Trigonometrically fitted multistep methods for solving oscillatory problems

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(1)UNIVERSITI PUTRA MALAYSIA TRIGONOMETRICALLY FITTED MULTISTEP METHODS FOR SOLVING OSCILLATORY PROBLEMS. AINI FADHLINA BINTI MANSOR. IPM 2018 1.

(2) PM. IG. H. T. U. TRIGONOMETRICALLY FITTED MULTISTEP METHODS FOR SOLVING OSCILLATORY PROBLEMS. R. By. ©. C. O. PY. AINI FADHLINA BINTI MANSOR. Thesis Submitted to the School of Graduate Studies, Universiti Putra Malaysia, in Fulfillment of the Requirements for the Degree of Master of Science. December 2017.

(3) ©. C T. H. IG. R. PY. O. U. PM.

(4) COPYRIGHT. PM. 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.. ©. C. O. PY. R. IG. H. T. U. Copyright © Universiti Putra Malaysia.

(5) ©. C. O. PY. R. IG. U. H. T. Dad: Mansor bin Abas Mum: Siti Alfah binti Mohammed Siblings: Zeti Marlyda binti Mansor Arman Faizul bin Mansor Muhammad Asraf bin Mansor Anuarul bin Mansor ... Lecturers Gadis-gadis INSPEM Friends. PM. DEDICATIONS.

(6) Abstract of thesis presented to the Senate of Universiti Putra Malaysia in fulfillment of the requirement for the degree of Master of Science. By. Fudziah binti Ismail, PhD Institute for Mathematical Research. H. : :. IG. Chairman Faculty. T. December 2017. U. AINI FADHLINA BINTI MANSOR. PM. TRIGONOMETRICALLY FITTED MULTISTEP METHODS FOR SOLVING OSCILLATORY PROBLEMS. PY. R. At the beginning of the thesis, we trigonometrically fitted the first point of the existing block multistep method which was purposely derived for solving special second order ordinary differential equations (ODEs). Based on the original multistep method, we construct both explicit and implicit trigonometrically fitted multistep methods of step number k=4 and develop code using both constant and variable step size. The trigonometrically fitting technique has been applied to the original method in order to construct the new methods. The numerical results show that the trigonometrically fitted multistep method is more efficient compared to the existing methods in solving special second order ordinary differential equations (ODEs) which are oscillatory in nature.. ©. C. O. Then, the 2-point explicit and implicit block multistep methods of step number k=3 and k=5 for solving special second order ODEs are derived using integration formula based on Newton-Gregory backward interpolation polynomial. The methods are implemented for constant step size by using the predictor-corrector technique, followed by the implementation using variable step size. The numerical results are given to show the efficiency of the new methods as compared to the existing methods. The 2-point explicit and implicit block multistep methods of step number k=3 and k=5 are then trigonometrically fitted so that they are suitable for solving special second order ordinary differential equations, which are highly oscillatory in nature. We developed codes based on the trigonometrically fitted methods using constant step size in predictor-corrector mode. The numerical results obtained show that trigonometrically fitted the methods give more accurate solutions than the existing i.

(7) methods.. ©. C. O. PY. R. IG. H. T. U. PM. In conclusion, trigonometrically fitted block and non-block multistep methods have been derived in this thesis for solving oscillatory problems. The illustrative examples are given in the form of both tables and graphs which clearly shown the advantage of the methods.. ii.

(8) Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagai memenuhi keperluan untuk ijazah Master Sains. Oleh. Fudziah binti Ismail, PhD Institut Penyelidikan Matematik. H. : :. IG. Pengerusi Fakulti. T. Disember 2017. U. AINI FADHLINA BINTI MANSOR. PM. KAEDAH MULTILANGKAH SUAI SECARA TRIGONOMETRI UNTUK MENYELESAIKAN MASALAH BERAYUN. O. PY. R. Pada permulaan tesis, kami suai secara trigonometri titik pertama kaedah multilangkah blok sedia ada yang diterbitkan dengan tujuan untuk menyelesaikan persamaan pembezaan biasa (PPB) peringkat kedua khas. Berdasarkan kepada kaedah multilangkah asal, kami membina kedua-dua kaedah multilangkah suai secara trigonometri secara tak tersirat dan tersirat bagi nombor langkah k=4 dan membangunkan kod menggunakan kedua-dua saiz langkah tetap dan boleh ubah. Kaedah suai secara trigonometri telah digunakan ke atas kaedah asal bagi membina kaedah baru tersebut. Keputusan berangka menunjukkan bahawa kaedah multilangkah suai secara trigonometri adalah lebih berkesan berbanding dengan kaedah sedia ada dalam menyelesaikan persamaan pembezaan biasa (PPB) peringkat kedua khas yang mana berayun secara semula jadi.. ©. C. Kemudian, 2-titik kaedah multilangkah blok secara tak tersirat dan tersirat bagi nombor langkah k=3 dan k=5 untuk menyelesaikan PPB peringkat kedua khas telah diterbitkan menggunakan formula kamiran berdasarkan kepada Newton-Gregory backward interpolation polynomial. Kaedah tersebut dilaksanakan untuk saiz langkah tetap dengan menggunakan teknik peramal-pembetul, diikuti dengan perlaksanaan menggunakan saiz langkah boleh ubah. Keputusan berangka diberikan untuk menunjukkan keberkesanan kaedah baru tersebut berbanding dengan kaedah sedia ada. 2-titik kaedah multilangkah blok secara tak tersirat dan tersirat bagi nombor langkah k=3 dan k=5 kemudiannya disuaikan secara trigonometri supaya ianya sesuai untuk menyelesaikan persamaan pembezaan biasa peringkat kedua khas, yang iii.

