much for everything. You are the source of my Ph.D completion and this is to you that this thesis is dedicated. v
I certify that a Thesis Examination Committee has met on 9 December 2015 to conduct the final examination of Nurul Huda Bt Abdul Aziz on her thesis entitled Block Multistep Methods for Solving First Order Retarded and Neutral Delay Differential Equations in accordance with the Universities and University Colleges Act 1971 and the Constitution of the Universiti Putra Malaysia [P.U.(A) 106] 15 March 1998. The Committee recommends that the student be awarded the Doctor of Philosophy. Members of the Thesis Examination Committee were as follows: Mohamad Rushdan Md Said, PhD Associate Professor Institute for Mathematical Research Universiti Putra Malaysia (Chairman) Lee Lai Soon, PhD Associate Professor Faculty of Sciences Universiti Putra Malaysia (Internal Examiner) Zarina Bibi Ibrahim, PhD Associate Professor Faculty of Sciences Universiti Putra Malaysia (Internal Examiner) Cemil Tunc, PhD Professor Faculty of Sciences Yuzuncu Yil University Turkey (External Examiner) ZULKARNAIN ZAINAL, PhD Professor and Deputy Dean School of Graduate Studies Universiti Putra Malaysia Date: vi
This thesis was submitted to the Senate of Universiti Putra Malaysia and has been accepted as fulfilment of the requirement for the degree of Doctor of Philosophy. The members of the Supervisory Committee were as follows: Zanariah Abdul Majid, PhD Professor Institute for Mathematical Research Universiti Putra Malaysia (Chairman) Fudziah Ismail, PhD Professor Faculty of Sciences Universiti Putra Malaysia (Member) Mohamed Suleiman, PhD Professor Institute for Mathematical Research Universiti Putra Malaysia (Member) Khairil Iskandar Othman, PhD Associate Professor Faculty of Computer and Mathematical Sciences Universiti Teknologi Mara (Member) BUJANG KIM HUAT, PhD Professor and Dean School of Graduate Studies Universiti Putra Malaysia Date: vii
Declaration by graduate student I hereby confirm that: this thesis is my original work; quotations, illustrations and citations have been duly referenced; this thesis has not been submitted previously or concurrently for any other degree at any other institutions; intellectual property from the thesis and copyright of thesis are fully-owned by Universiti Putra Malaysia, as according to the Universiti Putra Malaysia (Research) Rules 2012; written permission must be obtained from supervisor and the office of Deputy Vice- Chancellor (Research and Innovation) before thesis is published (in the form of written, printed or in electronic form) including books, journals, modules, proceedings, popular writings, seminar papers, manuscripts, posters, reports, lecture notes, learning modules or any other materials as stated in the Universiti Putra Malaysia (Research) Rules 2012; there is no plagiarism or data falsification/fabrication in the thesis, and scholarly integrity is upheld as according to the Universiti Putra Malaysia (Graduate Studies) Rules 2003 (Revision 2012-2013) and the Universiti Putra Malaysia (Research) Rules 2012. The thesis has undergone plagiarism detection software. Signature: Date: Name and Matric No: Nurul Huda Bt Abdul Aziz, GS30241 viii
Declaration by Members of Supervisory Committee This is to confirm that: the research conducted and the writing of this thesis was under our supervision; supervision responsibilities as stated in the Universiti Putra Malaysia (Graduate Studies) Rules 2003 (Revision 2012-2013) are adhered to. Signature: Name of Chairman of Supervisory Committee: Zanariah Abdul Majid, PhD Signature: Name of Member of Supervisory Committee: Mohamed Suleiman, PhD Signature: Name of Member of Supervisory Committee: Fudziah Ismail, PhD Signature: Name of Member of Supervisory Committee: Khairil Iskandar Othman, PhD ix
TABLE OF CONTENTS ABSTRACT ABSTRAK ACKNOWLEDGEMENTS APPROVAL LIST OF TABLES LIST OF FIGURES LIST OF ABBREVIATIONS CHAPTER Page i iii v vi xiii xvi xviii 1 INTRODUCTION 1 1.1 Introduction 1 1.2 Delay Differential Equations 2 1.3 Problem Statement 3 1.4 Objectives of the Thesis 3 1.5 Scope and Limitations 4 1.6 Outline of the Thesis 4 2 BACKGROUND AND LITERATURE REVIEW 6 2.1 Introduction 6 2.2 An Initial Value Problem 6 2.3 Assumptions and Definitions 7 2.4 Stability Properties of Numerical Method for Solving DDEs 13 2.5 Numerical Difficulties for Solving DDEs 14 2.5.1 Problem Arising From the Vanishing Delays 14 2.5.2 Derivative Discontinuities of the Solution 15 2.6 Review of Previous Works 16 2.6.1 Numerical methods for RDDE 16 2.6.2 Numerical methods for NDDE 18 2.6.3 Block Method 19 3 1-POINT MULTISTEP METHOD IN VARIABLE STEP SIZE FOR RDDE WITH NON-VANISHING AND VANISHING DELAYS 21 3.1 Introduction 21 3.2 Derivation of Predictor 1-Point Multistep Method 21 3.3 Derivation of Corrector 1-Point Multistep Method 23 3.4 Order, Zero-stable, Consistency and Convergence of the Method 25 3.5 P-Stability and Q-Stability of the Method 27 3.6 Implementation 28 3.7 Algorithm of R1PVS Code 30 3.8 Variable Step Size Strategy 32 x
