Legendre Spectral Methods for Fractional Differential Equations

Abstract

This dissertation aims to develop schemes for the numerical solutions of some types of fractional differential equations namely; solutions of Caputo Hadamard ordinary and partial fractional differential equations, and solutions of a fractional Sturm-Liouville problems. For better approximations of solutions, a modification of classical Legendre polynomials, which is suitable for the Hadamard fractional operators is preferred. So the generalized Legendre functions are introduced. These generalized functions are utilized by specifying an argument in each case. Based on these modified Legendre functions, a scheme for numerical solutions of linear and non linear Hadamard fractional differential equations is developed. Quasilinearization technique is employed to linearize non-linear Hadamard fractional differential equations. An upper bound for approximation error is derived. This is observed that the proposed method provides reasonably accurate results, even for relatively smaller order of Hadamard fractional Legendre functions. After dealing with ordinary differential equations, we proposed a method to obtain numerical solutions of Caputo-Hadamard fractional partial differential equations. Two-dimensional Hadamard fractional Legendre functions are utilized, which are actually the variants of two-dimensional shifted Legendre polynomials. Three different schemes for three classes of Caputo-Hadamard fractional partial differential equations are proposed and every scheme is accompanied by a numerical example to verify the applicability and efficiency of the suggested methods. Moreover, the estimates of upper bounds of error for the approximations have been derived. Lastly, we provided a scheme based on the normalized -Legendre functions for the solution of the fractional Sturm-Liouville problems. We have rewritten the standard form of the classical Sturm- Liouville equation in terms of an equivalent form of fractional operators with respect to another function. Our work also investigates some important properties of eigenvalues and eigenfunctions corresponding to a class of generalized fractional Sturm-Liouville operators. In order to deal with a variety of problems, we considered both the left and right -Legendre functions. While working on the schemes to approximate the solutions of the fractional Sturm-Liouville problems, we also proposed the schemes for the solution of some terminal and boundary value problems that involve the left Caputo differential operators. Furthermore, we present the upper bounds of the errors in approximations of derivatives of the unknown functions in terms of normalized -Legendre functions.