Abstract

This work investigates an initial-boundary value and an inverse coefficient problem of determining a space dependent coefficient in the fractional wave equation with the generalized Riemann–Liouville (Hilfer) time derivative. In the beginning, it is considered the initial boundary value problem (direct problem). By the Fourier method, this problem is reduced to equivalent integral equations, which contain Mittag-Leffler type functions in free terms and kernels. Then, using the technique of estimating these functions and the generalized Gronwall inequality, we get a priori estimate for solution via unknown coefficient which will be used to study the inverse problem. The inverse problem is reduced to the equivalent integral equation of Volterra type. To show existence unique solution to this equation the Schauder principle is applied. The local existence and uniqueness results are obtained.

中文翻译：

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用广义黎曼-刘维尔时间导数确定分数扩散方程的空间相关系数

摘要

这项工作研究了初始边界值和逆系数问题，用广义黎曼-刘维尔（希尔弗）时间导数确定分数波动方程中的空间相关系数。一开始考虑的是初始边值问题（直接问题）。通过傅立叶方法，该问题被简化为等价积分方程，其中包含自由项和核中的 Mittag-Leffler 型函数。然后，利用估计这些函数的技术和广义格朗沃尔不等式，我们通过未知系数得到解的先验估计，该估计将用于研究反问题。反问题被简化为Volterra型的等效积分方程。为了证明该方程存在唯一解，应用了 Schauder 原理。得到局部存在唯一性结果。