An Error Estimate of a Numerical Approximation to a Hidden-Memory Variable-Order Space-Time Fractional Diffusion Equation

Abstract

Variable-order space-time fractional diffusion equations, in which the variation of the fractional orders determined by the fractal dimension of the media via the Hurst index characterizes the structure change of porous materials, provide a competitive means to describe anomalously diffusive transport of particles through deformable heterogeneous materials. We develop a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation, which provides a physically more relevant variable-order fractional diffusion equation modeling. However, due to the impact of the hidden memory, the resulting L-1 discretization loses its monotonicity that was crucial in the error analysis of the widely used L-1 discretizations of constant-order fractional diffusion equations and even variable-order fractional diffusion equations without hidden memory. We develop a novel decomposition of the L-1 discretization weights to address the nonmonotonicity of the numerical approximation to prove its optimal-order error estimate without any (often untrue) artificial regularity assumption of its true solutions, but only under the regularity assumptions of the variable order, the coefficients, and the source term. Numerical experiments are performed to substantiate the theoretical findings.