In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian–Caputo fractional derivative in time. Theoretically, we prove a novel conditional Lipschitz stability result, and numerically, we develop an easy-to-implement fixed point iteration for recovering the unknown coefficient. In addition, we establish rigorous error bounds on the discrete approximation. These results are obtained by crucially using smoothing properties of the solution operators and suitable choice of a weighted Lp(0,T) norm. The efficiency and accuracy of the scheme are showcased on several numerical experiments in one- and two-dimensions.

中文翻译：

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次扩散中时间相关势的数值恢复 *

在这项工作中，我们研究了一个反问题，即从域上解的积分测量中恢复半线性次扩散模型中的时间相关势。该模型涉及时间的 Djrbashian-Caputo 分数阶导数。理论上，我们证明了一种新颖的条件 Lipschitz 稳定性结果，并且在数值上，我们开发了一种易于实现的定点迭代来恢复未知系数。此外，我们对离散近似建立了严格的误差界限。这些结果是通过关键地使用解算子的平滑特性和适当选择加权来获得的 Lp(0,时间） 规范。该方案的效率和准确性在多个一维和二维数值实验中得到了展示。