An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions

Abstract

In this paper, we develop an h-p version of finite element method for one-dimensional fractional differential equation $-_{0}D_{x}^{\alpha }u+Au=f(x)$ with Dirichlet boundary condition. First, we introduce the existence and uniqueness of the considered problem and show the wellposedness of the corresponding weak form. To solve the mentioned equation, the classical hierarchical polynomials are employed as the trial basis functions. Then, we develop a kind of novel test basis functions for the Petrov-Galerkin finite element method such that the stiffness matrix becomes an identity matrix and the coefficient matrix often has a small condition number. Moreover, we give some properties of the developed test basis functions, and discuss the implementation of the developed finite element method in detail. It is shown that the implementation of our method is easier than that of other finite (and spectral) element methods. Finally, we give a numerical example, and the numerical results show the effectiveness of the develped method.