(9) PM. mana sangat berayun secara semula jadi. Kami membangunkan kod berdasarkan kepada kaedah suai secara trigonometri menggunakan saiz langkah tetap dalam mod peramal-pembetul. Keputusan berangka yang diperolehi menunjukkan bahawa suai secara trigonometri kaedah tersebut memberikan lebih banyak penyelesaian yang tepat daripada kaedah sedia ada.. ©. C. O. PY. R. IG. H. T. U. Kesimpulannya, kaedah multilangkah blok dan bukan blok suai secara trigonometri telah diterbitkan di dalam tesis ini untuk menyelesaikan masalah berayun. Contohcontoh illustrasi diberikan di dalam kedua-dua bentuk jadual dan graf yang dengan jelas menunjukkan kelebihan kaedah-kaedah tersebut.. iv.

(10) ACKNOWLEDGEMENTS. PM. Bismillahirrahmanirrahim. In the Name of Allah, the Most Beneficent and the Most Merciful. Praised be to Allah the Almighty, Lord of the Worlds for His blessings showered upon me for making the writing of this thesis a successful one. However, it would not have been possible without the kind support and help of many individuals. I would like to express my great appreciation for the guidance and assistance received throughout the journey to complete this thesis writing.. U. I would like to express my deepest gratitude to my supervisor, Prof. Dr. Fudziah Ismail for her kindness, guidance, times and support. Her valuable guidance helped me a lot in completing this master thesis.. IG. H. T. I would like to thank my co-supervisor, Assoc. Prof. Dr. Norazak Senu for his help and guidance. A special thanks also to all the lecturers and staffs in the Department of Mathematics and Institute for Mathematical Research (INSPEM) for their kindness and assistance. Moreover, I want to thank School of Graduate Studies, Universiti Putra Malaysia, for giving me the opportunity to be a Graduate Research Fellow (GRF) and also Ministry of Higher Education for the financial support.. ©. C. O. PY. R. I would like to extend my sincere thanks to my family and friends for their continuous support and encouragement. I am really indebted to those who have helped me in facing this big challenge in completing this thesis. Thank you very much. May Allah shower all of you with His blessings.. v.

(11) Members of the Thesis Examination Committee were as follows:. T. U. Norihan binti Md Arifin, PhD Associate Professor Faculty of Science Universiti Putra Malaysia (Chairperson). PM. I certify that a Thesis Examination Committee has met on 21 December 2017 to conduct the final examination of AINI FADHLINA BINTI MANSOR on her thesis entitled ”TRIGONOMETRICALLY FITTED MULTISTEP METHODS FOR SOLVING OSCILLATORY PROBLEMS” 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 Master of Science.. IG. H. Zarina Bibi binti Ibrahim, PhD Associate Professor Faculty of Science Universiti Putra Malaysia (Internal Examiner). C. O. PY. R. Rokiah @ Rozita binti Ahmad, PhD Associate Professor Faculty of Science and Technology Universiti Kebangsaan Malaysia (External Examiner). ©. NOR AINI AB. SHUKOR, PhD Professor and Deputy Dean School of Graduate Studies Universiti Putra Malaysia Date: vi.

(12) This thesis was submitted to the Senate of Universiti Putra Malaysia and has been accepted as fulfillment of the requirement for the degree of Master of Science. The members of the Supervisory Committee were as follows:. PM. Fudziah binti Ismail, PhD Professor Faculty of Science Universiti Putra Malaysia (Chairperson). C. O. PY. R. IG. H. T. U. Norazak bin Senu, PhD Associate Professor Faculty of Science Universiti Putra Malaysia (Member). ©. ROBIAH BINTI YUNUS, PhD Professor and Dean School of Graduate Studies Universiti Putra Malaysia Date: vii.

