NUMERICAL SOLUTION VIA LAPLACE TRANSFORMS OF A FRACTIONAL ORDER EVOLUTION EQUATION
Artikel i vetenskaplig tidskrift, 2010

We consider the discretization in time if a fractional order diffusion equation. The approximation is based on a further development of the approach of using Laplace transformation to represent the solution as a contour integral, evaluated to high accuracy by quadrature. This technique reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. Three different methods, using 2N + 1 quadrature points, are discussed. The first has an error of order O (e(-cN)) away from t = 0, whereas the second and third methods are uniformly accurate of order O (e(-C root N)). Unlike the first and second methods, the third method does not use the Laplace transform of the forcing term. The basic analysis of the time discretizaiton takes place in a Banach space setting and uses a resolvent esitmate for the associated elliptic operator. The methods are combined with finite element discretization in the spatial variable to yield fully discrete methods.

Laplace transformation

spatially semidiscreted approximation

spaces

stability

diffusion

inversion

Fractional order diffusion equation

finite elements

quadrature

memory term

quadrature

parallel method

resolvent

time-discretization

Författare

William McLean

Vidar Thomee

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Journal of Integral Equations and Applications

0897-3962 (ISSN)

Vol. 22 57-94

Ämneskategorier

Beräkningsmatematik

DOI

10.1216/JIE-2010-22-1-57