NUMERICAL SOLUTION VIA LAPLACE TRANSFORMS OF A FRACTIONAL ORDER EVOLUTION EQUATION
Journal article, 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
Author
William McLean
Vidar Thomee
University of Gothenburg
Chalmers, Mathematical Sciences, Mathematics
Journal of Integral Equations and Applications
0897-3962 (ISSN)
Vol. 22 1 57-94Subject Categories
Computational Mathematics
DOI
10.1216/JIE-2010-22-1-57