In the present paper, we consider an inverse problem of recovering the space-dependent source for a multi-term time fractional diffusion equation from noisy final data. First, we proved that the direct problem has a unique solution. Second, we proved the existence and uniqueness for the inverse space-dependent source problem. We also prove the ill-posedness of the inverse problem by compactness of input–output mapping. Then, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source. Four different examples are presented to show the feasibility and efficiency of the proposed method.

中文翻译：

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多项时间分数扩散方程的逆空间相关源问题

在本文中，我们考虑从嘈杂的最终数据中恢复多项时间分数扩散方程的空间相关源的逆问题。首先，我们证明了直接问题有唯一解。其次，我们证明了逆空间相关源问题的存在唯一性。我们还通过输入-输出映射的紧凑性证明了逆问题的不适定性。然后，我们使用非平稳迭代 Tikhonov 正则化方法结合有限维近似来寻找稳定源。四个不同的例子展示了所提出方法的可行性和效率。