Numerical Solution of a Subdiffusion Equation with Variable Order Time Fractional Derivative and Nonlinear Diffusion Coefficient

Abstract

A grid approximation of the boundary value problem for the subdiffusion equation with a fractional time derivative of the order $\alpha(x,t)\in[\alpha_{0},\alpha_{1}]\subset(0,1)$ and a nonlinear diffusion coefficient $k(u)$ is studied theoretically and numerically. The only conditions imposed on $k(u)$ are its non-negativity and piecewise continuity, therefore, the class under consideration includes equations with a coefficient degenerate in nonlinearity. We prove the existence of a unique solution to a grid scheme approximating this problem, and establish stability estimate in the grid analogue of the norm $L^{\infty}((0,T);L^{1}(\Omega))$. The accuracy estimate is derived under the assumption of the existence of a smooth solution of the approximated differential problem. The asymptotic estimate of the accuracy with respect to the time grid step $\tau$ is equal to $O(\tau^{2-\alpha_{1}-(\alpha_{1}-\alpha_{0})})$, which coincides with the well-known estimate $O(\tau^{2-\alpha})$ in the case of constant order $\alpha$. A series of calculations was carried out for model 1D problems with degenerate and discontinuous coefficient $k(u)$. The results of the performed calculations confirmed the main theoretical results; moreover, the resulting accuracy estimates turned out to be of a higher order in $\tau$ than the proved theoretical estimate.