معادلات الاشتقاق الكسري وتطبيقاتها على الأنظمة الديناميكية

Abstract

Fractional calculus are introduced in this thesis. The Grunwald-Letnikov, Riemann- Liouville, and Caputo approaches to define derivatives and integrals will be discussed after a brief introduction and some preliminaries. Then some fundamental properties of fractional derivatives and integrals will be proven, such as linearity, the Leibniz rule, and composition. Following that, the fractional derivatives and integrals defini- tions will be applied to a few examples. There will also be a discussion of fractional differential equations and one approach for solving them, which is, Laplace transform. Equilibrium point has a long history of being used in dynamical systems, in order to demonstrate the stability. Lyapunov have made the most extensive contribution to the stability study of nonlinear dynamic systems. We will study how to apply Lyapunov theorems on equilibrium points to determine stability of the systems under sufficient conditions. Without solving the systems themselves, Lyapunov functional approaches have controlled the determination of stability for general nonlinear sys- tems. The thesis surveys with a few examples stability and stabilization of linear, non-linear and finite-time non-linear fractional differential equations in the sense of Caputo approach.