| Titre : | Essential of conformable fractional calculus |
| Auteurs : | Hamlet Tayeb, Auteur ; Idrissi Soumia, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | Université de Saida - Dr Moulay Tahar. Faculté des Sciences. Département de Mathématiques., 2020/2021 |
| Format : | 74ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 008448 |
| Catégories : | |
| Mots-clés: | Fractional derivatives, Conformable fractional derivatives, Fractional calculus, Conformable fractional calculus |
| Résumé : |
The concept of derivatives of non-integer order, known as fractional derivatives, first ap-
peared in the letter between L’Hôpital and Leibniz in which the question of a half- order derivative was posed. Since then, many formulations of fractional derivatives have ap- peared. Recently, a new definition of fractional derivative, named "conformable fractional derivative", has been introduced. This new fractional derivative is compatible with the classical derivative and it has attracted attention in domains such as mechanics, electron- ics and anomalous diffusion. Motivated by the considerable attention and the wide resonance in the scientific com- munity that conformable fractional derivative have received it. This master thesis is devoted to the theory of conformable fractional calculus, it summarizes the most recent contributions in this area, and vastly expands on them to create a comprehensive theory conformable fractional calculus. |
| Note de contenu : |
Contents
Acknowledgments 3 Abstract/Resume 6 Notations And Symbols 7 Introduction 9 1 Preliminaries. 12 1.1 Basics tools for stochastic calculus. . . . . . . . . . . . . . . . . . . . . . . 12 1.1.1 Stochastic processes. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 Stochastic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.1 Itô integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.2 One dimentionel Ito formula . . . . . . . . . . . . . . . . . . . . . . 22 1.2.3 Stochastic differential equations . . . . . . . . . . . . . . . . . . . . 22 1.3 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 Some Results from Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . 26 2 Fractional Calculus 27 2.1 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.1 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Basic fractional approche . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 Grunwald-Letnikov derivative . . . . . . . . . . . . . . . . . . . . . 29 2.2.2 Riemann-Liouville approche . . . . . . . . . . . . . . . . . . . . . . 30 2.2.3 Caputo fractional derivative . . . . . . . . . . . . . . . . . . . . . . 33 2.2.4 Main properties of fractional operator . . . . . . . . . . . . . . . . . 33 4 2.3 Ordinary fractional differential equation . . . . . . . . . . . . . . . . . . . 39 2.3.1 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 Stochastic fractional differential equation . . . . . . . . . . . . . . . . . . . 42 2.4.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . 44 3 Conformable Fractional Calculus 49 3.1 Conformable Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.1 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.2 Conformable Fractional Derivative . . . . . . . . . . . . . . . . . . 51 3.1.3 Conformable fractional integral . . . . . . . . . . . . . . . . . . . . 57 3.2 Ordinary Conformable Fractional Differential equations . . . . . . . . . . . 64 3.2.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.3 Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Stochastic Conformable Fractional Differential equations . . . . . . . . . . 66 3.3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3.2 Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . 68 Conclusion 71 Bibliography 72 |
Exemplaires
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BUC-M 008448 Adobe Acrobat PDF |