(13) Declaration by graduate student. Signature:. IG. H. T. U. PM. 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. Date:. ©. C. O. PY. R. Name and Matric No: Aini Fadhlina Binti Mansor, GS44313. viii.

(14) Declaration by Members of Supervisory Committee. PM. 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.. T H. ©. C. O. PY. R. IG. Signature: Name of Member of Supervisory Committee: Norazak bin Senu. U. Signature: Name of Chairman of Supervisory Committee: Fudziah binti Ismail. ix.

(15) TABLE OF CONTENTS. PM. Page ABSTRACT. i. ABSTRAK. iii. ACKNOWLEDGEMENTS. v. APPROVAL. vi. LIST OF TABLES. U. xiii. LIST OF FIGURES. xvii. LIST OF ABBREVIATIONS. T. xix. CHAPTER. 1 1 1 2 2 3 3 4 5 6 6 7. 2 LITERATURE REVIEW 2.1 Introduction of Linear Multistep Method 2.2 Introduction of Block Method 2.3 Introduction of Trigonometrically Fitted Method and Oscillatory Problems. 8 8 8. O. PY. R. IG. H. 1 INTRODUCTION 1.1 Differential Equations 1.2 The Initial Value Problem 1.3 Numerical Methods for Ordinary Differential Equations 1.4 Single Step and Multistep Methods 1.5 Block Method 1.6 Trigonometrically Fitted Method 1.7 Order Conditions 1.8 Stability of the Method 1.9 The Objectives of the Thesis 1.10 Outline of the Thesis 1.11 Scope of the Research. ©. C. 3 TRIGONOMETRICALLY FITTED MULTISTEP METHODS OF STEP NUMBER k=4 FOR SOLVING OSCILLATORY PROBLEMS 3.1 Introduction 3.2 Derivation of Trigonometrically Fitted Explicit and Implicit Methods for Step Number k=4 3.3 Stability Analysis 3.3.1 Zero Stability 3.3.2 Absolute Stability 3.4 Implementation of Constant Step Size 3.4.1 Problems Tested and Numerical Results x. 9. 12 12 12 19 19 19 21 21.

(16) 3.4.2 Discussion Implementation of Variable Step Size 3.5.1 Problems Tested and Numerical Results 3.5.2 Discussion. 39 40 41 47. PM. 3.5. 48 48 48 48 51 52 54 56 56 60 64 64 66 68 69 82 83 84 96. R. IG. H. T. U. 4 2-POINT BLOCK MULTISTEP METHODS FOR SPECIAL SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS 4.1 Introduction 4.2 Derivation of Block Methods for Step Number k=3 and k=5 4.2.1 First Point of the Explicit Block Methods 4.2.2 Second Point of the Explicit Block Methods 4.2.3 First Point of the Implicit Block Methods 4.2.4 Second Point of the Implicit Block Methods 4.3 Order of The Method 4.3.1 Explicit 2-point Block Methods 4.3.2 Implicit 2-point Block Methods 4.4 Stability Analysis 4.4.1 Zero Stability 4.4.2 Absolute Stability 4.5 Implementation of Constant Step Size 4.5.1 Problems Tested and Numerical Results 4.5.2 Discussion 4.6 Implementation of Variable Step Size 4.6.1 Problems Tested and Numerical Results 4.6.2 Discussion. ©. C. O. PY. 5 TRIGONOMETRICALLY FITTED 2-POINT BLOCK MULTISTEP METHODS FOR SOLVING OSCILLATORY PROBLEMS 98 5.1 Introduction 98 5.2 Derivation of Trigonometrically Fitted Methods for Step Number k=3 and k=5 98 5.2.1 First and Second Point of Explicit Block Methods (k=3) 98 5.2.2 First and Second Point of Implicit Block Methods (k=3) 100 5.2.3 First and Second Point of Explicit Block Methods (k=5) 102 5.2.4 First and Second Point of Implicit Block Methods (k=5) 105 5.3 Implementation of the Method 108 5.3.1 Numerical Results 108 5.3.2 Discussion 124 6 CONCLUSION AND FUTURE WORK 6.1 Conclusion 6.2 Future Work. 126 126 126. REFERENCES. 127. BIODATA OF STUDENT. 129 xi.

(17) LIST OF PUBLICATIONS. ©. C. O. PY. R. IG. H. T. U. PM. 131. xii.