3.9 Test Problems 33 3.10 Numerical Results 35 3.11 Discussions 41 4 2-POINT AND 3-POINT BLOCK MULTISTEP METHODS IN VARIABLE STEP SIZE FOR RDDE WITH NON-VANISHING AND VANISHING DE- LAYS 43 4.1 Introduction 43 4.2 2-Point Block Multistep Method 43 4.2.1 Derivation of Predictor 2-Point Block Multistep Method 43 4.2.2 Derivation of Corrector 2-Point Block Multistep Method 46 4.2.3 Order, Zero-stable, Consistency and Convergence of the Method 48 4.2.4 P-Stability and Q-Stability of the Method 49 4.2.5 Implementation 51 4.2.6 Algorithm of R2PVS Code 52 4.3 3-Point Block Multistep Method 54 4.3.1 Derivation of Predictor 3-Point Block Multistep Method 54 4.3.2 Derivation of Corrector 3-Point Block Multistep Method 56 4.3.3 Order, Zero-stable, Consistency and Convergence of the Method 59 4.3.4 P-Stability and Q-Stability of the Method 61 4.3.5 Implementation 63 4.3.6 Algorithm of R3PVS Code 63 4.4 Variable Step Size Strategy 65 4.5 Numerical Results 66 4.6 Discussions 72 5 2-POINT BLOCK MULTISTEP METHOD IN VARIABLE STEP SIZE AND VARIABLE ORDER FOR RDDE WITH NON-VANISHING AND VANISHING DELAYS 74 5.1 Introduction 74 5.2 2-Point Block Multistep Method 74 5.2.1 Derivation of the Method with Different Orders 74 5.2.2 Order, Zero-stable, Consistency and Convergence of the Method 88 5.2.3 P-Stability and Q-Stability of the Method 88 5.2.4 Implementation 92 5.2.5 Algorithm of R2PVSVO Code 92 5.3 Variable Step Size and Variable Order Strategy 94 5.4 Numerical Results 95 5.5 Discussion 105 6 2-POINT AND 3-POINT BLOCK MULTISTEP METHODS IN VARIABLE STEP SIZE FOR RDDE WITH DISCONTINUITY CASE 107 6.1 Introduction 107 6.2 Derivative Discontinuity of RDDE 107 6.3 Implementation 109 6.4 Flow Chart of R2PVS(DC) Code 109 6.5 Variable Step Size Strategy 110 6.6 Test Problems 112 xi
6.7 Numerical Results 113 6.8 Discussion 116 7 2-POINT AND 3-POINT BLOCK MULTISTEP METHODS IN VARIABLE STEP SIZE FOR NDDE WITH NON-DISCONTINUITY AND DISCONTI- NUITY CASES 118 7.1 Introduction 118 7.2 NP-Stability of the Method 118 7.2.1 2-Point Block Multistep Method 118 7.2.2 3-Point Block Multistep Method 121 7.3 Case 1: Non-Discontinuity Case 123 7.3.1 Implementation 123 7.3.2 Algorithm of N2PVS Code 123 7.4 Case 2: Discontinuity Case 125 7.4.1 Implementation 126 7.4.2 Flow Chart of N2PVS(DC) Code 126 7.5 Test Problems 128 7.6 Numerical Results 130 7.7 Discussion 138 8 CONCLUSION 139 8.1 Summary 139 8.2 Future Works 140 BIBLIOGRAPHY 142 BIODATA OF STUDENT 146 LIST OF PUBLICATIONS 149 xii
LIST OF TABLES Table Page 1.1 The differences between ODEs and DDEs 2 3.1 Numerical results of Problem 3.1 36 3.2 Numerical results of Problem 3.2 36 3.3 Numerical results of Problem 3.3 37 3.4 Numerical results of Problem 3.4 37 3.5 Numerical results of Problem 3.5 37 3.6 Numerical results of Problem 3.6 38 3.7 Numerical results of Problem 3.7 38 4.1 Numerical results of Problem 3.1 66 4.2 Numerical results of Problem 3.2 67 4.3 Numerical results of Problem 3.3 67 4.4 Numerical results of Problem 3.4 68 4.5 Numerical results of Problem 3.5 68 4.6 Numerical results of Problem 3.6 69 4.7 Numerical results of Problem 3.7 69 5.1 Interpolation points for different orders 75 5.2 Predictor coefficients for 2-point block multistep method of order 4 75 5.3 Predictor coefficients for 2-point block multistep method of order 6 76 5.4 Predictor coefficients for 2-point block multistep method of order 7 77 5.5 Predictor coefficients for 2-point block multistep method of order 8 78 5.6 Predictor coefficients for 2-point block multistep method of order 9, j = 0,...,4 79 xiii
5.7 Predictor coefficients for 2-point block multistep method of order 9, j = 5,...,8 80 5.8 Corrector coefficients for 2-point multistep block method of order 4 81 5.9 Corrector coefficients for 2-point multistep block method of order 6 82 5.10 Corrector coefficients for 2-point multistep block method of order 7 83 5.11 Corrector coefficients for 2-point multistep block method of order 8, j = 0,...,3 84 5.12 Corrector coefficients for 2-point multistep block method of order 8, j = 4,...,8 85 5.13 Corrector coefficients for 2-point multistep block method of order 9, j = 0,...,3 86 5.14 Corrector coefficients for 2-point multistep block method of order 9, j = 4,...,8 87 5.15 Order and error constant for 2-point block multistep method 88 5.16 Numerical results of Problem 3.1 96 5.17 Numerical results of Problem 3.2 97 5.18 Numerical results of Problem 3.3 98 5.19 Numerical results of Problem 3.4 99 5.20 Numerical results of Problem 3.5 100 5.21 Numerical results of Problem 3.6 101 5.22 Numerical results of Problem 3.7 102 6.1 Numerical results of Problem 6.1 113 6.2 Numerical results of Problem 6.2 114 6.3 Numerical results of Problem 6.3 114 7.1 Numerical results of Problem 7.1 131 7.2 Numerical results of Problem 7.2 131 7.3 Numerical results of Problem 7.3 132 7.4 Numerical results of Problem 7.4 132 xiv
