iv AUTHOR’S STATEMENT I hereby agree to give the University of Lagos through University of Lagos Library, a non-exclusive, worldwide right to reproduce and distribute my thesis and abstract (hereinafter “the Work”) in whole or in part, by any and all media of distribution, in its present form or style or in any form or style as it may be translated for the purpose of future preservation and accessibility provided that such translation does not change its content. By the grant of non-exclusive rights of the University of Lagos are royalty free and that I am free to publish the Work in its present version or future versions elsewhere. Warranties I further agree as follows: i. That I am the author of the Work and I hereby give the University of Lagos the right to make available the Work in the way described above after a three (3) year period of the award of my doctorate degree in compliance with the regulation established by the University of Lagos Senate. ii. That the Work does not contain confidential information which should not be divulged to any third party without written consent. iii. That I have exercised reasonable care to ensure that the work is original and it does not to the best of my knowledge breach any Nigerian law or infringe any third party’s copyright or other Intellectual Property Right. iv. That to the extent that the Work contains material for which I do not hold copyright, I represent that I have obtained the unrestricted permission of the copyright holder to grant this license to the University of Lagos Library and that such third party material is clearly identified and acknowledged in the Work. v. In the event of a subsequent dispute over the copyrights to material contained in the work, I agree to indemnify and hold harmless the University of Lagos and all of its officers, employees and agenda for any uses of the material authorized by this agreement. vi. That the University of Lagos has no obligation whatsoever to take legal action on my behalf as the Depositor, in the event of breach of Intellectual property rights, or any other right, in the material deposited. Author’s Name Signature/Date Email Supervisor’s Name Signature/Date Email Supervisor’s Name Signature/Date Email v DEDICATION This Thesis is solely dedicated to my parents who supported me enough to reach this pinnacle of education. vi ACKNOWLEDGEMENTS All praises and adorations are due to Almighty Allah who taught by the pen and taught man that which he knew not. His name be praised always for His guidance through the completion of this program. To You alone be all the glory. My unquantifiable appreciation goes to my supervisor and academic mentor, Professor S. A. Okunuga, for his thorough supervision, invaluable advice, guidance, encouragements and constructive criticisms. His enormous contributions to my academic pursuit is highly acknowledged and appreciated. I would like to appreciate the immense contribution of my co-supervisor, Dr. O. A. Akinfenwa, towards the success of this work. I appreciate his thoroughness, constructive criticisms and computational contributions to the research work. A sincere appreciation goes to the HOD and all academic staff of the Department of Mathematics for their constructive criticisms and suggestions in the various seminar presentations which greatly improve the research work. I am indebted to Professor J. O. Olaleru and Dr. A. Adeniyan for their fatherly roles during the course of the program. Special thanks to the Departmental postgraduate coordinator, Dr. A. A. Mogbademu, for his brotherly role throughout the program. Also, to the members of the Numerical group which I belong, I thank you all, for you have been