| Titre : | Introduction to Evolutionary Algorithms on Numerical Calculations |
| Auteurs : | Fatiha Ghedjemis, Auteur ; Naceur Khelil, Directeur de thèse |
| Type de document : | Thése doctorat |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2025 |
| Format : | 1 vol. (137 p.) |
| Langues: | Anglais |
| Mots-clés: | Di¤erentialequations ; Metaheuristicalgorithms ; Chebyshevpolynomials ; FlowerPollinationAlgorithm. |
| Résumé : |
This thesis proposes a new hybrid computational method that combines the accuracy of spectral methods with the optimization abilities of the Flower Pollination Algorithm to find solutions of differential equations, particularly boundary value problems. The approach uses Chebyshev polynomials for spectral approximation and combines FPA to minimize residual errors and optimize the coefficients, leading to accurate numerical solutions. The study begins by exploring the structures of spectral methods and metaheuristic algorithms, concentrating on their mathematical properties and practical roles in optimization. It then introduces a new three-step hybrid methodology: extracting an initial approximation, calculating the residual error, and optimizing undetermined coefficients via FPA. The efficiency of this method is confirmed through several case studies, involving linear and nonlinear boundary value problems. Experimental results validate that the proposed hybrid approach improves solution accuracy and computational efficiency contrast classical methods. The findings highlight the method’s adaptability and potential in broader applications such as fluid dynamics, structural analysis, and data-driven modelling. This work contributes a robust and flexible approach for solving complex differential problems, paving the way for future research in advanced numerical and optimization strategies. |
| Sommaire : |
Contents Dedicace i Acknowledgements ii Abstract iii Achieved work v Abreviations and Notations vi Contents vii List of Figures ix List of Tables x Introduction 1 1 Spectral Methods 6 1.1 Di¤erential Equations and Mathematical Formulation............ 6 1.2 Di¤erential equation sand types........................ 8 1.3 Chebyshev Polynomials............................. 10 1.3.1First-Kind Chebyshev Polynomials.................. 11 1.3.2Second-KindChebyshevPolynomials................. 12 viii 1.3.3Chebyshev Polynomialsin[a,b].................... 14 1.3.4ShiftedChebyshevPolynomials.................... 15 1.4NumericalMethods............................... 16 1.4.1LocalMethods............................. 16 1.4.2GlobalMethods............................. 16 1.4.3CollocationMethodUsingChebyshevPolynomials.......... 18 1.5Conclusion.................................... 20 2 EvolutionaryAlgorithms:AnIntroductiontoMetaheuristicOptimiza- tion 22 2.1Optimization.................................. 22 2.2SearchforOptimality.............................. 24 2.3UnderstandingEvolutionaryandMetaheuristicApproaches......... 25 2.4Classi…cationofMetaheuristicAlgorithmsBasedonTheirNature..... 26 2.4.1Deterministic.............................. 26 2.4.2Stochastic................................ 27 2.4.3HybridofStochasticandDeterministicAlgorithms......... 27 2.5Classi…cationofMetaheuristicAlgorithmsBasedonTheirWorkingSystem 29 2.5.1Procedure-BasedAlgorithms...................... 29 2.5.2Equation-BasedAlgorithms...................... 30 2.6OtherClassi…cations.............................. 34 2.7SearchMechanismsandTheoreticalFoundations............... 35 2.7.1Gradient-GuidedMoves........................ 36 2.7.2RandomPermutation.......................... 36 2.7.3Direction-basedPerturbations..................... 36 2.7.4IsotropicRandomWalks........................ 36 2.7.5Long-tailed,Scale-freeRandomWalks................ 37 2.8RandomWalksandLévyFlights....................... 39 ix 2.8.1RandomVariables........................... 39 2.8.2RandomWalks............................. 40 2.8.3LévyFlight............................... 41 2.9Intensi…cationandDiversi…cation:....................... 44 2.10WaysforIntensi…cationandDiversi…cation:................. 45 2.11ABriefHistoryofMetaheuristicandEvolutionaryAlgorithms....... 47 2.12Conclusion.................................... 50 3 FlowerPollinationAlgorithm 51 3.1FlowersandFlowering............................. 51 3.1.1Cross-PollinationandSelf-Pollination................. 52 3.1.2FlowerConstancy............................ 52 3.2TheAlgorithm................................. 53 3.2.1NumericalResults........................... 55 3.3VariantsofFlowerPollinationAlgorithm................... 62 3.3.1HybridizedVariantsofFlowerPollinationAlgorithm........ 66 3.4Conclusion.................................... 73 4 ChebyshevMetaheuristicSolverApproach 74 4.1ConstructionoftheChebyshevMetaheuristicSolverApproach....... 75 4.2ParametersofFlowerPollinationAlgorithm................. 78 4.3PseudocodeofChebyshevMeatheuristicSolverApproach.......... 79 4.4Results...................................... 80 4.4.1LinearBoundaryValueProblems................... 80 4.4.2Non-LinearBoundaryValueProblems:................ 99 4.4.3InitialValueProblem.......................... 108 4.5Conclusion.................................... 114 GeneralConclusion.................................116 x Contents Bibliography...................119 AppendixA:MATLAB............125 AppendixB:MATLAB’sCodeUsed126 4.6MATLABCodeoftheFirstChapter..................... 127 4.6.1GenerationofChebyshevPolynomialsoftheFirstKind....... 127 4.6.2GenerationofChebyshevPolynomialsoftheFirstKindin[1,4].. 128 4.6.3GenerationofShiftedChebyshevPolynomials............ 129 4.6.4MATLABCodetoSolvetheFirstExampleUsingChebyshevCol- locationMethod............................ 131 4.6.5MATLABCodetoSolvetheSecondExampleUsingChebyshevCol- locationMethod............................ 134 4.7CodeMATLABfortheFourthChapter.................... 137 4.7.1FlowerPollinationAlgorithm..................... 137 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/172 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