7.5 Numerical results of Problem 7.5 133 7.6 Numerical results of Problem 7.6 133 7.7 Numerical results of Problem 7.7 134 xv
LIST OF FIGURES Figure Page 2.1 General Block Method 8 2.2 Vanishing Delay of RDDE 14 2.3 Small Delay of RDDE 15 3.1 1-Point Multistep Method 21 3.2 1-Point Multistep Method for r = q = m = 1 2 25 3.3 P-Stability for 1-Point Multistep Method 28 3.4 Q-Stability for 1-Point Multistep Method 29 3.5 Comparison of FNC versus MAXERR for Problem 3.1 38 3.6 Comparison of TS versus MAXERR for Problem 3.2 39 3.7 Comparison of TS versus MAXERR for Problem 3.3 39 3.8 Comparison of TS versus MAXERR for Problem 3.4 39 3.9 Comparison of FNC versus MAXERR for Problem 3.5 40 3.10 Comparison of FNC versus MAXERR for Problem 3.6 40 3.11 Comparison of FNC versus MAXERR for Problem 3.7 40 4.1 2-Point Block Multistep Method 43 4.2 2-Point Block Multistep Method for r = 1 2 48 4.3 P-Stability for 2-Point Block Multistep Method 50 4.4 Q-Stability for 2-Point Block Multistep Method 51 4.5 Non-Vanishing Delay in 2-Point Block Multistep Method 52 4.6 Vanishing Delay in 2-Point Block Multistep Method 52 4.7 3-Point Block Multistep Method 54 4.8 3-Point Block Multistep Method for r = 2 1 59 4.9 P-Stability for 3-Point Block Multistep Method 62 xvi
4.10 Q-Stability for 3-Point Block Multistep Method 62 4.11 Non-Vanishing Delay in 3-Point Block Multistep Method 63 4.12 Vanishing Delay in 3-Point Block Multistep Method 63 4.13 Comparison of FNC versus MAXERR for Problem 3.1 70 4.14 Comparison of TS versus MAXERR for Problem 3.2 70 4.15 Comparison of TS versus MAXERR for Problem 3.3 70 4.16 Comparison of TS versus MAXERR for Problem 3.4 71 4.17 Comparison of FNC versus MAXERR for Problem 3.5 71 4.18 Comparison of FNC versus MAXERR for Problem 3.6 71 4.19 Comparison of FNC versus MAXERR for Problem 3.7 72 5.1 P-Stability for 2-Point Multistep Block Method at r = 0.5 89 5.2 Q-Stability for 2-Point Multistep Block Method at r = 0.5 89 5.3 P-Stability for 2-Point Multistep Block Method at r = 1.0 90 5.4 Q-Stability for 2-Point Multistep Block Method at r = 1.0 90 5.5 P-Stability for 2-Point Multistep Block Method at r = 2.0 91 5.6 Q-Stability for 2-Point Multistep Block Method at r = 2.0 91 5.7 Comparison of TS versus MAXERR for Problem 3.1 103 5.8 Comparison of TS versus MAXERR for Problem 3.2 103 5.9 Comparison of TS versus MAXERR for Problem 3.3 103 5.10 Comparison of TS versus MAXERR for Problem 3.4 104 5.11 Comparison of TS versus MAXERR for Problem 3.5 104 5.12 Comparison of FNC versus MAXERR for Problem 3.6 104 5.13 Comparison of FNC versus MAXERR for Problem 3.7 105 6.1 Flow chart for discontinuity treatment of RDDE 110 6.2 Comparison of FNC versus MAXERR for Problem 6.1 115 6.3 Comparison of FNC versus MAXERR for Problem 6.2 115 xvii
6.4 Comparison of TS versus MAXERR for Problem 6.3 116 7.1 NP-Stability of 2-Point Block Multistep Method for c = 0.25 120 7.2 NP-Stability of 2-Point Block Multistep Method for c = 1.0 120 7.3 NP-Stability of 3-Point Block Multistep Method for c = 0.25 122 7.4 NP-Stability of 3-Point Block Multistep Method for c = 1.0 122 7.5 Flow chart for discontinuity treatment of NDDE 127 7.6 Comparison of TS versus MAXERR for Problem 7.1 134 7.7 Comparison of TS versus MAXERR for Problem 7.2 135 7.8 Comparison of TS versus MAXERR for Problem 7.3 135 7.9 Comparison of TS versus MAXERR for Problem 7.4 136 7.10 Comparison of TS versus MAXERR for Problem 7.5 136 7.11 Comparison of FNC versus MAXERR for Problem 7.6 137 7.12 Comparison of FNC versus MAXERR for Problem 7.7 137 xviii
LIST OF ABBREVIATIONS DDEs RDDE NDDE NFDE ODEs RODEs IVP VS VSVO PPT SLB SLPB DTE ERKM LMM MOS LOB UPB Delay Differential Equations Retarded Delay Differential Equation Neutral Delay Differential Equation Neutral Functional Differential Equation Ordinary Differential Equations Retarded Ordinary Differential Equations Initial Value Problem Variable Step Size Variable Step Size and Variable Order Persamaan Pembezaan Tunda Saiz Langkah Berubah Saiz Langkah dan Peringkat Berubah Discontinuity Tracking Equation Embedded Runge-Kutta Method Linear Multistep Method Method of Steps Lower Bound Upper Bound xix