wonderful. To all graduate students of the Department of Mathematics University of Lagos, thank you for all your supports and encouragement. I will like to thank Professor R. B. Adeniyi of University of Ilorin for facilitating my contact with Dr. T. A. Biala. I would also like to appreciate the following Numerical analysts who did not only provide me with their research work but also found time to reply all my e-mails and offer their professional advice, Professor S. N. Jator, Professor H. Ramos, Dr. T.A. Biala and Dr. F. F. Ngwane. To Professor A. P. Akinola, Dr. P. O. Layeni and Dr. B. S. Ogundare for your warm receptions, encouragement and advice during my visits to Obafemi Awolowo University. I say a big thank you. Special thanks to the vii Clement’s family for the hospitality received while I was at Ilesha during this program. Abdulkareem Ridwan you are wonderful for taking part in the typing of this research work. To Mr. Oshunkayode Abdulkabir and Mr. A. O. Adeniran, I say thank you for putting me through the use of Maple Software. To all maple user on mapleprime.com, thank you for your computational contributions, corrections and suggestions. Also, I appreciate the encouragement of Ustadh Sulaymon AbdusSalam Alageshinwo throughout this program. Many thanks to my colleagues in the Distance Learning Institute, University of Lagos. I also will like to thanks my friends, home and abroad, for their immeasurable support throughout this program. This appreciation will not be completed without thanking those that supported me financially through this program, the Amusat family particularly, Alh. Azeez Amusat and Musilihudeen Amusat, Alh. Taofeek Fagbohun (Dearest Uncle), Mr. Tijani, Miss Shuhudah AbdulJabar, University of Lagos Muslim Alumni (UMA), Mr. Adekunle Hassan and the host of others. I am very grateful to my parents, my parents-in-law and to the rest of my family members for all their support, prayers and faith in me. My soulmate Afusat Adesewa and my beautiful daughters: Umukulthum and Maymunah, I appreciate your cooperation and endurance throughout this program. I love you all. Oh Allah grant me the power and ability to be grateful for all Your Favours upon me and upon my parent and to do righteousness which You approve, and admit me by Your mercy among Your righteous servants. viii TABLE OF CONTENTS Page Cover page i Title page ii Certification iii Author’s Statement iv Dedication v Acknowledgements vi Table of contents viii List of Figures xv List of Tables xvii List of Abbreviations and Acronyms xxii Abstract xxv CHAPTER ONE INTRODUCTION 1.1 Background to the Study 1 1.2 Statement of the Problem 2 1.3 Objectives of Study 3 1.4 Scope and Delimitation of the Study 3 1.5 Significance of the Study 4 1.6 Operational Definition of Terms 4 1.7 Basic Definitions 4 1.8 Some Important Definitions 5 ix CHAPTER TWO LITERATURE REVIEW 2.1 Numerical Methods for Initial Value Problems 7 2.2 One step Methods 8 2.2.1 Euler Method 8 2.2.2 Runge-Kutta Method 8 2.3 Linear Multistep Methods 10 2.4 Backward Differentiation Formula 12 2.5 Hybrid Methods 13 2.6 Enright Second Derivative Formulae 14 2.7 Numerical Scheme for Solving Oscillatory Initial Value Problem 14 2.7.1 Non-Fitted Methods 14 2.7.2 Methods Based on Exponential Fitting 17 2.7.3 Trigonometric Fitted Methods 19 2.8 Choice of Parameter 𝑢 in the Stability Analysis of the Trigonometrically 23 Fitted Schemes 2.9 Estimation of Computational Frequency 24 2.9.1 Method due to Vander-Berghe et al. (2001) 25 2.9.2 Methods due to Holleovet, Van Daele and Vander-Berghe (2009) 25 and D’Ambrosio et al. (2.012a and 2012b) 2.9.3 Method due to Ngwane and Jator (2013a and 2013b) 26 x CHAPTER THREE METHODOLOGY 3.1 