Second-order approximations for variable order fractional derivatives: Algorithms and applications

Fractional calculus allows variable-order of fractional operators, which can be ex ploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and/or time. In this paper, we derive two secondorder approximation formulas for the variable-order fractional time derivatives involved in anomalous diffusion and wave propagation. We then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous subdiffusion and super-diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable-order function. Finally, we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super-diffusion, and also simu lations of the Burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and/or time.