<p style='text-indent:20px;'>The aim of this paper is to investigate an inverse problem of recovering a space-dependent source term governed by distributed order time-fractional diffusion equations in Hilbert scales. Such a problem is ill-posed and has important practical applications. For this problem, we propose a general regularization method based on the idea of the filter method. With a suitable source condition, we prove that the method is of optimal order under various choices of regularization parameter. One is based on the a priori regularization parameter choice rule and another one is the discrepancy principle. Finally, the capabilities of our method are illustrated by both the Tikhonov and the Landweber method.</p>

中文翻译：

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识别分布阶时间分数扩散方程中的空间相关源项

<p style='text-indent:20px;'>本文的目的是研究在希尔伯特尺度中恢复由分布阶时间分数扩散方程控制的空间相关源项的逆问题。这样的问题是不适定的并且具有重要的实际应用。针对这个问题，我们提出了一种基于滤波方法思想的通用正则化方法。在合适的源条件下，我们证明了该方法在各种正则化参数选择下的最优阶。一种是基于先验正则化参数选择规则，另一种是差异原则。最后，Tikhonov 和 Landweber 方法都说明了我们方法的能力。</p>