(18) PM. LIST OF TABLES. Table. 3.3. Comparison between ETFMM with the existing methods for problem 3.2. 25. Comparison between ETFMM with the existing methods for problem 3.3. 26. 27. Comparison between ETFMM with the existing methods for problem 3.5. 28. Comparison between ITFMM with the existing methods for problem 3.1. 32. Comparison between ITFMM with the existing methods for problem 3.2. 33. Comparison between ITFMM with the existing methods for problem 3.3. 34. Comparison between ITFMM with the existing methods for problem 3.4. 35. 3.10 Comparison between ITFMM with the existing methods for problem 3.5. 36. 3.11 Comparison between ITFMM with the existing methods for problem 3.3 with xn = 100. 43. 3.12 Comparison between ITFMM with the existing methods for problem 3.6 with xn = 100. 43. 3.13 Comparison between ITFMM with the existing methods for problem 3.7 with xn = 10. 44. 3.14 Comparison between ITFMM with the existing methods for problem 3.8 with xn = 100. 44. 3.6. 3.7. PY. 3.8. IG. 3.5. H. Comparison between ETFMM with the existing methods for problem 3.4. R. 3.4. 24. U. 3.2. Comparison between ETFMM with the existing methods for problem 3.1. T. 3.1. Page. ©. C. O. 3.9. xiii.

(19) 4.1. Integration coefficients of the first point of the explicit block method. 4.2. Integration coefficients of the second point of the explicit block method 52. 4.3. Integration coefficients of the first point of the implicit block method. 4.4. Integration coefficients of the second point of the implicit block method 56. 4.5. Comparison between 2PBM4 with the existing methods for problem 3.2 with xn = 10. 71. Comparison between 2PBM4 with the existing methods for problem 3.3 with xn = 10. 71. Comparison between 2PBM4 with the existing methods for problem 3.5 with xn = 10. 72. PM. Comparison between 2PBM4 with the existing methods for problem 3.6 with xn = 10. 73. Comparison between 2PBM4 with the existing methods for problem 4.1 with xn = 10. 74. 4.10 Comparison between 2PBM6 with the existing methods for problem 3.2 with xn = 10. 77. 4.11 Comparison between 2PBM6 with the existing methods for problem 3.3 with xn = 10. 78. 4.12 Comparison between 2PBM6 with the existing methods for problem 3.5 with xn = 10. 78. 4.13 Comparison between 2PBM6 with the existing methods for problem 3.6 with xn = 10. 79. 4.14 Comparison between 2PBM6 with the existing methods for problem 4.1 with xn = 10. 79. 4.15 Comparison between 2PBM4 with Adams method for problem 3.1 with xn = 10. 86. 4.16 Comparison between 2PBM4 with Adams method for problem 3.6 with xn = 10. 86. 4.17 Comparison between 2PBM4 with Adams method for problem 4.1 with xn = 5. 87. 4.18 Comparison between 2PBM4 with Adams method for problem 4.2 with xn = 2. 87. ©. C. O. PY. R. IG. 4.9. H. 4.8. 54. U. 4.7. T. 4.6. 51. xiv.

(20) 87. 4.20 Comparison between 2PBM4 with Adams method for problem 4.6 with xn = 2. 88. 4.21 Comparison between 2PBM6 with Adams method for problem 4.2 with xn = 2. 91. 4.22 Comparison between 2PBM6 with Adams method for problem 4.3 with xn = 5. 92. U. PM. 4.19 Comparison between 2PBM4 with Adams method for problem 4.4 with xn = 2. 92. 4.24 Comparison between 2PBM6 with Adams method for problem 4.5 with xn = 10. 93. 4.25 Comparison between 2PBM6 with Adams method for problem 4.6 with xn = 2. 93. 5.2. 109. Comparison between TF2PBM4 with the existing methods for problem 3.3. 110. Comparison between TF2PBM4 with the existing methods for problem 3.5. 111. Comparison between TF2PBM4 with the existing methods for problem 3.6. 112. Comparison between TF2PBM4 with the existing methods for problem 4.1. 113. Comparison between TF2PBM6 with the existing methods for problem 3.2. 117. Comparison between TF2PBM6 with the existing methods for problem 3.3. 118. Comparison between TF2PBM6 with the existing methods for problem 3.5. 119. Comparison between TF2PBM6 with the existing methods for problem 3.6. 120. PY. 5.3. Comparison between TF2PBM4 with the existing methods for problem 3.2. R. 5.1. IG. H. T. 4.23 Comparison between 2PBM6 with Adams method for problem 4.4 with xn = 2. 5.4. O. 5.5. ©. C. 5.6. 5.7. 5.8. 5.9. xv.

(21) 121. ©. C. O. PY. R. IG. H. T. U. PM. 5.10 Comparison between TF2PBM6 with the existing methods for problem 4.1. xvi.

