On the dissipativity of some Caputo time-fractional subdiffusion models in multiple dimensions: Theoretical and numerical investigations

Abstract

In this work, we consider multidimensional diffusion–reaction equations with time-fractional partial derivatives of the Caputo type and orders of differentiation in (0,1). The models are extensions of various well-known equations from mathematical physics, biology, and chemistry. In the present manuscript, we will impose initial–boundary data on a closed and bounded spatial multidimensional domain. Single-term and multi-term fractional systems are considered in this work. In the first stage, we show that the fractional models possess energy-like functionals which are dissipated in L2(Ω) with respect to time. The systems are investigated rigorously from the analytical point of view, and dissipative numerical models to approximate their solutions are proposed and rigorously analyzed. Our discretizations will make use of the uniform L1 approximation scheme to estimate the time-fractional derivatives, and the usual central-difference operators to approximate the spatial Laplacian. To that end, various results of the literature will be crucial, including some useful discrete forms of Paley–Wiener inequalities. Some numerical examples are included to show the asymptotic behavior of the numerical methods and, ultimately, their dissipative character.