Spectral Methods for Fractional Differential Equations

Abstract

The mathematical analysis and solution of fractional differential equations are like a bridge connecting physical assumptions and concepts with interpretations, effects and conclusions. Numerical methods are helpful in overcoming the problem related to the shortage of analytical methods in the solution of some type of fractional differential equations. This thesis aims at numerical solutions of a certain type of fractional differential equations. The particular focus is the formulation of numerical methods to solve fractional boundary value problems. After the main introduction, the proofs of different lemmas and results in generalized fractional calculus are simply presented by using related properties of operators in classical fractional calculus. We modify classical orthogonal polynomials, shifted-Chebyshev and Laguerre polynomials for a better approximation of the solutions. The second part of the thesis is the main core in which we have specifically formulated and analyzed different numerical schemes to solve certain types of linear and non-linear fractional differential equations. We successfully apply modified polynomials for the formulation of operational matrices of integer and non-integer order integration and differentiation. The projection of these operational matrices reduces the fractional differential equations in a system of algebraic equations and the solution is carried out simply. The solution of non-linear fractional differential equations involves the Quasilinearization technique and the proposed method. The convergence analysis of all proposed numerical schemes is discussed in detail. The analysis of the solutions for integer and non-integer order derivatives demonstrates the convergence of non-integer order solution to integer order solution. Error estimation of approximation and solution is analyzed. Finally, several examples are presented to check the applicability, reliability and efficiency of the proposed numerical schemes.