Theoretical Framework 27 3.1.1 Existence and Uniqueness of Solution of Ordinary Differential 27 Equations 3.1.2 Approximation of Function 28 3.1.3 Collocation Technique 29 3.2 Theoretical Procedure for the Second Derivative 30 Trigonometrically Fitted Block Backward Differentiation Formula (TFBBDF) 3.3 Theoretical Properties of TFBBDF 35 3.3.1 Local Truncation Error of TFBBDF 35 3.3.2 Zero Stability of 𝑘 −step TFBBDF 37 3.3.3 Consistency of 𝑘 −step TFBBDF 38 3.3.4 Convergence of 𝑘 −step TFBBDF 38 3.3.5 Linear Stability of TFBBDF 40 3.4 Specification of 𝑘 −step Second Derivative 41 Trigonometrically-Fitted Block Backward Differentiation Formula 3.4.1 Derivation of Second Derivative Trigonometrically Fitted 42 Block Backward Differentiation Formula for 𝑘 = 2 3.4.2 Derivation of Second Derivative Trigonometrically-Fitted 47 Block Backward Differentiation Formula for 𝑘 = 3 3.4.3 Derivation of Second Derivative Trigonometrically-Fitted 55 Block Backward Differentiation Formula for 𝑘 = 4 xi 3.5 Theoretical Procedure for the Second Derivative 65 Trigonometrically-Fitted Block Schemes of Simpson Type (TFBSST) 3.6 Theoretical Properties of TFBSST 68 3.6.1 Local Truncation Error of TFBSST 68 3.6.2 Linear Stability of TFBSST 69 3.6.3 Consistency of 𝑘 −step TFBSST 70 3.6.4 Convergence of 𝑘 −step TFBSST 70 3.7 Specification of the 𝑘 −step Second Derivative 73 Trigonometrically Block Scheme of Simpson Type 3.7.1 Derivation of Second Derivative Trigonometrically Block 73 Scheme of Simpson Type for 𝑘 = 2 3.7.2 Derivation of Second Derivative Trigonometrically Block 77 Scheme of Simpson Type for 𝑘 = 3 3.7.3 Derivation of Second Derivative Trigonometrically Block 86 Scheme of Simpson Type for 𝑘 = 4 3.8 Analysis of Second Derivative Trigonometrically-Fitted 93 Block Backward Differentiation Formula (TFBBDF) 3.8.1 Analysis of 2 −Step TFBBDF 93 3.8.1.1 Local Truncation Error, Error Constant and Order 93 3.8.1.2 Order of 2 −step TFBBDF 93 3.8.1.3 Consistency of TFBBDF 93 3.8.1.4 Stability of TFBBDF 93 3.8.1.5 Zero Stability 94 xii 3.8.1.6 Convergence of 2 −step TFBBDF 94 3.8.1.7 Linear Stability and Region of Absolute Stability of 2 −step 94 TFBBDF 3.8.1.8 Test for 𝐿 stability 95 3.8.2 Analysis of 3 −Step TFBBDF 96 3.8.2.1 Local Truncation Error 96 3.8.2.2 Order and Error Constant of 3 −step TFBBDF 96 3.8.2.3 Consistency of TFBBDF 96 3.8.2.4 Stability of 3 −step TFBBDF 96 3.8.2.5 Zero Stability 97 3.8.2.6 Convergence of 3 −step TFBBDF 97 3.8.2.7 Linear Stability and Region of Absolute Stability of 97 3 −Step TFBBDF 3.8.2.8 Test for 𝐿(𝛼) −Stability 98 3.8.3 Analysis of 4 − step SDTFBBDF 99 3.8.3.1 Local Truncation Error 99 3.8.3.2 Order and Error Constant of 4 −step TFBBDF 99 3.8.3.3 Consistency of TFBBDF 99 3.8.3.4 Stability of 4 −step TFBBDF 99 3.8.3.5 Zero Stability 100 3.8.3.6 Convergence of 4 −step TFBBDF 100 3.8.3.7 Linear Stability and Region of Absolute Stability of 4 −step 100 TFBBDF 3.8.3.8 Test for 𝐿(𝛼) −Stability 102 xiii 3.9 Analysis of Second Derivative Trigonometrically-Fitted 102 Block Scheme of Simpson Type (TFBSST) 3.9.1 Analysis of 2 −Step TFBSST 102 3.9.1.1 Local Truncation Error 102 3.9.1.2 Order and Error of 2 −step TFBSST 102 3.9.1.3 Consistency of TFBSST 102 3.9.1.4 Stability of 2 −step TFBSST 103 3.9.1.5 Zero Stability 103 3.9.1.6 Convergence of 2 −step TFBSST 103 3.9.1.7 Linear Stability and Region of Absolute Stability of 103 2 −Step TFBSST 3.9.2 Analysis of 3 −Step TFBSST 105 3.9.2.1 Local Truncation Error 105 3.9.2.2 Order and Error Constant of 3 −step TFBSST 105 3.9.2.3 Consistency of TFBSST 106 3.9.2.4 Stability of 3 −step TFBSST 106 3.9.2.5 Zero Stability 106 3.9.2.6 Convergence of 3 −step TFBSST 106 3.9.2.7 Linear Stability and