Numerical Approximation and Error Estimation of a Time Fractional Order Diffusion Equation

Abstract

Finite element method is used to approximately solve a class of linear time-invariant, time-fractional-order diffusion equation formulated by the non-classical Fick law and a “long-tail” power kernel. In our derivation, “long-tail” power kernel relates the matter flux vector to the concentration gradient while the power-law relates the mean-squared displacement to the Gauss white noise. This work contributes a numerical analysis of a fully discrete numerical approximation using the space Galerkin finite element method and the approximation property of the Caputo time fractional derivative of an efficient fractional finite difference scheme. Both approximate schemes and error estimates are presented in details. Numerical examples are included to validate the theoretical predictions for various values of order of fractional derivatives.

Error estimates of a high order numerical method for solving linear fractional differential equations

“…Adolfsson et al [1], [2] considered an efficient numerical method to integrate the constitutive response of fractional order viscoelasticity based on the finite element method. Li et al [25] considered a time fractional partial differential equation by using the finite element method and obtain the error estimates in both semidiscrete and fully discrete cases. Jiang et al [21] considered a high-order finite element method for the time fractional partial differential equaions and proved the optimal order error estimates.…”

Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations

“…In recent years, the numerical treatment and supporting analysis of fractional order differential equations has become an important research topic that offers great potential. The FEMs for fractional partial differential equations have been studied by many authors (see [1][2][3]). All of these papers only considered single-term fractional equations, where they only had one fractional differential operator.…”

A finite element method for time fractional partial differential equations

“…Adolfsson et al [1], [2] considered an efficient numerical method to integrate the constitutive response of fractional order viscoelasticity based on the finite element method. Li et al [16] considered a time fractional partial differential equation by using the finite element method and obtained error estimates in both semidiscrete and fully discrete cases. Jiang et al [17] considered a high-order finite element method for the time fractional partial differential equations and proved the optimal order error estimates.…”