Numerical Approximation of Space-Time Fractional Parabolic Equations Academic Article uri icon


  • Abstract 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 ) {E(t)} for the initial value problem can be written as a DunfordTaylor integral involving the Mittag-Leffler function e , 1 {e_{alpha,1}} and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here 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 ) {W(t)} and the forcing function F ( t ) {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 DunfordTaylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of 1 h {frac{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 DunfordTaylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size k involves k

published proceedings


author list (cited authors)

  • Bonito, A., Lei, W., & Pasciak, J. E.

citation count

  • 14

complete list of authors

  • Bonito, Andrea||Lei, Wenyu||Pasciak, Joseph E

publication date

  • October 2017