Region of Absolute Stability of 106 3 −Step TFBSST 3.9.3 Analysis of 4 −Step TFBSST 107 3.9.3.1 Local Truncation Error 107 3.9.3.2 Order and Error Constant of 4 −step TFBSST 108 3.9.3.3 Consistency of TFBSST 108 xiv 3.9.3.4 Stability of 4 −step TFBSST 108 3.9.3.5 Zero Stability 109 3.9.3.6 Convergence of 4 −step TFBSST 109 3.9.3.7 Linear Stability and Region of Absolute Stability of 109 4 −Step TFBSST CHAPTER FOUR RESULTS AND DISCUSSION 4.1 Implementation of the Derived Methods 111 4.2 Numerical Examples 111 4.3 Implementation of Coefficients Trigonometrically Versus 184 Implementation of Coefficients in Power Series Form CHAPTER FIVE SUMMARY AND CONCLUSION 5.1 Summary of Findings 188 5.2 Conclusion 188 5.3 Contributions to Knowledge 189 5.4 Suggestion for Further Study 190 References 191 Appendices 204 xv LIST OF FIGURES Page Figure 2.1 Categories of Numerical Schemes for IVP 7 Figure 2.2 Region of stability of Adams Bashfort Methods 12 Figure 2.3 Region of stability of Adams Moulton Method 12 Figure 3.1 Region of Absolute Stability of 2 −step TFBBDF 95 Figure 3.2 Region of Absolute Stability of 3 −step TFBBDF 98 Figure 3.3 Region of Absolute Stability of 4 −step TFBBDF 101 Figure 3.4 Region of Absolute Stability of 2 −step TFBSST 104 Figure 3.5 Region of Absolute Stability of 3 −step TFBSST 107 Figure 3.6 Region of Absolute Stability of 4 −step TFBSST 110 Figure 4.1 Line Graph Comparison of Absolute Error 114 Figure 4.2 Bar Chart Comparison of TFBBDF and TFBSST 115 Figure 4.3 Bar Chart of Absolute Error of TFBBDF and TFBSST 117 Figure 4.4a Bar Chart Comparison of TFBBDF and TFBSST with ℎ = 𝜋 30 119 Figure 4.4b Bar Chart Comparison of BBDF1 and BBDF2 with ℎ = 𝜋 30 120 Figure 4.5 Bar Chart Comparison of TFBBDF and TFBSST with ℎ = 𝜋 60 121 Figure 4.6 Comparison of Efficiency Curves 124 Figure 4.7 Efficiency Curves for TFBBDF and TFBSST 125 Figure 4.8 Bar Chart Comparison of Global Errors 127 Figure 4.9 Bar Chart Comparison of Global Error of TFBBDF and TFBSST 128 Figure 4.10 Comparison of Efficiency Curves 132 Figure 4.11 Comparison of Efficiency Curves 132 Figure 4.12 Bar Chart Comparison of TFBBDF and TFBSST with 𝛽 = 𝜋 30 133 xvi Figure 4.13 Bar Chart Comparison of TFBBDF and TFBSST with 𝛽 = −1000 134 Figure 4.14 Bar Chart Comparison of Maximum Absolute Errors 144 Figure 4.15 Line Graph Comparison of End Point Absolute Error 147 Figure 4.16 Bar Chart Comparison of Euclidean Norm and End Point Global Error 150 Figure 4.17 Bar Chart Comparison of Euclidean Norm Global Error of TFBBDF 151 and TFBSST Figure 4.18 Comparison of Efficiency Curves 152 Figure 4.19 Comparison of Efficiency Curve of TFBBDF and TFBSST 153 Figure 4.20 Comparison of Efficiency Curves 156 Figure 4.21 Comparison of Efficiency Curves 163 Figure 4.22 Comparison of Efficiency Curves 164 Figure 4.23 Comparison of Efficiency Curves 168 Figure 4.24 Comparison of Efficiency Curves 168 Figure 4.25 Comparison of Efficiency Curves 169 Figure 4.26 Comparison of Efficiency Curves 176 Figure 4.27 Comparison of Efficiency Curves 177 Figure 4.28 Comparison of Efficiency Curves 180 Figure 4.29 Comparison of Efficiency Curves 181 Figure 4.30 Comparison of Efficiency Curves 183 xvii LIST OF TABLES Page Table 2.1 Butcher Tableau for classical fourth order Runge-Kutta method 9 Table 2.2 Butcher Tableau for two-stage implicit Runge-Kutta method 10 for order four Table 2.3 Coefficients and error constants of Adams Bashforth methods 11 Table 2.4 Coefficients and error constants of Adams Moulton methods 11 Table 2.5 Coefficients and error constants of BDF (Lambert, 1973) 13 Table 2.6 Coefficients and error constants of ESDF 14 Table 4.1a Exact Solution and Computed Solution for Equation 4.1 112 Table 