(22) LIST OF FIGURES. Page. PM. Figure 2-point block multistep method.. 3.1. Stability region of trigonometrically fitted multistep method for k=4. 3.2. Efficiency curves for problem 3.1 with xn = 1, 000 and h = 1 − 0.125i, i = 3, ..., 6.. 29. Efficiency curves for problem 3.2 with xn = 1, 000 and h = 0.125/2i , i = 2, ..., 5.. 29. Efficiency curves for problem 3.3 with xn = 1, 000 and h = 0.125/2i , i = 2, ..., 6.. 30. Efficiency curves for problem 3.4 with xn = 1, 000 and h = 0.125/2i , i = 3, ..., 7.. 30. Efficiency curves for problem 3.5 with xn = 1, 000 and h = 0.5/2i , i = 0, ..., 4.. 31. 3.6. T. Efficiency curves for problem 3.1 with xn = 1, 000 and h = 1 − 0.125i, i = 3, ..., 6.. PY. 3.7. H. 3.5. IG. 3.4. R. 3.3. 3. U. 1.1. 3.8. C ©. 37. Efficiency curves for problem 3.2 with xn = 1, 000 and h = 0.125/2i , i = 2, ..., 5.. 37. Efficiency curves for problem 3.3 with xn = 1, 000 and h = 0.125/2i , i = 2, ..., 6.. 38. O. 3.9. 20. 3.10 Efficiency curves for problem 3.4 with xn = 1, 000 and h = 0.125/2i , i = 3, ..., 7.. 38. 3.11 Efficiency curves for problem 3.5 with xn = 1, 000 and h = 0.5/2i , i = 0, ..., 4.. 39. 3.12 Efficiency curves for problem 3.3 with xn = 100.. 45. 3.13 Efficiency curves for problem 3.6 with xn = 100.. 45. 3.14 Efficiency curves for problem 3.7 with xn = 10.. 46. 3.15 Efficiency curves for problem 3.8 with xn = 100.. 46. xvii.

(23) Stability region of 2-point block multistep method for k=3. 67. 4.2. Stability region of 2-point block multistep method for k=5. 68. 4.3. Efficiency curves for problem 3.2 with xn = 10 and h = 0.125/2i , i = 2, ..., 5.. 75. Efficiency curves for problem 3.3 with xn = 10 and h = 0.125/2i , i = 3, ..., 7.. 75. Efficiency curves for problem 3.5 with xn = 10 and h = 0.5/2i , i = 0, ..., 4.. 76. Efficiency curves for problem 3.6 with xn = 10 and h = 0.125/2i , i = 0, ..., 5.. 76. Efficiency curves for problem 4.1 with xn = 10 and h = 0.5/2i , i = 0, ..., 4.. 77. 4.7. 4.8. U. T. 4.6. H. 4.5. Efficiency curves for problem 3.2 with xn = 10 and h = 0.125/2i , i = 2, ..., 5.. IG. 4.4. PM. 4.1. Efficiency curves for problem 3.3 with xn = 10 and h = 0.125/2i , i = 2, ..., 5.. 80. 4.10 Efficiency curves for problem 3.5 with xn = 10 and h = 0.5/2i , i = 1, ..., 4.. 81. 4.11 Efficiency curves for problem 3.6 with xn = 10 and h = 0.5/2i , i = 1, ..., 5.. 81. 4.12 Efficiency curves for problem 4.1 with xn = 10 and h = 0.5/2i , i = 1, ..., 4.. 82. 4.13 Efficiency curves for problem 3.1 with xn = 10.. 88. O. PY. R. 4.9. 4.14 Efficiency curves for problem 3.6 with xn = 10.. 89. 4.15 Efficiency curves for problem 4.1 with xn = 5.. 89. 4.16 Efficiency curves for problem 4.2 with xn = 2.. 90. 4.17 Efficiency curves for problem 4.4 with xn = 2.. 90. 4.18 Efficiency curves for problem 4.6 with xn = 2.. 91. 4.19 Efficiency curves for problem 4.2 with xn = 2.. 94. 4.20 Efficiency curves for problem 4.3 with xn = 5.. 94. C ©. 80. xviii.

