کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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519741 | 867680 | 2015 | 17 صفحه PDF | دانلود رایگان |
Fractional calculus allows variable-order of fractional operators, which can be exploited 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 second-order 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 sub-diffusion 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 simulations 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. Taken together, our results demonstrate that variable-order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long-standing issues of outflow boundary conditions and monotonicity of integer-order PDEs.
Journal: Journal of Computational Physics - Volume 293, 15 July 2015, Pages 184–200