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Inverse Problem of Determining the Heat Source Density for the Subdiffusion Equation
Differential Equations ( IF 0.6 ) Pub Date : 2021-01-21 , DOI: 10.1134/s00122661200120046
R. R. Ashurov , A. T. Mukhiddinova

Abstract

We study the inverse problem of determining the right-hand side of a subdiffusion equation with Riemann–Liouville fractional derivative whose elliptic part has the most general form and is defined in an arbitrary multidimensional domain (with sufficiently smooth boundary). The Fourier method is used to prove theorems on the existence and uniqueness of the classical solution of the initial–boundary value problem and on the unique reconstruction of the unknown right-hand side of the equation. The concept of generalized solution is introduced and a theorem on its existence is proved. The stability of classical and generalized solutions is proved. Requirements for the initial function and for the additional condition are established under which the classical Fourier method can be applied to the inverse problem under consideration. The results obtained are also new for the classical diffusion equation.



中文翻译:

确定子扩散方程热源密度的反问题

摘要

我们研究了用黎曼-利维尔分数阶导数确定子扩散方程右侧的逆问题,该分数阶椭圆形部分具有最一般的形式,并定义在任意多维域(具有足够光滑的边界)中。傅里叶方法用于证明关于初边值问题经典解的存在性和唯一性,以及等式未知右手边的唯一重构的定理。介绍了广义解的概念,并证明了它的存在性。证明了经典解和广义解的稳定性。建立了对初始函数和附加条件的要求,在该条件下可以将经典的傅里叶方法应用于所考虑的逆问题。

更新日期:2021-01-21
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