| Titre : | Application of Metaheuristics in Solving Initial Value Problems (IVPs) |
| Auteurs : | Fatima Ouaar, Auteur ; Naceur Khelil, Directeur de thèse ; Redouane Boudjemaa, Directeur de thèse |
| Type de document : | Monographie imprimée |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2021 |
| Format : | 1 vol. (167 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Initial Value Problem (IVP),Optimization problem,Exponential problem,Logistic problem,FLFBA,Numerical methods,Metaheuristic algorithms |
| Résumé : |
Some differential equations admit analytic solutions given by explicit formulas. However, in most other case only approximated solutions can be found. Several methods are available in the literature to find approximate solutions to differential equations. Numerical methods form an important part of solving IVP in ODE, most especially in cases where there is no closed form of solutions.
The present dissertation focus the attention toward solving IVP by transforming it to an optimization approach which can be solved through the application of non-standard methods called Metaheuristic. By transforming the IVP into an optimization problem, an objective function, which comprises both the IVP and initial conditions, is constructed and its optimum solutions represents an approximative solution of the IVP. The main contribution of the present thesis is divided in twofold. In the one hand, we consider IVPs as an optimization problem when the search of the optimum solution is performed by means of MAs including ABC, BA and FPA and a set of numerical methods including Euler methods, Runge–Kutta methods and predictor corrector methods. On the other hand, we propose a new MA called Fractional L´evy Flight Bat Algorithm (FLFBA) (which is an improvement of the BA, based on velocity update through fractional calculus and local search procedure based on a L´evy distribution random walk). We illustrates its computational efficiency by comparing its performance with the previous methodds in solving the bacterial population growth models ( both the logistic growth model and the exponential growth model). |
| Sommaire : |
Dedication i
Acknowledgements ii Summary iv Summary in Arabic v Achieved Works v Table of Acronyms vii Table of Contents ix List of Figures xiii List of Tables xiv General Introduction 1 I Preliminary Theory 10 1 First Order Initial Value Problems 11 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Classification of Differential Equations . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Partial vs. Ordinary . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 First Order, Second Order . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.3 Linear vs. Nonlinear . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ix1.2.4 Homogeneous vs. Non-homogeneous . . . . . . . . . . . . . . . . . 15 1.3 Solutions to Differential Equations . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Initial Value Problems (IVPs) . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.1 Definition of IVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.2 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . 18 1.5 Numerical Solutions of First-Order IVP . . . . . . . . . . . . . . . . . . . . 19 1.5.1 Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.2 Implicit methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.3 Higher-order methods . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.4 Multistep methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.6 Real-life applications of IVP . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Metaheuristics As Optimization Algorithms 29 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.2 Search for optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.3 Optimization algorithms . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.4 Parameters of an Optimization Algorithm . . . . . . . . . . . . . . 33 2.3 Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.1 Definition of metaheuristics . . . . . . . . . . . . . . . . . . . . . . 34 2.3.2 Properties of metaheuristics . . . . . . . . . . . . . . . . . . . . . . 35 2.3.3 Classification of metaheuristics . . . . . . . . . . . . . . . . . . . . 35 2.3.4 Applications of Metaheuristics . . . . . . . . . . . . . . . . . . . . . 38 2.4 Metheuristic Methods for ODEs . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 II Main Results 41 3 Fractional L´evy Flight Bat Algorithm (FLFBA) 42 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 x3.2 Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.1 Basic bat algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.2 Fractional-order calculus . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.3 Bat algorithm modified equations . . . . . . . . . . . . . . . . . . . 46 3.2.4 L´evy flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.5 DE-based location update formula . . . . . . . . . . . . . . . . . . . 48 3.3 Fractional L´evy Flight Bat Algorithm . . . . . . . . . . . . . . . . . . . . . 49 3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.1 Parameters settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.2 Results Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.3 Post-hoc Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Application of FLFBA in Optimizing IVP 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Population Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.1 Exponential growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.2 Logistic growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.1 Application example . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.2 Parameters adopted to solve IVP . . . . . . . . . . . . . . . . . . . 63 4.4.3 Comparison of FLFBA with numerical methods . . . . . . . . . . . 65 4.4.4 Comparison of FLFBA with metaheuristic algorithms . . . . . . . . 68 4.4.5 Time taken for the algorithms . . . . . . . . . . . . . . . . . . . . . 71 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 General Conclusion 73 4.6 Bilan of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 xiTable of Contents 4.7 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Bibliography 76 A What is MATLAB? 86 B FLFBA tables 101 B.1 Benchmark functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 B.2 Multiple comparison tests tables . . . . . . . . . . . . . . . . . . . . . . . . 101 B.3 Post-hoc procedures tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 C Statistical tests 125 C.1 Multiple comparison tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 C.2 Post-hoc procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 C.3 Unadjusted p-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 D FLFBA matlab code 143 E Background 163 E.1 Artificial Bee Colony Algorithm (ABCA) . . . . . . . . . . . . . . . . . . . 163 E.2 Bat Algorithm (BA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 E.3 Flower Pollination Algorithm (FPA) |
| En ligne : | http://thesis.univ-biskra.dz/5356/1/OUAAR%20Fatima%20Doctoral%20Thesis.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
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| TM/109 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