(24) 95. 4.22 Efficiency curves for problem 4.5 with xn = 10.. 95. 4.23 Efficiency curves for problem 4.6 with xn = 2.. 96. 5.5 5.6 5.7. U. Efficiency curves for problem 3.5 with xn = 1, 000 and h = 0.5/2i , i = 0, ..., 4.. 115. Efficiency curves for problem 3.6 with xn = 1, 000 and h = 0.125/2i , i = 0, ..., 5.. 115. Efficiency curves for problem 4.1 with xn = 1, 000 and h = 0.5/2i , i = 0, ..., 4.. 116. Efficiency curves for problem 3.2 with xn = 1, 000 and h = 0.125/2i , i = 2, ..., 5.. 122. Efficiency curves for problem 3.3 with xn = 1, 000 and h = 0.125/2i , i = 3, ..., 7.. 122. Efficiency curves for problem 3.5 with xn = 1, 000 and h = 0.5/2i , i = 0, ..., 4.. PY. 5.8. 114. T. 5.4. Efficiency curves for problem 3.3 with xn = 1, 000 and h = 0.125/2i , i = 3, ..., 7.. H. 5.3. 114. IG. 5.2. Efficiency curves for problem 3.2 with xn = 1, 000 and h = 0.125/2i , i = 2, ..., 5.. R. 5.1. PM. 4.21 Efficiency curves for problem 4.4 with xn = 2.. 123. Efficiency curves for problem 3.6 with xn = 1, 000 and h = 0.125/2i , i = 0, ..., 5.. 123. 5.10 Efficiency curves for problem 4.1 with xn = 1, 000 and h = 0.5/2i , i = 0, ..., 4.. 124. ©. C. O. 5.9. xix.

(25) LIST OF ABBREVIATIONS. EMSM. PM. U. T. H. PY. IMSM. IG. ITFMM. Initial value problem Linear multistep methods Ordinary differential equation Left-hand side Right-hand side Local truncation error Tolerance Order of the method Frequency weight dependent of the problem Step size Step number End of interval Maximum error Total number of function evaluations Total number of successful steps Total number of fail steps Total number of steps Time taken to compute the method in second The four-step explicit trigonometrically fitted multistep method proposed in chapter 3 The four-step implicit trigonometrically fitted multistep method proposed in chapter 3 The four-step explicit multistep method by Yap et al. (2011) The four-step implicit multistep method by Yap et al. (2011) The fourth order explicit two-step hybrid method by Franco (2006) The fourth order improved Runge-Kutta-Nystrom method with three stages by Rabiei et al. (2012) The fourth order phase fitted Runge-KuttaNystrom method by Papadopoulos et al. (2008) The four-step multistep hybrid method by Li and Wang (2016) The sixth order optimized embedded RungeKutta-Nystrom pair by Anastassi and Kosti (2015) The fourth order embedded Runge-KuttaNystrom method by Van de Vyver (2005) The fourth order Merson embedded Runge-Kutta method by Butcher (2008) The fourth order 2-point block multistep method derived in chapter 4 The sixth order 2-point block multistep method derived in chapter 4. R. IVP LMMs ODE LHS RHS LTE TOL p ω h k xn MAXERR FE SSTEP FSTEP NOS TIME(s) ETFMM. ETSHMs IRKNM. O. PFRKN. C. MSHMs. ERKN6(4). ©. ERKN4(3) MERK4(3) 2PBM4 2PBM6. xx.

(26) VSAM4 VSAM6 TF2PBM4. The fourth order variable step size Adams method The sixth order variable step size Adams method The fourth order trigonometrically fitted 2-point block multistep method derived in chapter 5 The sixth order trigonometrically fitted 2-point block multistep method derived in chapter 5 The sixth order explicit two-step hybrid method by Franco (2006) The sixth order phase fitted hybrid method by Senu et al. (2015) The sixth order explicit Numerov-type method by Tsitouras (2003) The three-step implicit Stormer-Cowell method obtained from Dormand (1996) The five-step implicit Stormer-Cowell method obtained from Dormand (1996) The fourth order explicit Runge-Kutta-Nystrom method by Dormand et al. (1987). PM. TF2PBM6 ETSHM6 PFHM6. U. NTM6 ISCM3. T. ISCM5. ©. C. O. PY. R. IG. H. DRKN4. xxi.

(27) ©. C T. H. IG. R. PY. O. U. PM.

(28) CHAPTER 1. 1.1. PM. INTRODUCTION Differential Equations. The Initial Value Problem. IG. 1.2. H. T. U. A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change and the equation defines a relationship between the two. Differential equations play a prominent role in many disciplines including engineering, physics, economics and biology. In mathematics, differential equations are studied from several different perspectives but mostly concerned with their solutions. Ordinary differential equation (ODE) is a differential equation contains only derivatives of one or more unknown functions with respect to a single independent variable.. Differential equation together with its initial value is called initial value problem (IVP). The initial value problem for a special second order ordinary differential equation is defined in the form of. R. y = f (x, y),. y(a) = η ,. y (a) = η ,. (1.2.1). PY. where the first derivative does not appear explicitly.. O. Theorem 1.1 (Existence and Uniqueness) Let f(x,y) be defined and continuous for all points (x,y) in the region D defined by a ≤ x ≤ b, −∞ < y < ∞, where a and b finite, and let there exist a constant L such that for any x ∈ [a, b] and any two numbers y and y*, | f (x, y) − f (x, y∗)| ≤ L|y − y ∗ |.. (1.2.2). ©. C. Then, if η is any given number, there exists a unique solution y(x) of the initial value problem (1.2.1), where y(x) is continuous and differentiable for all (x,y) in D.. The condition (1.2.2) is known as a Lipschitz condition, and the constant L as a Lipschitz constant. For proof, see (Henrici (1962)). In this work, we shall assume the theorem establishes the existence of a unique solution of (1.2.1). Such problems often arise in many scientific areas of engineering and applied science such as celestial mechanics, molecular dynamics and quantum mechanics. The solution of (1.2.1) also often exhibits a pronounced oscillatory character. It is well.

