A spectral method for time-fractional diffusion systems

The time fractional derivative of a function y(t) depends on the past history of the function y(t), and so time fractional differential systems are naturally suitable to describe evolutionary processes with memory. Fractional models are increasingly used in many modelling situations including, for example, viscoelastic materials in mechanics, anomalous diffusion in transport dynamics of complex systems and some biological processes in rheology. Here we consider a time-fractional reaction diusion problem [2]. This is a non-local model and as the solution depends on all its past history, numerical step-by-step methods are computationally expensive. We propose a mixed method, which consists of a finite difference scheme through space and a spectral collocation method through time. The spectral method considerably reduces the computational cost with respect to step-by-step methods and is exponentially convergent [3]. Some classes of spectral bases are considered, which exhibit different convergence rates and some numerical results based on time diffusion reaction diffusion equations are given [1]. References [1] Burrage, K., Cardone, A., D'Ambrosio, R. and Paternoster, B. 2017 Numerical solution of time fractional diffusion systems. Appl. Numer. Math. 116 8294. [2] Gafiychuk, V., Datsko, B. and Meleshko, V. 2008 Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220(1-2) 215225. [3] Zayernouri, M. and Karniadakis, G. Em 2014 Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1) A40A62.