Numerical Studies for Fractional Order Differential and Integro-Differential Equations

Abstract

This thesis is a contribution on numerical studies for fractional order differential and integra-differential equations. Three different defini­ tions of the fractional derivative with their famous properties are stud­ ied (Riemann-Liouville, Griinwald-Letnikov, Caputo). A class of nu­ merical methods for solving the fractional wave equations is presented, this class of methods is very close to the weighted average finite differ­ ence method. Two theorems with their proofs are presented to study the stability analysis and the truncation error of this method. Also, a numerical method based on shifted Chebyshev polynomials for solv­ ing the linear and nonlinear fractional integra-differential equation of Volterra type, is studied. An approximate formula for Caputo deriva­ tive ba..sed on Chebyshev expansion is derived. Morever, special at­ tention is given to study boundary stabilization of a fractional wave equation via fractional order controller and to make a comparison be­ tween the fractional order boundary controller and the integer order boundary controller.