FRACTIONAL ORDER PARTIAL DIFFERENTIAL EQUATIONS: SPECTRAL ANALYSIS AND NUMERICAL SIMULATION

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.