(29) 1.3. Numerical Methods for Ordinary Differential Equations. PM. known that it is rather difficult to get the accurate numerical results if the initial value problems are oscillatory in nature. A lot of research has been focused on developing methods to address the problem.. H. T. U. Numerical analysis involves the study of methods of computing numerical data. In many problems this implies producing a sequence of approximations by repeating the procedure again and again. Many differential equations cannot be solved analytically. For practical purposes, however, such as in engineering, a numeric approximation to the solution is often sufficient. Numerical methods for ODEs are methods used to find numerical approximations to the solutions of ODEs where this tecnique usually used by scientists and engineers to solve their problems. The implementation of a numerical method with an appropriate convergence check in a C Programming Language is called as a numerical algorithm.. Single Step and Multistep Methods. R. 1.4. IG. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. Generally, numerical methods often fall into one of two large categories that are single step and multistep methods. The methods also can be divided into explicit and implicit methods.. PY. Single step methods refer to only one previous point and its derivative to determine the current value. There are several well-known single step methods such as Euler method and Runge-Kutta method. Methods such as Runge-Kutta methods take some intermediate steps to obtain a higher order method and discard all the previous information before taking a second step.. ©. C. O. Methods that require more than one previous point to compute the approximation solution at the next point we called them as linear multistep methods (LMMs). Based on Lambert (1973), linear multistep method for special second order ODEs is defined by, k. k. j=0. j=0. ∑ α j yn+ j = h2 ∑ β j fn+ j ,. (1.4.1). where α j and β j are constant and assume αk = 1. Linear multistep method (1.4.1) is said to be explicit if βk = 0 and implicit if βk = 0. In this research, we are going to focus on the linear multistep method in the form of equation (1.4.1). Both Adams Bashforth and Adams Moulton methods are well-known explicit and implicit LMMs. 2.

(30) respectively.. PM. Definition 1.1 [Lambert (1973)] The linear multistep method (1.4.1) is said to be convergent if, for all initial value problems (1.2.1) subject to the hypothesis of the Theorem 1.1, we have that lim yn = y(x). (1.4.2). h→0 nh=x−a. h→0. Block Method. T. 1.5. U. holds for all x ∈ [a, b], and for all solutions yn of the difference equation (1.4.1) satisfying starting conditions yμ = ημ (h) for which lim ημ (h) = η , μ = 0, 1, 2, . . . , k − 1.. O. PY. R. IG. H. Block method is the approximation of solution for initial value problems at r points simultaneously. There are two types of block methods called as one-step block and multi-block. In one-step block method, the new block yn+1 , yn+2 , . . . , yn+r is computed from the value yn meanwhile in multi-block method, the new block is computed using the information from one or more previous blocks. Block method is an efficient method compared to the non-block method in terms of accuracy, total number of steps and execution times since it computes more than one value of the solutions at one time. Thus, the time period used and cost in solving the problems can be reduced.. Figure 1.1: 2-point block multistep method.. ©. C. In Figure (1.1), the approximation solutions of yn+1 and yn+2 are simultaneously computed at the points xn+1 and xn+2 where xn becomes the starting point and xn+2 is the last point in the block that has the step size 2h. The final values of yn+2 at the point xn+2 will then be taken as the initial values for the next iteration. The interval [a, b] is divided into a series of blocks that contained two points at each block.. 1.6. Trigonometrically Fitted Method. In order to find efficient methods for solving oscillatory problems, a lot of research has been focused on developing methods with reduced phase-lag and amplifica3.

