Numerical methods based on resolvent kernal and Legendre functions for fractional differential equations.

Abstract

In this thesis, we present the resolvent kernal method and the Legendre approximation method to obtain the numerical solution of fractional differential equations. For this purpose, we convert the fractional differential equation to a fractional integral equation. The solution of the equivalent fractional integral equation is derived by the resolvent kernal method. When the exact fractional integration of the forcing function in differential equations is not possible, we approximate it by fractional Legendre polynomials. We obtained the numerical solution to the Darboux problem by using the resolvent kernal method. The convergence of the resolvent kernal method is also provided. Furthermore, we presented a Legendre approximation method for the numerical solution of the Darboux problem. Every method is accompanied by a numerical example to verify the applicability and effectiveness of the proposed methods.