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

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.

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