CHAPTER 1 1.1 Introduction INTRODUCTION Delay Differential Equations (DDEs) play an important roles in the history of scientific areas. It is known as a central of mathematical models that used to describe many phenomena in real life. For instance, the delay can be represented as a transport delay (a signal to travel to the controlled object; driver reaction time), dengue fever epidemics (the delay exist from the time of a bite to the time at which the human is infective), biological processes (the time for the body to produce red blood cells and the cell devision time), and physiological and pharmaceutical kinetics (the body s reaction to CO 2 ) (Lumb, 2004). Indeed, the use of the time delays are found in many processes whether natural or man made, as quoted by Kuang (1993), Like it or not, time delays occur so often, in almost every situation, that to ignore them is to ignore the reality. In mathematics, DDEs are defined as a differential equations in which the derivatives of some unknown functions at present time are dependent on the values of the functions at previous time. It is also called as a time-delay systems, hereditary systems, equations with deviating argument or differential-difference equations. Some authors refer to DDEs as ordinary differential equations with time lags or retarded ordinary differential equations (RODEs) where the terminology for DDEs has yet to be standardized. However, all names of DDEs above belong to the class of functional differential equations (Driver, 1977). In the elementary theory of initial value problems, ordinary differential equations (ODEs) and DDEs are said to have a few similarities in terms of obtaining a unique solution and both problems originating from the study of physical phenomena that can be solved exactly. Despite the obvious similarities, there are some important differences between ODEs and DDEs as has been highlighted in Table 1.1. From Table 1.1, it shows that the numerical solution of ODEs require only the initial value at initial point, x = a in order to obtain the sequence of the solution y(x). Unlike DDEs, two extra steps are required in the process before obtained the approximate solution y(x). The numerical solution of DDEs start by finding the location of the delay terms, α i = x i τ(x i,y(x i )) that may lies within the interval at point, x = x i for i = 0,1,2,...,n. After the location of the delays are known, the decision in approximating y(α i ) = y(x i τ(x i,y(x i ))) will be made. If the delays, α i lies in between [ τ,0], the initial function will be used as y(α i ) = φ(α i ). Otherwise, if the delays are in between [a,b] where b is the end point, then an
Table 1.1: The differences between ODEs and DDEs ODEs Standard form: y (x) = f (x,y(x)), a system that ignores the presence of delays Solution: The differential equations are all evaluated at certain time x Needs an initial value at point y(a) = y 0 to determine a unique solution y(x) DDEs Standard form: y (x) = f (x,y(x),y(x τ(x,y(x)))), a system with the presence of delays Solution: The differential equations are all evaluated at both current time x and prior time (x τ(x,y(x))) Needs an initial function y(x) = φ(x) to determine a unique solution y(x) interpolation procedure must be performed in finding the solution of the delays. The detailed description in which the properties of DDE systems are different from those of ODE systems can be seen in Martin and Ruan (2001) and Raghothama and Narayanan (2002). 1.2 Delay Differential Equations DDEs can be divided into four different classes which are retarded DDE (Baker, 2000), distributed DDE (Augeraud-Veron and Leandri, 2014), neutral DDE (Jackiewicz and Lo, 2006) and stochastic DDE (Fan, 2011) where the retarded type has become the most well-known class of DDEs. In this thesis, the DDEs of retarded and neutral types will be considered for solving in block multistep methods. A retarded delay differential equations (RDDE) is an ordinary differential equation that involved the solution of the delay term y(x τ(x,y(x))), given by y (x) = f (x,y(x),y(x τ(x,y(x)))), x [a,b], y(x) = φ(x), x a. (1.1) A neutral delay differential equations (NDDE) is an ordinary differential equation that involved both solutions of the delay term y(x τ(x,y(x))) and its derivative y (x σ(x,y(x))), given by y (x) = f (x,y(x),y(x τ(x,y(x))),y (x σ(x,y(x)))), x [a,b], y(x) = φ(x), x a, y (x) = φ (x), x a. (1.2) Here, the delays or lags τ and σ are measurable as a physical quantities that is a scalar in a function. It is always non-negative and the function f is assumed to be continuous and satisfies the Lipschitz condition in y(x) for all x [a,b]. φ(x) is the given initial function which is understood to be defined in [ρ,x 0 ], where 2
