Numerical solutions of Hadamard fractional differential equations by generalized Legendre functions

Abstract

This paper aims to develop a numerical method for the solutions of Hadamard fractional differential equations. We introduced Hadamard fractional Legendre functions by modifying classical Legendre polynomials and used them to approximate the numerical solutions of linear and nonlinear Hadamard fractional differential equations. The solution is approached by approximating the appropriate terms in Hadamard fractional differential equations using Hadamard fractional Legendre functions and converting the problem to a system of algebraic equations. Quasilinearization technique is employed to linearize the nonlinear Hadamard fractional differential equations. An upper bound for approximation error is derived. This method provides reasonably accurate results for a relatively smaller order of Hadamard fractional Legendre functions.