This paper is devoted to identify a space-dependent source function in a multiterm time-fractional diffusion equation with nonhomogeneous boundary condition from a part of noisy boundary data. The well-posedness of a weak solution for the corresponding direct problem is proved by the variational method. We firstly investigate the uniqueness of an inverse initial problem by the analytic continuation technique and the Laplace transformation. Then, the uniqueness of the inverse source problem is derived by employing the fractional Duhamel principle. The inverse problem is solved by the Levenberg-Marquardt regularization method, and an approximate source function is found. Numerical examples are provided to show the effectiveness of the proposed method in one- and two-dimensional cases.

中文翻译：

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具有非均匀边界条件的多项式时间分数阶扩散方程的逆源问题

本文致力于从一部分嘈杂的边界数据中，确定具有非均匀边界条件的多时分形扩散方程中与空间有关的源函数。通过变分方法证明了相应直接问题的弱解的适定性。我们首先通过解析连续技术和拉普拉斯变换研究初始反问题的唯一性。然后，通过采用分数Duhamel原理导出反源问题的唯一性。通过Levenberg-Marquardt正则化方法解决了反问题，并找到了一个近似源函数。数值算例表明了该方法在一维和二维情况下的有效性。