Boundary Value Problems for Fractional Differential Equations: Existence Theory and Numerical Solutions

Abstract

Fractional calculus can be considered as supper set of conventional calculus in the sense that it extends the concepts of integer order differentiation and integration to an arbitrary (real or complex) order. This thesis aims at existence theory and numerical solutions to fractional differential equations. Particular focus of interest are the boundary value problems for fractional order differential equations. This thesis begins with the introduction to some basic concepts, notations and definitions from fractional calculus, functional analysis and the theory of wavelets. Existence and uniqueness results are established for boundary value problems that include, two–point, three–point and multi–point problems. Sufficient conditions for the existence of positive solutions and multiple positive solutions to scalar and systems of fractional differential equations are established using the Guo–Krasnoselskii cone expansion and compression theorems. Owning to the increasing use of fractional differential equations in basic sciences and engineering, there exists strong motivation to develop efficient, reliable numerical methods. In this work wavelets are used to develop a numerical scheme for solution of the boundary value problems for fractional ordinary and partial differential equations. Some new operational matrices are developed and used to reduce the boundary value problems to system of algebraic equations. Matlab programmes are developed to compute the operational matrices. The simplicity and efficiency of the wavelet method is demonstrated by aid of several examples and comparisons are made between exact and numerical solutions.