ρ = min 1 i n {min x x 0 (x τ i )}. (1.3) There are three conditions that the delay can be represent which are a constant (the constant delay case), a functions of x, τ i = τ i (x) (the variable or time-dependent delay case) and a functions of both x and y, τ i = τ i (x,y(x)) (the state-dependent delay case) (Bellen and Zennaro, 2003; Hayashi, 1996). 1.3 Problem Statement In the numerical solution of DDEs, there are three essential issues that one needs to be considered. First, the approximation of the retarded argument y(x τ(x, y(x))); second, handling the problem arising from the vanishing delays; third, the numerical treatment of derivative discontinuities in the solution. Some authors has classified the second and third issues as a main challenges in solving DDEs numerically (Yagoub et al., 2011; Carver, 1978; Oberle and Pesch, 1981). The vanishing delays occur when τ(x,y(x)) 0 as x x for some x. This is due to the delay values x τ(x,y(x)) that are smaller than the step size and caused it lies in the current step. The difficulties arise when there is no current solution of y(x) available at the current point to approximate the solution of the delay term y(x τ(x,y(x))). The particular issue that caused to inaccurate or inefficient in the numerical method are mostly originate from the nature of discontinuity in DDEs. It is occurs when the local truncation error that form the basis of most step size control algorithms may no longer be valid in the region of such discontinuities and derivative discontinuities (Paul, 1991). This observation motivated us to study in more detail associated with the numerical solution of retarded and neutral type of DDEs with the difficulties treatment in block multistep methods. 1.4 Objectives of the Thesis This thesis will be focused on the following objectives: 1. to develop new algorithms for solving non-vanishing and vanishing delays of RDDE in 1-point multistep method, 2-point block multistep method and 3-point block multistep method. 2. to develop new algorithms for discontinuity treatment of RDDE in 2-point block multistep method and 3-point block multistep method. 3. to analyse the properties of RDDE in 1-point multistep method, 2-point block multistep method and 3-point block multistep method including order, consistency, 3
zero-stability, convergence and stability. 4. to develop new algorithms for solving non-discontinuity and discontinuity cases of NDDE in 2-point block multistep method and 3-point block multistep method. 5. to analyse stability of NDDE in 2-point block multistep method and 3-point block multistep method. 1.5 Scope and Limitations This thesis concentrates on the numerical solution of block multistep methods for solving first order retarded and neutral delay differential equations. There are three methods that will be implemented to solve both types of DDEs which are 1-point multistep method, 2- point block multistep method and 3-point block multistep method. The analysis of these methods including order, consistency, zero-stability, convergence and stability are also discussed in this thesis. In implementing the step size control mechanism, two strategies of the variable step size (VS) and the variable step size and variable order (VSVO) will be adapted in the numerical solution. 1.6 Outline of the Thesis This thesis is organized as follows. In Chapter 1, a brief introduction associated with the DDEs in the real life applications and its difference with ODEs are presented. In Chapter 2, some of the basic assumptions and definitions that are necessary for the numerical method of DDEs are introduced. Then, followed by the discussion of the numerical difficulties that may arise in solving DDEs. A review of previous works in both retarded and neutral type of DDE and the block method are also presented in this chapter. The numerical solution of 1-point multistep method for solving the non-vanishing delay and vanishing delay of retarded DDE is discussed in Chapter 3. In this chapter, the strategy of Newton divided difference interpolation and assigning the current predictorcorrector solution for handling the non-vanishing and vanishing delays, respectively are highlighted. The developed algorithm is implemented in the variable step size strategy that requires the step size ratios to be constant, halved or doubled. The analysis of the order, consistency, zero-stability, convergence and stability are also been discussed. In Chapter 4, the 2-point and 3-point block multistep methods are presented. These methods are used to solve problems with non-vanishing delay and vanishing delay presented in Chapter 3. Comparisons are made in order to illustrate the advantage of this block multistep methods. The same strategy of Newton divided difference interpolation and assigning the current predictor-corrector solution are adapted in 2-point and 3-point block multistep methods. 4
