Abstract
We consider a boundary value problem for a nonlinear subdiffusion equation with Caputo fractional time derivative of the order \(\alpha(u)\in(0,1)\) and a diffusion coefficient \(k(u)>0\) that depend on the solution \(u\). For this problem, an implicit finite-difference scheme is constructed and studied. Conditions are given for the function \(\alpha(u)\) under which its values lie in the interval \((0,1)\) for acceptable argument values. The existence of a solution to the grid scheme is proved, uniqueness and stability are established under an additional assumption on the function \(\alpha(u)\). Assuming the existence of a smooth solution to the differential problem, accuracy estimates are obtained. Iterative methods for solving the constructed nonlinear mesh scheme are proposed. A series of calculations for one-dimensional problems was carried out and analyzed.
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Funding
This work was supported by the Russian Science Foundation, project no. 22-71-10087.
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Lapin, A., Yanbarisov, R. Approximation of the Subdiffusion Equation with Solution-dependent Fractional Time Derivative and Diffusion Coefficient. Lobachevskii J Math 45, 287–298 (2024). https://doi.org/10.1134/S1995080224010323
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DOI: https://doi.org/10.1134/S1995080224010323