Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-02-22 DOI:10.1007/s13540-024-00240-0

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Abstract

This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form \((D_{t}^{\rho })^{2}u(t)+2\alpha D_{t}^{\rho }u(t)+Au(t)=p( t)q+f(t)\) , where \(0<t\le T\) , \(0<\rho <1\) and \(D_{t}^{\rho }\) is the Caputo derivative. The equation contains a self-adjoint positive operator A and a time-varying multiplier p(t) in the source function, which, like the solution of the equation, is unknown. To solve the inverse problem, an additional condition \(B[u(t)] = \psi (t)\) is imposed, where B is an arbitrary bounded linear functional. The existence and uniqueness of a solution to the problem are established and stability inequalities are derived. It should be noted that, as far as we know, such an inverse problem for the telegraph equation is considered for the first time. Examples of the operator A and the functional B are discussed.

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带有卡普托导数的分式电报方程的时间相关识别问题

Abstract 本研究探讨了在希尔伯特空间中确定电报方程右边的逆问题。所考虑的主要方程的形式为((D_{t}^{\rho })^{2}u(t)+2\alpha D_{t}^{\rho }u(t)+Au(t)=p( t)q+f(t)\)其中 \(0<tle T\) , \(0<\rho <1\) 和 \(D_{t}^{\rho }\) 是 Caputo 导数。方程包含一个自交正算子 A 和源函数中的时变乘数 p(t)，后者与方程的解一样是未知的。为了解决逆问题，需要附加一个条件（B[u(t)] = \psi (t)\），其中 B 是一个任意有界线性函数。建立了问题解的存在性和唯一性，并导出了稳定性不等式。需要指出的是，据我们所知，这种电报方程的逆问题是第一次被考虑。讨论了算子 A 和函数 B 的示例。

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