Abstract:
Fractional calculus is a generalization of the basic calculus in the sense that it extends the concepts
of integer order differentiations and integrations to arbitrary order (real or complex) and include
the basic calculus as special case. Due to the increasing applications of fractional order differential
equations and fractional order partial differential equations in sciences and engineering, there exists
strong motivation to develop efficient and reliable numerical methods for solutions of fractional
order differential and integral equations. In this thesis, we develop numerical schemes for numerical
solutions of fractional differential equations, fractional partial differential equations and their coupled
systems. Particularly, we focus on different types of boundary value problems such as n-point local
boundary value problems, n-point nonlocal boundary value problems and boundary value problems
with integral type nonlocal boundary conditions. This thesis begins with the introduction to some
basic concepts, notations and definitions from fractional calculus and approximation theory.
In this work shifted Jacobi polynomials are used to develop numerical schemes for solution of the
boundary value problems for coupled system of fractional ordinary and partial differential equations.
Some new operational matrices are developed and applied to transform the boundary value problems
to system of algebraic equations. The idea of operational matrix technique is extended to two-
dimensional and three-dimensional cases and reliable techniques are developed to solve fractional
partial differential equations in two and three dimensions.
Matlab programmes are developed to compute the operational matrices. The simplicity and
efficiency of the proposed methods is demonstrated by aid of several test problems and comparisons
are made between exact and approximate solutions. Some of the results are also compared with
other standard methods like Haar wavelets collocation method, Homotopy perturbation method,
Radial base functions method, Adomain decomposition method and Reproducing Kernel method.
