Analytical and Numerical Treatment of Nonlinear Fractional Differential Equations Involving Caputo Fractional Operator / Lina Chetioui

Public ISBD Titre : Analytical and Numerical Treatment of Nonlinear Fractional Differential Equations Involving Caputo Fractional Operator Type de document : document électronique Auteurs : Lina Chetioui, Auteur ; Khalouta,Ali, Directeur de thèse Editeur : Setif:UFA Année de publication : 2025 Importance : 1 vol (98 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Fractional differential equations Caputo fractional derivative Approximate solution Analytical solution Index. décimale : 510 - Mathématique Résumé : Nonlinear fractional differential equations play an important role in applied mathematics and physics. It is difficult to obtain the exact solution for these problems due to the complexity of the nonlinear terms included. In recent decades, there has been great development in the numerical analysis and exact solution for nonlinear fractional differential equations. The main objective of this thesis is to study the solutions of nonlinear fractional differential equations involving Caputo fractional operator by proposing new technique. To demonstrate the validity and reliability of this technique, it is applied to several numerical examples. Note de contenu : Sommaire Introductionii 1 Basicconceptsoffractionalcalculus1 1.1Applicationsoffractionalsystems........................ 1 1.1.1Automatic................................. 1 1.1.2Physics................................... 2 1.1.3Mechanicsofcontinuousmedia..................... 2 1.1.4Acoustic.................................. 3 1.2Functionalspaces................................. 3 1.2.1Spacesofintegrablefunctions...................... 3 1.2.2Spacesofcontinuousandabsolutelycontinuousfunctions....... 4 1.2.3Spacesofcontinuousfunctionswithweight............... 5 1.2.4BanachÂ…xedpointtheorem....................... 5 1.3SpeciÂ…cfunctionsforfractionalderivation................... 6 1.3.1Gammafunction............................. 6 1.3.2Betafunction............................... 7 1.3.3Mittag-Le­erfunction.......................... 7 1.4Fractionalintegralsandderivatives....................... 8 1.4.1FractionalintegralintheRiemann-Liouvillesense........... 8 1.4.2FractionalderivativeintheRiemann-Liouvillesense......... 12 1.4.3SomepropertiesoffractionalderivationinthesenseofRiemann- Liouville.................................. 15 1.4.4FractionalderivativeinthesenseofCaputo.............. 17 1.4.5SomepropertiesoffractionalderivationinthesenseofCaputo.... 22 1.4.6RelationbetweentheRiemann-LiouvilleapproachandthatofCaputo 23 2 Fractionaldi¤erentialequationsinthesenseofCaputo25 2.1EquivalenceresultbetweentheCauchyproblemandtheVolterraintegral equation...................................... 25 2.2Resultofexistenceanduniquenessofthesolution............... 27 3 Semi-analyticalmethodsandtheirconvergence33 3.1Adomiandecompositionmethod(ADM).................... 33 3.1.1Descriptionofthemethod........................ 33 3.1.2Adomianpolynomials........................... 35 3.1.3ConvergenceoftheADM......................... 36 3.2Homotopyperturbationmethod(HPM)..................... 39 3.2.1Descriptionofthemethod........................ 40 3.2.2Convergenceanalysis........................... 41 3.3Variationaliterationmethod(VIM)....................... 48 3.3.1Descriptionofthemethod........................ 49 3.3.2AlternativeapproachtoVIM...................... 49 3.3.3Convergenceanalysis........................... 51 3.4Newiterativemethod(NIM)........................... 56 3.4.1Descriptionofthemethod........................ 56 3.4.2ConvergenceofNIM........................... 57 4 Onthesolutionofnonlinearfractionaldi¤erentialequations61 4.1ApplicationoftheADM............................. 61 4.2ApplicationoftheHPM............................. 65 4.3ApplicationoftheVIM.............................. 69 4.4ApplicationoftheNIM.............................. 72 5 NewcombinationmethodforsolvingnonlinearfractionalLienardequa- tion 77 5.1Lienardequation................................. 77 5.2Khaloutatransform................................ 78 5.3Di¤erentialtransformmethod.......................... 82 5.4DescriptionoftheKHDTM........................... 84 5.5ConvergenceoftheKHDTM........................... 85 5.6Illustrativeexamples............................... 