4.1b Comparison of Absolute Error 113 Table 4.2 Comparison of Absolute Error 114 Table 4.3 Absolute Error s of Classes of TFBBDF and TFBSST 114 Table 4.4a Exact Solution and Computed Solution for equation 4.3 for 115 ℎ = 10−2 and 𝜔 = 0.5 Table 4.4b Comparison of Absolute Errors for ℎ = 10−2 and 𝜔 = 0.5 116 Table 4.5 Comparison of Absolute Errors of TFBBDF and TFBSST 116 for different values of ℎ and for ℎ = 10−2 and 𝜔 = 0.5 Table 4.6a Exact Solution and Computed Solution for Equation 4.5 for 117 ℎ = 0.01 Table 4.6b Comparison of Maximum Errors 118 Table 4.7a Exact Solution and Computed Solution for Equation 4.7 for 119 ℎ = 𝜋 30 at 𝑥 = 2𝜋 Table 4.7b Comparison of Absolute Errors with ℎ = 𝜋 30 119 Table 4.8 Comparison of Absolute Errors with ℎ = 𝜋 60 120 xviii Table 4.9 Comparison of Absolute Errors with ℎ = 0.02 121 Table 4.10 Comparison of End Point Absolute Errors 122 Table 4.11 Comparison of Maximum Errors 123 Table 4.12 Comparison of Maximum Errors of TFBBDF and TFBSST 123 Table 4.13 Comparison of Maximum Errors 124 Table 4.14 Comparison of Maximum Errors of TFBBDF and TFBSST 125 Table 4.15a Exact Solution and Computed Solution for Equation 4.13 for ℎ = 0.1 126 Table 4.15b Comparison of Global Errors with ℎ = 0.1 127 Table 4.16 Comparison of Global Error of Classes of TFBBDF and TFBSST 128 Table 4.17 Comparison of Absolute Errors of TFBBDF2, TFBSST2 and EF 129 for 𝜔 = 1 , ℎ = 0.1 2𝑖 , 𝑖 = 0(1)2 and 𝛽 = −10 (a non stiff case) Table 4.18 Comparison of Absolute Errors of TFBBDF2, TFBSST2 and EF 130 for 𝛽 = −1000 (a stiff case) Table 4.19 Comparison of Absolute Errors of TFBBDF2, TFBSST2 and TRK 130 for 𝛽 = −1000 (a stiff case) Table 4.20 Comparison of Absolute Errors for 𝛽 = −3 130 Table 4.21 Comparison of Absolute Errors for 𝛽 = −1000 131 Table 4.22 Comparison of Absolute Errors for 𝛽 = −3 132 Table 4.23 Comparison of Absolute Errors for 𝛽 = −1000 133 Table 4.24 Comparison of Global Absolute Errors 135 Table 4.25 Comparison of Rate of Convergence and Relative Errors 136 Table 4.26 Comparison of Maximum Absolute Errors 136 Table 4.27 Comparison of Maximum Endpoint Global Errors 137 Table 4.28 Comparison of Absolute Errors for 𝜔 = 8 139 xix Table 4.29 Comparison of Absolute Errors of TFBBDF and TFBSST 140 for 𝜔 = 8 Table 4.30 Comparison of Absolute for 𝜔 = 10 and ℎ = 𝜋 200 141 Table 4.31 Comparison of Absolute Errors for 𝜔 = 10 and ℎ = 𝜋 12 of TFBBDF 141 and TFBSST with TFBTDM of Jator (2015) of order 6 Table 4.32 Comparison of Absolute Errors 142 Table 4.33 Comparison of 𝐿∞ −norms errors with ℎ = ( 1 2 ) 𝑚 , 𝑚 = −1(1)10 142 Table 4.34 Comparison of Maximum Absolute Error for 𝜔 = 1, ℎ = 𝜋 2𝑖 𝑖 = 2(1)9 143 Table 4.35 Comparison of Absolute Error 145 Table 4.36 Comparison of Global Errors for ℎ = 1 10𝑚 𝑚 = 1(1)4 146 Table 4.37a Exact Solution and Computed Solution for Equation 4.31 146 Table 4.37b Comparison of End Point Absolute Errors 147 Table 4.38 Global Error of Classes of TFBBDF and TFBSST 148 Table 4.39 Comparison of Sdmax values 148 Table 4.40 Comparison of Sdmax for the classes of TFBBDF and TFBSST 149 Table 4.41a Exact Solution and Computed Solution for Equation 4.35 for 𝑥 = 10 149 Table 4.41b Comparison of Euclidean Norms of End Point Global Error 150 Table 4.42 Comparison of Euclidean Norms of End Point Global Error 150 of TFBBDF and TFBSST Table 4.43a Exact Solution and Computed Solution for Equation 4.37 at 𝑥 = 10 152 Table 4.43b Comparison of Absolute Error and Number of Functions Evaluation 152 Table 4.44 Comparison of Absolute Error and Number of Functions Evaluations 153 of TFBBDF and TFBSST xx Table 4.45a Exact Solution and Computed Solution for Equation 4.39 at 𝑥 = 10 154 Table 4.45b Comparison of Global Errors 155 Table 