Abstract

In this article, two dimensional inverse problem of determining convolution kernel in the fractional diffusion equation with the time-fractional Caputo derivative is studied. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation with Riemann–Liouville derivative is constructed. Using the formulas of asymptotic expansions for the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel function, which was used for studying the inverse problem. The inverse problem is reduced to the equivalent integral equation of the Volterra type. The local existence and global uniqueness results are proven by the aid of fixed point argument in suitable functional classes. Also the stability estimate is obtained.

中文翻译：

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时间分数阶扩散方程的二维卷积核确定问题

 抽象的

本文研究了利用时间分数阶Caputo导数确定分数扩散方程中卷积核的二维反问题。为了表示直接问题的解，构造了具有黎曼-刘维尔导数的时间分数扩散方程的基本解。利用基本解及其导数的渐近展开式，根据未知核函数的范数得到正问题解的估计，用于研究反问题。反问题被简化为Volterra型的等效积分方程。借助适当函数类中的不动点论证，证明了局部存在性和全局唯一性结果。还获得了稳定性估计。