A fast collocation approximation to a two-sided variable-order space-fractional diffusion equation and its analysis

We develop a fast indirect collocation method for two-sided variable-order space-fractional diffusion equation, which models, e.g., the superdiffusive transport of solute in heterogeneous porous medium. Due to impact variable fractional order, stiffness matrix loses Toeplitz structure that is common context constant-order sFDEs. Consequently, discrete Fourier transform technique or convolution quadrature used development methods equations no longer apply. approximate by combination Toeplitz-like matrices via an entrywise expansion. prove approximated system asymptotically consistent with original problem only O(NlogN) memory and O(Nlog2N) operations are required per iteration solvers. numerical solution has same order accuracy as underlying does, without any artificial regularity assumption true solution. Numerical experiments presented demonstrate utility method.