4.46 Comparison of Absolute Errors for ℎ = ( 1 2 ) 𝑚 and 𝜔 = 10 157 Table 4.47 Comparison of Absolute Errors 158 Table 4.48 Comparison of Absolute Errors 158 Table 4.49 Comparison of Absolute Errors 159 Table 4.50 Comparison of End Point Absolute Errors 160 Table 4.51 Comparison of End Point Absolute Errors of TFBBDF and TFBSST 160 Table 4.52a Exact Solution and Computed Solution for Equation 4.41 for ℎ = 𝜋 5 161 Table 4.52b Comparison of Absolute Errors for ℎ = 𝜋 5 162 Table 4.53 Comparison of Absolute Errors for ℎ = 𝜋 12 162 Table 4.54 Comparison of End Point Absolute Errors for 𝑚 = 40.5𝜋 1.01 163 Table 4.55 Comparison of CPU Time 164 Table 4.56 Comparison of End Point Global Error 165 Table 4.57 Comparison of Maximum Errors 166 Table 4.58 Comparison of End Point Global Errors 167 Table 4.59 Comparison of 𝐿∞Norm Errors 168 Table 4.60 Computed Values for 𝛾, 𝑥 = 40𝜋 , 𝑦(𝑥) = 1.001972 170 Table 4.61 Computed of Absolute Errors Values for 𝐸𝑟𝑟(𝛾) and 𝐸𝑟𝑟 (𝑍), 171 𝑦(𝑥) = 1.001972 , 𝑥 = 40𝜋 Table 4.62 Comparison of Maximum Global Errors 172 Table 4.63 Comparison of Maximum Absolute Error and Absolute Errors 173 of the Real Part xxi Table 4.64 Comparison of Maximum Absolute Error and Absolute Errors of the 174 Imaginary Part Table 4.65 Comparison of Maximum Errors 175 Table 4.66 Comparison of Maximum Errors and Number of Functions Evaluation 176 Table 4.67 Comparison of Maximum Errors and CPU Time 177 Table 4.68 Comparison of Absolute Errors 178 Table 4.69 Comparison of Maximum Errors and Number of Functions Evaluation 179 Table 4.70 Comparison of Maximum Errors and Number of Functions Evaluation 180 Table 4.71 Comparison of Maximum Errors and Number of Functions Evaluation 182 Table 4.72 Comparison of Maximum Absolute Error 183 Table 4.73 Comparison of Absolute Error 184 Table 4.74 Comparison of Maximum Error 184 Table 4.75 Comparison of Absolute Error with 𝛽 = −10 (a non stiff case) 185 Table 4.76 Comparison of Absolute Error with 𝛽 = −1000 (a non stiff case) 185 Table 4.77 Comparison of Maximum Absolute Error 186 Table 4.78 Comparison of Global Errors 187 Table 5.1 Summary of Findings 188 xxii LIST OF ABBREVIATIONS AND ACRONYMS IIb Mixed Collocation Methods of Algebraic order 4 with 2 stages IIIa Mixed Collocation Methods of Algebraic order 4 with 3 stages IIIb Mixed Collocation Methods of Algebraic order 6 with 3 stages IVb Extended Mixed Collocation Methods of Algebraic order 4 with 2 stages 2PBOSM Two-point block one-step method ARKC Absolute stable Runge-Kutta Collocation method BBDF1 Block Backward Differentiation Formula of order four BBDF2 Block Backward Differentiation Formula of order five BBDF4 Block Backward Differentiation Formula of order four BBDF5 Block Backward Differentiation Formula of order five BDF Backward Differentiation Formula BHMTB Block Hybrid Method with Trigonometric Basis BHSM2 Block Hybrid Simpson’s Method with Two off-grid points BHT Block Hybrid Trigonometrically-Fitted Method BHTFM Block Hybrid Method with Trigonometrically-Fitted Method BTFEBDM Block Trigonometrically-Fitted Extended Backward Differentiation Method CHEB I Zeros of Chebyshev polynomial of the first kind CHEB II Zeros of Chebyshev polynomial of the second kind CHEBY24 Dissipative Chebyshev Exponential-Fitted Methods CTDBM Continuous Third Derivative Block methods DIRK Diagonally Implicit Runge-Kutta method DIRKN Diagonally Implicit Runge-Kutta Nystrom method E2PBN Explicit 2-Point 1-Block Method xxiii E3PBN Explicit 3-Point 1-Block Method EF Exponentially Fitted EFRK Exponentially Fitted Runge-Kutta EOPM Explicit One-step P-Stable Method EQPTS Equidistance points in [𝑥0, 𝑏] ERK4 Explicit phase fitted Runge Kutta of order four ESDM Enright Second