Efficient numerical algorithms for Riesz-space fractional partial differential equations based on finite difference/operational matrix
The paper is concerned with initial-boundary value problems for linear PDEs with first order time derivative and fractional space derivatives of Riesz type on a bounded interval. Depending on whether one or (a sum of) two such fractional derivatives of appropriate orders occur, the equation is interpreted as a ``Riesz space fractional diffusion equation'' or a ``Riesz space fractional advection-dispersion equation'', respectively. Numerical methods are proposed to compute approximations to both types of problems, using a finite difference scheme based on the ``matrix transform method'' for spatial discretizations and an ``operational matrix method'' for time discretization. Questions of convergence order and stability are discussed, and some numerical experiments are carried out for illustration.
Riesz-space fractional diffusion equation
Riesz-space fractional advection-dispersion equation
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