An essential feature of the subdiffusion equations with the $\alpha$-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter $\sigma\in (0,1)\cup(1,2)$. Under this general regularity assumption, we here obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. To the end, we present a refined discrete fractional-type Gr\"onwall inequality and a rigorous analysis for the truncation errors. Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.
翻译:使用 $\ alpha$- 顺序时间分解衍生物的子扩散方程式的一个基本特征是初始时的微弱单数性。 溶液的薄弱常规性通常以常规参数$\sigma\ in (0, 1,\cup(1, 2)$为特征。 根据这个一般的常规假设, 我们在这里获得对非线性分解方程式广泛使用的 L1 方案在时间上的点误差估计。 最后, 我们提出了一个精细的离散分分分数型 Gr\\" 墙壁不平等和对脱轨错误的严格分析。 提供了数值实验, 以证明我们理论分析的有效性 。