We consider an inverse time-dependent source problem governed by a distributed time-fractional diffusion equation using interior measurement data. Such a problem arises in some ultra-slow diffusion phenomena in many applied areas. Based on the regularity result of the solution to the direct problem, we establish the solvability of this inverse problem as well as the conditional stability in suitable function space with a weak norm. By a variational identity connecting the unknown time-dependent source and the interior measurement data, the conjugate gradient method is also introduced to construct the inversion algorithm under the framework of regularizing scheme. We show the validity of the proposed scheme by several numerical examples.

中文翻译：

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分布阶次时间分数扩散方程的一个逆源问题

我们考虑由使用内部测量数据的分布式时间分数扩散方程控制的反时限源问题。这样的问题出现在许多应用领域的一些超慢扩散现象中。基于正问题解的正则性结果，我们建立了该反问题的可解性以及在弱范数的合适函数空间中的条件稳定性。通过连接未知时变源和内部测量数据的变分恒等式，还引入了共轭梯度法，在正则化方案的框架下构造了反演算法。我们通过几个数值例子证明了所提出方案的有效性。