The main purpose of this paper is to study a problem of recovering a parabolic equation with fractional derivative from its time averaging. This problem can be established as a new boundary value problem where a Cauchy condition is replaced by a prescribed time average of the solution. By applying some properties of the Mittag–Leffler function, we set some of the results above existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given, and convergence rate between the regularized solution and the exact solution in L^p space is also derived.

中文翻译：

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用分数阶导数从时间平均值中识别抛物线扩散的初始状态

本文的主要目的是研究从时间平均中恢复具有分数阶导数的抛物线方程的问题。这个问题可以被建立为一个新的边界值问题，其中 Cauchy 条件被解决方案的规定时间平均值所取代。通过应用 Mittag–Leffler 函数的一些性质，我们在一些合适的空间中设置了一些关于所提出问题的温和解的存在性、唯一性和规律性的结果。此外，我们还展示了 Hadamard 意义上的问题的不适定性。给出了正则化解，并推导了正则化解与L ^p空间中精确解的收敛速度。