(31) PM. tion error. Phase-lag or dispersion error is the angle between the true and the approximated solution while dissipation (amplification) error is the distance of the computed solution from the cyclic solution. The performance of numerical methods for solving oscillatory problems can be enhanced by phase fitting the method. Trigonometrically fitting the method means we assumed that the true solution is in the form of trigonometric functions and the methods are derived based on this assumption.. Order Conditions. H. 1.7. T. U. Using the ideas of trigonometrically fitting as in Fang and Wu (2008), the linear multistep method is required to integrate exactly the linear combination of (1.4.1) the functions e−iω t , eiω t or equivalently with {sin(ω t), cos(ω t)} for ω ∈ R and consider the test equation y = −ω 2 y with ω > 0. Let yn = eiω t , we have yn+1 = eiω (t+h) and by substituting eiH = cos(H) + i sin(H) where H = ω h to the (1.4.1), the fitted values will be obtained.. IG. Order conditions of the method are important to determine the order of the method. Based on linear multistep method (1.4.1), the linear difference operator L is defined as: L [y(x); h] =. k. ∑ [α j y(x + jh) − h2 β j y (x + jh)],. (1.7.1). j=0. R. where y(x) is an arbitrary function, continuously differentiable on [a,b]. By expanding y(x + jh) and y (x + jh) as Taylor series about point x, it gives. PY. L [y(x); h] = C0 y(x) +C1 hy(1) (x) + . . . +Cq hq y(q) (x) + . . . ,. (1.7.2). where Cq are constants.. ©. C. O. Based on equation (1.7.2), the order conditions for special second order ODEs are defined as follows:. 4.

(32) C0. =. k. ∑ αj. j=0. =. k. PM. C1. ∑ ( jα j ). j=0. C2. =. k. j2. ∑ ( 2! α j − β j ). j=0. =. k. j3. ∑ ( 3! α j − jβ j ). U. C3. j=0. .. . =. k. jq. jq−2. ∑ ( q! α j − (q − 2)! β j ),. q = 3, 4, . . .. (1.7.3). H. j=0. T. Cq. Stability of the Method. R. 1.8. IG. Following Henrici (1962), the method is said to has order p if C0 = C1 = · · · = C p = C p+1 = 0, C p+2 = 0; C p+2 is then the error constant of the method.. PY. Based on Lambert (1973), the first and second characteristic polynomial of linear multistep method (1.4.1) are defined as ρ (ζ ) and σ (ζ ) respectively:. ρ (ζ ) =. k. ∑ α jζ j,. σ (ζ ) =. j=0. k. ∑ β jζ j.. (1.8.1). j=0. y = λ y.. (1.8.2). C. O. Based on Lambert (1973), the test equation used for LMM (1.4.1) is,. ©. Next, definitions on certain types of stability of LMM will be given. Definition 1.2 [Lambert (1973)] The linear multistep method (1.4.1) is said to be zero-stable if no root of the first characteristic polynomial ρ (ζ ) has modulus greater than one, and if every root with modulus one has multiplicity not greater than two. Definition 1.3 [Fatunla (1991)] The block method is zero stable provided the roots R j , j = 1(1)k of the first charac5.

(33) teristic polynomial ρ (R) specified as k. ρ (R) = det[ ∑ A(i) Rk−i ] = 0, A(0) = −I. (1.8.3). PM. i=0. satisfies |R j | ≤ 1, and for those roots with |R j | = 1, the multiplicity must not exceed 2.. U. Definition 1.4 [Lambert (1973)] The linear multistep method (1.4.1) is said to be absolutely stable for a given h¯ if, for that h¯ , all the roots rs of stability polynomial, π (r, h¯ ) = ρ (r) − h¯ 2 σ (r) = 0, where h¯ = hλ satisfy |rs | < 1, s = 1, 2, . . . , k, and to be absolutely unstable for that h¯ otherwise.. IG. H. T. Definition 1.5 [Yap et al. (2011)] The linear multistep method (1.4.1) is said to have region of absolute stability D where D is a region of the complex h¯ -plane, if it is absolutely stable for all h¯ ∈ D. The intersection of D with the real axis is called the interval of absolute stability.. 1.9. The Objectives of the Thesis. R. Objectives of this research are as follows:. PY. 1. To construct the trigonometrically fitted explicit and implicit multistep methods of step number k=4 based on the existing methods derived by Yap et al. (2011) and to develop constant and variable step size based on the methods for solving oscillatory problems.. O. 2. To derive 2-point block multistep method of step number k=3 and k=5 and developing codes based on the methods using both constant and variable step size techniques for solving special second order ODEs. 3. To construct the trigonometrically fitted 2-point block multistep methods of step number k=3 and k=5 for solving oscillatory problems.. C. 4. To determine the zero stability and absolute stability of the new methods.. ©. 1.10. Outline of the Thesis. This thesis is arrange as follows: In Chapter 1, a brief introduction to the differential equation, initial value problem and numerical methods for ordinary differential equations are given. Definition 6.

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