In Chapter 5, the implementation of variable step size and variable order strategy in 2-point block multistep method is developed for solving the problems of retarded DDE. The range of the order is started from order 4 and up until order 9 depending on the local truncation error condition. In this implementation, the algorithm is designed to detect the number of interpolation points involved is proportional to the current order of the method. A detailed discussion associated with this strategy is discussed in this chapter. The investigation associated with the numerical treatment of derivative discontinuities and its propagation for retarded DDE is discussed in Chapter 6. There are three steps in treating the discontinuities such as detect the derivative discontinuities, locate the discontinuity points to include in mesh points and treat the discontinuities are described. This strategy is adapted in 2-point and 3-point block multistep methods and the detailed of the implementation is described in Section 6.4. Then, in Chapter 7, the numerical solution of neutral DDE for the case of nondiscontinuity and discontinuity using the variable step size implementation are presented. In this chapter, the strategy of how to approximate the derivative solution of the delay term y (x σ(x,y(x))) using the interpolation of function evaluation is described. Meanwhile, the strategy in treating the discontinuities in neutral DDE are adapted from the strategy that has been implemented in Chapter 6. Finally, Chapter 8 summarizes the findings and highlights some potential future researches in this study. 5
BIBLIOGRAPHY Al-Mutib, A. N. (1977). Numerical Methods for Solving Delay-Differential Equations. PhD thesis, Mathematics Department, Victoria University of Manchester. Arndt, H. (1984). Numerical solution of retarded initial value problems: Local and global error and stepsize control. Numerische Mathematik, 43:343 360. Augeraud-Veron, E. and Leandri, M. (2014). Optimal Pollution Control with Distributed Delays. Journal of Mathematical Economics, 55:24 32. Baker, C. (2000). Retarded differential equations. Journal of Computational and Applied Mathematics, 125:309 335. Baker, C. T. H. and Paul, C. A. H. (2006). Discontinuous solutions of neutral delay differential equations. Applied Numerical Mathematics, 56:284 304. Bellen, A. and Guglielmi, N. (2009). Solving neutral delay differential equations with state-dependent delays. Journal of Computational and Applied Mathematics, 229:350 362. Bellen, A., Jackiewicz, Z., and Zennaro, M. (1988). Stability analysis of one-step methods for neutral delay-differential equations. Numerische Mathematik, 52:605 619. Bellen, A. and Zennaro, M. (2003). Numerical Methods for Delay Differential Equations. Oxford University Press Inc., New York. Bellman, R. (1961). On the Computational Solution of Differential-Difference Equations. Journal of Mathematical Analysis and Application, 2:108 110. Bellman, R., Buell, J., and Kalaba, R. (1965). Numerical Integration of a Differential- Difference Equation with a Decreasing Time-Lag. Communications of the ACM, 8:227 228. Bellman, R. and Cooke, K. L. (1965). On the Computational Solution of a Class of Functional Differential Equations. Journal of Mathematical Analysis and Application, 12:495 500. Butcher, J. C. (1965). A modified multistep method for the numerical integration of ordinary differential equations. Journal of the ACM, 12:124 135. Carver, M. B. (1978). Efficient Integration Over Discontinuities in Ordinary Differential Equation Simulations. Mathematics and Computers in Simulation, pages 190 196. Corwin, S. P., Sarafyan, D., and Thompson, S. (1997). DKLAG6: A Code Based on Continuously Imbedded Sixth-Order Runge-Kutta Methods for the Solution of State- Dependent Functional Equations. Applied Numerical Mathematics, 24:319 330. Dalquist, G. (1956). Convergence and stability in the numerical integration of ordinary differential equations. Mathematica Scandinavica, 4(7):33 53. Driver, R. D. (1977). Ordinary and Delay Differential Equations. ISBN 0-387-90231-7. Springer-Verlag, New York. 142
Enright, W. and Hayashi, H. (1997). A delay differential equation solver based on a continuous runge-kutta method with defect control. Numerical Algorithms, 16:349 364. Enright, W. H. and Hu, M. (1995). Interpolating Runge-Kutta Methods for Vanishing Delay Differential Equations. Computing, 55:223 236. Fan, Z. (2011). Waveform Relaxation Method for Stochastic Differential Equations with Constant Delay. Applied numerical Mathematics, 61:229 240. Gear, C. W. (1964). Hybrid method for initial value problems in ordinary differential equations. Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, 2:69 86. Gragg, W. B. and Stetter, H. J. (1964). Generalized Multistep Predictor-Corrector Methods. Journal of the ACM, 11:188 209. Hairer, E., Norsett, S. P., and Wanner, G. (1987). Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. Hayashi, H. (1996). Numerical Solution of Retarded and Neutral Delay Differential Equations Using Continuous Runge-Kutta Methods. PhD thesis, Department of Computer Science, University of Toronto. Henrici, P. (1962). Discrete Variable Methods in Ordinary Differential Equations. John Willey and Sons, New York. Henrici, P. (1963). Error Propagation for Difference Methods. John Willey and Sons, New York. Higham, D. J. (1993). Error control for initial value problems with discontinuities and delays. Applied Numerical Mathematics, 12:315 330. Ibrahim, Z. B., Suleiman, M., Iskandar, K., and Majid, Z. A. (2005). Block method for generalised multistep adams and backward differentiation formulae in solving first order odes. MATEMATIKA, 21(1):25 33. In t Hout, K. J. (1992). A New Interpolation Procedure for Adapting Runge-Kutta Methods to Delay Differential Equations. BIT, 32:634 649. Ishak, F., Majid, Z. A., and Suleiman, M. (2010). Two point block method in variable step size technique for solving delay differential equations. Journal of Materials Science and Engineering, 4(12):86 90. Ishak, F., Suleiman, M., and Majid, Z. (2013). Block Method for Solving Pantograph- Type Functional Differential Equations. In Proceedings of the World Congress on Engineering 2013, volume 2. Ishak, F., Suleiman, M., and Majid, Z. (2014). Numerical Solution and Stability of Block Method for Solving Functional Differential Equations. In Transactions on Engineering Technologies, pages 597 609. Springer Netherlands. Ishak, F., Suleiman, M., Majid, Z. A., and Othman, K. I. (2011). Development of Variable Stepsize Variable Order Block Method in Divided Difference Form for the Numerical Solution of Delay Differential Equations. In World Academy of Science, Engineering and Technology, volume 53. 143
Ishak, F., Suleiman, M., and Omar, Z. (2008). Two-point predictor-corrector block method for solving delay differential equations. MATEMATIKA, 24(2):131 140. Ismail, F. and Alkhasawneh, R. A. A. (2008). Solving Delay Differential Equations Using Embedded Singly Diagonally Implicit Runge-Kutta Methods. Acta Mathematica Vietnamica, 33(2):95 105. Ismail, F., Alkhasawneh, R. A. A., Aung, S. L., and Suleiman, M. (2002). Numerical Treatment of Delay Differential Equations by Runge-Kutta Method Using Hermite Interpolation. Matematika, 18(2):79 90. Ismail, F., Alkhasawneh, R. A. A., and Suleiman, M. (2003). Comparison of interpolations used in solving delay differential equations by runge-kutta method. International Journal of Computer Mathematics, 80(7):921 930. Ismail, F. and Suleiman, M. (2001). Solving Delay Differential Equations Using Intervalwise Partitioning by Runge-Kutta Method. Applied Mathematics and Computation, 121:37 53. Jackiewicz, Z. (1982). Adams methods for neutral functional differential equations. Numerische Mathematik, 39:221 230. Jackiewicz, Z. (1984). One-step methods of any order for neutral functional differential equations. SIAM J. Numerical Analysis, 21:486 511. Jackiewicz, Z. (1986). Quasilinear multistep methods and variable step predictorcorrector methods for neutral functional differential equations. SIAM Journal on Numerical Analysis, 23(2):423 452. Jackiewicz, Z. (1987). Variable-step variable-order algorithm for the numerical solution of neutral functional differential equations. Applied Numerical Mathematics, 3:317 329. Jackiewicz, Z. and Lo, E. (1991). The numerical solution of neutral functional differential equations by adams predictor-corrector methods. Applied Numerical Mathematics, 8:477 491. Jackiewicz, Z. and Lo, E. (2006). Numerical solution of neutral functional differential equations by adams methods in divided difference form. Journal of Computational and Applied Mathematics, 189:592 605. Kappel, F. and Kunisch, K. (1981). Spline approximations for neutral functionaldifferential equations. SIAM Journal on Numerical Analysis, 18:1058 1080. Karoui, A. (1992). On the Numerical Solution of Delay Differential Equations. PhD thesis, University of Ottawa. Karoui, A. and Vaillancourt, R. (1994). Computer Solutions of State-Dependent Delay Differential Equations. Computers and Mathematics with Applications, 27(4):37 51. Karoui, A. and Vaillancourt, R. (1995). A Numerical Method for Vanishing-Lag Delay Differential Equations. Applied Numerical Mathematics, 17:383 395. Kuang, Y. (1993). Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc., London. 144
Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations. John Willey and Sons, New York. Lambert, J. D. (1991). Numerical Methods for Ordinary Differential Equations: The Initial Value Problem. John Willey and Sons, New York. Lim, T. H. (2014). A Third Order Nakashima Type Implicit Pseudo Runge-Kutta Method for Delay Differential Equations. Sains Malaysiana, 43(8):1259 1262. Lumb, P. M. (2004). Delay Differential Equations: Detection of Small Solutions. PhD thesis, Mathematics Department, University of Liverpool. Majid, Z. A. (2004). Parallel Block Methods for Solving Ordinary Differential Equations. PhD thesis, Universiti Putra Malaysia, Malaysia. Majid, Z. A. and Suleiman, M. (2006). 1-point implicit code of adams moulton type for solving first order ordinary differential equations. Chiang Mei Journal of Sciences, 33(2):153 159. Majid, Z. A. and Suleiman, M. (2007). Implementation of four-point fully implicit block method for solving ordinary differential equations. Applied Mathematics and Computation, 184:514 522. Majid, Z. A. and Suleiman, M. (2008). Variable step variable order two point block fully implicit method for solving ordinary differential equations. European Journal of Scientific Research, 21(3):521 529. Majid, Z. A. and Suleiman, M. (2011). Predictor-corrector block iteration method for solving ordinary differential equations. Sains Malaysiana, 40(6):659 664. Majid, Z. A., Suleiman, M., Ismail, F., and Othman, M. (2003). 2-point implicit block one-step method half gauss-seidel for solving first order ordinary differential equations. Matematika, 19(2):91 10. Majid, Z. A., Suleiman, M., and Omar, Z. (2006). 3-point implicit block method for solving ordinary differential equations. Bulletin of the Malaysian Mathematical Sciences Society, 29(1):23 31. Martin, A. and Ruan, S. (2001). Predator-prey models with delay and prey harvesting. Journal of Mathematical Biology, 43:247 267. Mehrkanoon, S., Majid, Z. A., and Suleiman, M. (2010). Implementation of 2-point 2-step methods for the solution of first order odes. Applied Mathematical Sciences, 4(7):305 316. Neves, K. and Feldstein, M. (1984). Higher order methods for state-dependent delay differential equations with non-smooth solutions. SIAM Journal on Numerical Analysis, 21:844 863. Neves, K. W. (1975). Automatic Integration of Functional Differential Equations: An Approach. ACM Transactions on Mathematical Software, 1:357 368. Neves, K. W. and Feldstein, A. (1976). Characterization of jump discontinuities for state-dependent delay differential equations. Journal of Mathematical Analysis and Applications, 56(3):689 707. 145
Neves, K. W. and Thompson, S. (1992). Solution of systems of functional differential equations with state dependent delays. Applied Numerical Mathematics, 9:385 401. Nur Ain, A. S. and Mustafa, M. (2013). Solving Delay Differential Equations (DDEs) using Nakashima s 2 Stages 4th Order Pseudo-Runge-Kutta Methods. World Applied Sciences Journal, 21:181 186. Oberle, H. J. and Pesch, H. J. (1981). Numerical Treatment of Delay Differential Equations by Hermite Interpolation. Numerische Mathematik, (37):235 255. Paul, C. A. H. (1991). Developing a Delay Differential Equation Solver. Applied Numerical Mathematics, 9:403 414. Paul, C. A. H. (1992). Runge-Kutta Methods for Functional Differential Equations. PhD thesis, Manchester University, England. Raghothama, A. and Narayanan, S. (2002). Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dynamics, 27:341 365. Rosser, J. B. (1967). A runge-kutta for all seasons. SIAM Review, 9(3):417 452. San, H. C., Majid, Z. A., and Othman, M. (2011). Solving Delay Differential Equations Using Coupled Block Method. In 4th Int. Confer. on Modelling, Simulation and Applied Optimization. Shampine, L. F. (2005). Solving ODEs and DDEs with Residual Control, Applied Numerical Mathematics. Applied Numerical Mathematics, 52:113 127. Shampine, L. F. and Gordon, M. K. (1975). Computer Solutions of Ordinary Differential Equations. Freeman. Shampine, L. F. and Thompson, S. (2001). Solving ddes in matlab. Applied Numerical Mathematics, 37:441 458. Stetter, H. J. (1965). Numerische Losung von Differentialgleichungen mit nacheilendem Argument. ZAMM, 45:79 80. Thompson, S. (1995). Stepsize control for Delay Differential Equations using Continuously Imbedded Runge-Kutta Methods of Sarafyan. Computing, (55):223 236. Thompson, S. and Shampine, L. F. (2006). A friendly fortran dde solver. Applied Numerical Mathematics, 56:503 516. Wille, D. R. (1989). The Numerical Solution of Delay-Differential Equations. PhD thesis, Department of Mathematics, University of Manchester. Wille, D. R. and Baker, C. T. H. (1992). Delsol - a numerical code for the solution of systems of delay-differential equations. Applied Numerical Mathematics, 9:223 234. Wille, D. R. and Baker, C. T. H. (1994). Stepsize control and continuity consistency for state-dependent delay differential equations. Journal of Computational and applied mathematics, 53:163 170. Yagoub, H., Nguyen-Ba, T., and Vaillancourt, R. (2011). Variable-step Variable-order 3-Stage Hermite-Birkhoff-Obrechkoff DDE Solver of Order 4 to 14. Applied Mathematics and Computation, 217:10247 10255. 146