Derivative Method ETSHM5TF Explicit Two Step Hybrid Modified Fifth order Trigonometrically-Fitted ETSHM6TF Explicit Two Step Hybrid Modified Sixth order Trigonometrically-Fitted ETSHM7TF Explicit Two Step Hybrid Modified Seventh order Trigonometrically-Fitted HLMM Hybrid Linear Multistep Method of order seven HMB Exponentially Fitted Hybrid Method HSDM Hybrid Second Derivative Methods IVP Initial Value Problem LMM Linear Multistep Methods LTE Local Truncation Error MRK Modified Runge-Kutta-Nyström N4 Fourth order Runge-Kutta-Nyström Method NTDRK New Trigonometrically-Fitted Two-derivative Runge-Kutta NFE Number of function evaluation NVSBBDF New Variable Step size Block Backward Differentiation Formula ODE Ordinary Differential Equation PDE Partial Differential Equation PSM Power Series Method xxiv RK Runge-Kutta RKN Runge-Kutta-Nyström SDBBDF Second Derivative Block Backward Differentiation Formula SDTFF Second Derivative Trigonometrically-Fitted Block Backward Differentiation Formula of Adams Type SSDM Simpson Second Derivative Method TBNM Trigonometrically-Fitted Block Numerov Type Method TFBBDF2 Second Derivative Trigonometrically-Fitted Block Backward Differentiation Formula for 𝑘 = 2 TFBBDF3 Second Derivative Trigonometrically-Fitted Block Backward Differentiation Formula for 𝑘 = 3 TFBBDF4 Second Derivative Trigonometrically-Fitted Block Backward Differentiation Formula for 𝑘 = 4 TFBSST2 Second Derivative Trigonometrically-Fitted Block Scheme of Simpson’s Type for 𝑘 = 2 TFBSST3 Second Derivative Trigonometrically-Fitted Block Scheme of Simpson’s Type for 𝑘 = 3 TFBSST4 Second Derivative Trigonometrically-Fitted Block Scheme of Simpson’s Type for 𝑘 = 4 TFBTDM Trigonometrically-Fitted Block Third Derivative Method TIRK3 Three stages Trigonometrically Implicit Runge-Kutta TS Total Step TSDM Trigonometrically-Fitted Second Derivative Method TTRKNM Trigonometrically-Fitted implicit Third Derivative Runge-Kutta-Nystrom Method VSSBBDF Variable Stepsize Superclass Block Backward Differentiation Formula xxv ABSTRACT This study develops two classes of Second Derivative Trigonometrically-Fitted Block Schemes for the numerical integration of oscillatory IVPs using collocation techniques. The two classes of methods are the Second Derivative Trigonometrically-Fitted Block Backward Differentiation Formula (TFBBDF) and Second Derivative Trigonometrically-Fitted Block Scheme of Simpson Type (TFBSST). The Trigonometrically-Fitted Methods for each scheme depend on the step size and frequency that are constructed using trigonometric basis function. The continuous Second Derivative Trigonometrically- Fitted Method for each scheme is used to generate the main method. The additional 𝑘 − 1 complementary methods for TFBBDF are obtained from the second differentiation of its continuous form, while the complementary methods of TFBSST are obtained from the same continuous method as main method. The main and complimentary methods in their converted power series form are combined and applied in block form as simultaneous numerical integrators. The stability properties for both classes are investigated using boundary locus plot. It is found that both classes are zero stable, consistent and convergent. The class of 𝑘`-step TFBBDF is of order 2𝑘 + 1 while that of 𝑘`-step TFBSST of order 2𝑘 + 2. Both classes of the methods are applied on a number of numerical examples and the results showed that they are more accurate and more efficient for oscillatory problems when compared with existing methods in the literature reviewed in this work. Keywords: Backward Differentiation Formula, Collocation technique, Oscillatory Problems, Simpsons Type, Trigonometrically-Fitted methods.