Abstract

A quasilinear parabolic problem with a time fractional derivative of the Caputo type and mixed boundary conditions is considered. The coefficients of the elliptic operator depend on the gradient of the solution, and this operator is uniformly monotone and Lipschitz-continuous. For this problem, unconditionally stable linear regularized semi-discrete scheme is constructed based on the \(L1\)-approximation of the fractional time derivative. Stability estimates are obtained by the variational method. Accuracy estimates are given provided that the initial data and the solution to the differential problem are sufficiently smooth. The proved result of stability of the semi-discrete scheme is valid for the mesh scheme obtained from the semi-discrete problem using the finite element method in spatial variables.

中文翻译：

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用时间分数导数求解拟线性抛物型方程的正则化网格方案

摘要

考虑了具有 Caputo 型时间分数阶导数和混合边界条件的拟线性抛物线问题。椭圆算子的系数取决于解的梯度，并且这个算子是一致单调和 Lipschitz 连续的。对于这个问题，无条件稳定线性正则化半离散方案是基于\(L1\) - 分数时间导数的近似构建的。稳定性估计是通过变分方法获得的。如果初始数据和微分问题的解足够平滑，则给出准确度估计。半离散方案稳定性的证明结果对于在空间变量中使用有限元方法从半离散问题获得的网格方案是有效的。