Numerical solution of time-fractional reaction-diffusion systems

Fractional differential systems model many dynamical phenomena all associated with memory aspects. These include anomalous diffusion in transport dynamics, the response of viscoelastic materials under mechanical stress, some biological processes in rheology and the kinetics of complex systems in spatially crowded domains. In recent years, considerable attention has been paid to fractional reaction-diffusion systems, where the fractional derivative index alpha produces new effects with respect to the classical model. For example, in the nonlinear model [1], when 0<alpha<1, a relaxation process arises, and when 1<alpha<2 periodic solutions may occur. In this talk we analyse the numerical treatment of time-fractional reaction-diffusion systems. As the solution depends on all its past history, numerical step-by-step methods are computationally expensive. On the other hand spectral methods can avoid the discretization of the ‘heavy tail’ and are exponentially convergent [2]. 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. Finally we illustrate preliminary numerical results on some significant test equations. [1] V. Gafiychuk, B. Datsko, and V. Meleshko. Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math., 220(1-2):215–225, 2008. [2] M. Zayernouri and G. Em Karniadakis. Fractional spectral collocation method. SIAM J. Sci. Comput., 36(1):A40–A62, 2014. Keywords: reaction-diffusion systems, fractional differential equations, spectral methods, finite-difference schemes.