Numerical Approximation of Space-Time Fractional Parabolic Equations

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e. an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator E(t) for the initial value problem can be written as a Dunford-Taylor integral involving the Mittag-Leffler function e(alpha), 1 and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here alpha denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator W(t) and the forcing function F(t). We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford-Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of 1/h. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying a sinc quadrature technique to approximate the Dunford-Taylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size k involves k(-2) nodes and results in an additional O(exp(- c/k)) error. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo-midpoint quadrature. This involves the average of W-h(s), (the semi-discrete approximation to W(s)) over the quadrature interval. This average can also be written as a Dunford-Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford-Taylor integrals for computing the averages. We show that for a refined grid in time with a mesh of O(N log(N)) intervals, the error between the semi-discrete and fully discrete approximation is O(N-2 + log(N) exp(-c/k)). We also report the results of numerical experiments that are in agreement with the theoretical error estimates.