87 Conclusionandresearchperspectives92 Bibliography92 Côte titre : DM/0205 Analytical and Numerical Treatment of Nonlinear Fractional Differential Equations Involving Caputo Fractional Operator [document électronique] / Lina Chetioui, Auteur ; Khalouta,Ali, Directeur de thèse . - [S.l.] : Setif:UFA, 2025 . - 1 vol (98 f.) ; 29 cm. Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Fractional differential equations Caputo fractional derivative Approximate solution Analytical solution Index. décimale : 510 - Mathématique Résumé : Nonlinear fractional differential equations play an important role in applied mathematics and physics. It is difficult to obtain the exact solution for these problems due to the complexity of the nonlinear terms included. In recent decades, there has been great development in the numerical analysis and exact solution for nonlinear fractional differential equations. The main objective of this thesis is to study the solutions of nonlinear fractional differential equations involving Caputo fractional operator by proposing new technique. To demonstrate the validity and reliability of this technique, it is applied to several numerical examples. Note de contenu : Sommaire Introductionii 1 Basicconceptsoffractionalcalculus1 1.1Applicationsoffractionalsystems........................ 1 1.1.1Automatic................................. 1 1.1.2Physics................................... 2 1.1.3Mechanicsofcontinuousmedia..................... 2 1.1.4Acoustic.................................. 3 1.2Functionalspaces................................. 3 1.2.1Spacesofintegrablefunctions...................... 3 1.2.2Spacesofcontinuousandabsolutelycontinuousfunctions....... 4 1.2.3Spacesofcontinuousfunctionswithweight............... 5 1.2.4BanachÂ…xedpointtheorem....................... 5 1.3SpeciÂ…cfunctionsforfractionalderivation................... 6 1.3.1Gammafunction............................. 6 1.3.2Betafunction............................... 7 1.3.3Mittag-Le­erfunction.......................... 7 1.4Fractionalintegralsandderivatives....................... 8 1.4.1FractionalintegralintheRiemann-Liouvillesense........... 8 1.4.2FractionalderivativeintheRiemann-Liouvillesense......... 12 1.4.3SomepropertiesoffractionalderivationinthesenseofRiemann- Liouville.................................. 15 1.4.4FractionalderivativeinthesenseofCaputo.............. 17 1.4.5SomepropertiesoffractionalderivationinthesenseofCaputo.... 22 1.4.6RelationbetweentheRiemann-LiouvilleapproachandthatofCaputo 23 2 Fractionaldi¤erentialequationsinthesenseofCaputo25 2.1EquivalenceresultbetweentheCauchyproblemandtheVolterraintegral equation...................................... 25 2.2Resultofexistenceanduniquenessofthesolution............... 27 3 Semi-analyticalmethodsandtheirconvergence33 3.1Adomiandecompositionmethod(ADM).................... 33 3.1.1Descriptionofthemethod........................ 33 3.1.2Adomianpolynomials........................... 35 3.1.3ConvergenceoftheADM......................... 36 3.2Homotopyperturbationmethod(HPM)..................... 39 3.2.1Descriptionofthemethod........................ 40 3.2.2Convergenceanalysis........................... 41 3.3Variationaliterationmethod(VIM)....................... 48 3.3.1Descriptionofthemethod........................ 49 3.3.2AlternativeapproachtoVIM...................... 49 3.3.3Convergenceanalysis........................... 51 3.4Newiterativemethod(NIM)........................... 56 3.4.1Descriptionofthemethod........................ 56 3.4.2ConvergenceofNIM........................... 57 4 Onthesolutionofnonlinearfractionaldi¤erentialequations61 4.1ApplicationoftheADM............................. 61 4.2ApplicationoftheHPM............................. 65 4.3ApplicationoftheVIM.............................. 69 4.4ApplicationoftheNIM.............................. 72 5 NewcombinationmethodforsolvingnonlinearfractionalLienardequa- tion 77 5.1Lienardequation................................. 77 5.2Khaloutatransform................................ 78 5.3Di¤erentialtransformmethod.......................... 82 5.4DescriptionoftheKHDTM........................... 84 5.5ConvergenceoftheKHDTM........................... 85 5.6Illustrativeexamples............................... 87 Conclusionandresearchperspectives92 Bibliography92 Côte titre : DM/0205

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