Fractional differential systems arise in many fields, and are particularly suitable to model processes with memory. For example, the wall-friction through the fluid boundary layer exhibits some cumulative memory effects, and may be modeled by fractional partial derivatives in Navier–Stokes equations. Moreover reaction–diffusion phenomena with anomalous diffusion such as occurs in spatially inhomogeneous environments, are modeled by fractional reaction–diffusion equations. The numerical approximation to fractional differential systems is not trivial, since one has to take into account the non-local nature and the long the long-range history dependence of such problems. Here we consider the numerical solution of time-fractional reaction-diffusion equations. We propose a numerical scheme consisting of a spectral method through time, on a basis of functions suitably chosen for the problem, and a finite-difference method through space, whose coefficients are adapted according to the qualitative behaviour of the solution. The aim is to reduce the computational cost and to obtain an exponential superconvergence of the time discretization. On the other hand, a finite-difference scheme along space would simplify the implementation of the overall method, is enough accurate for most applications and is flexible, since its coefficients may be specially tuned on the problem.

A mixed spectral method for time-fractional reaction-diffusion systems

BURRAGE, KEVIN;CARDONE, Angelamaria;D'AMBROSIO, RAFFAELE;PATERNOSTER, Beatrice
2015

Abstract

Fractional differential systems arise in many fields, and are particularly suitable to model processes with memory. For example, the wall-friction through the fluid boundary layer exhibits some cumulative memory effects, and may be modeled by fractional partial derivatives in Navier–Stokes equations. Moreover reaction–diffusion phenomena with anomalous diffusion such as occurs in spatially inhomogeneous environments, are modeled by fractional reaction–diffusion equations. The numerical approximation to fractional differential systems is not trivial, since one has to take into account the non-local nature and the long the long-range history dependence of such problems. Here we consider the numerical solution of time-fractional reaction-diffusion equations. We propose a numerical scheme consisting of a spectral method through time, on a basis of functions suitably chosen for the problem, and a finite-difference method through space, whose coefficients are adapted according to the qualitative behaviour of the solution. The aim is to reduce the computational cost and to obtain an exponential superconvergence of the time discretization. On the other hand, a finite-difference scheme along space would simplify the implementation of the overall method, is enough accurate for most applications and is flexible, since its coefficients may be specially tuned on the problem.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4654755
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