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.
