Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Fractional Calculus and Applied Analysis, Vol. 8, No 3, (2005), 323p-341p
This paper provides a new method and corresponding numerical schemes
to approximate a fractional-in-space diffusion equation on a bounded domain
under boundary conditions of the Dirichlet, Neumann or Robin type.
The method is based on a matrix representation of the fractional-in-space
operator and the novelty of this approach is that a standard discretisation
of the operator leads to a system of linear ODEs with the matrix raised
to the same fractional power. Numerical results are provided to gauge the
performance of the proposed method relative to exact analytical solutions
determined using a spectral representation of the fractional derivative. Initial
results for a variety of one-dimensional test problems appear promising.
Furthermore, the proposed strategy can be generalised to higher dimensions.