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Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. (English) Zbl 1241.65077

Anomalous subdiffusion partial differential equations arise in a number of applications related with mathematical models in biology (cytoplasmic crowding in lipid bilayers, fluorescence photobleaching recovery, diffusion of proteins and lipids in membranes, etc.). The theoretical analysis of equations of this type in two space dimensions is based on the use of the variable-order operator calculus and specially on the variable-order Riemann-Liouville fractional partial derivative.

In this paper, new numerical methods (implicit and explicit) are proposed to solve one such equation in two space dimensions with initial and boundary conditions, carrying out an stability, convergence and solvability analysis with Fourier techniques. It is shown that the new implicit numerical scheme is unconditionally stable, uniquely solvable and convergent with order $𝒪\left({{\Delta }}_{t}+{{\Delta }}_{x}^{2}+{{\Delta }}_{y}^{2}\right)$, where ${{\Delta }}_{t}$ denotes the time step size and ${{\Delta }}_{x}$, ${{\Delta }}_{y}$ are the spatial steps. Concerning the explicit method, it has the same order of convergence but it is only conditionally stable.

The theoretical analysis of both methods is illustrated by several numerical examples, where the implicit scheme is thoroughly tested.

##### MSC:
 65M20 Method of lines (IVP of PDE) 65L06 Multistep, Runge-Kutta, and extrapolation methods 35K20 Second order parabolic equations, initial boundary value problems 35R11 Fractional partial differential equations 65M12 Stability and convergence of numerical methods (IVP of PDE)