A Galerkin finite element method for the space Hadamard fractional partial differential equation

Abstract

In this paper, we study the Galerkin finite element approximation for the space Hadamard fractional partial differential equation. We first introduce a modified Fourier transform to analyse the Hadamard fractional calculus, construct the fractional derivative spaces and fractional Sobolev space. Furthermore, we investigate the existence and uniqueness of the weak solution in the fractional Sobolev space. Then using a newly defined log-Lagrangian polynomial as shape function, we discuss the convergence analysis of the semi-discrete scheme. Together with the Crank–Nicolson scheme in time, we present a fully discrete scheme, analyse the stability and convergence. Finally a numerical example is displayed which support the theoretical analysis.