Sergei A. Avdonin, Sergei A. Ivanov,, Jun-Min Wang

1. 
   Sergei A. Avdonin, Sergei A. Ivanov, , Jun-Min Wang - Inverse Problems For The Heat Equation With Memory, Inverse Problems and Imaging, Volume 13, No. 1, 2019, 31–38, doi:10.3934/ipi.2019002,
   

We study inverse boundary problems for one dimensional linear integro-differential equation of the Gurtin–Pipkin type with the Dirichlet-to Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator, we give the explicit formula for the solution of the problem with the observation on the semiaxis t>0. For the observation on finite time interval, we prove the uniqueness result, which is similar to the local Borg–Marchenko theorem for the Schr¨odinger equation.

2. 
   Zhilin Li & Kewang Zheng, An Inverse Problem in a Parabolic Equation, Differential Equations and Computational Simulations III J. Electronic Journal of Differential Equations, Conference 01, 1997, pp. 193–199.ISSN: 1072-6691. https://www.researchgate.net/publication/26393471_An_inverse_problem_in_a_parabolic_equation .
   

An inverse problem in a parabolic equation is studied. An unknown function in the equation is related to two integral equations in terms of heat kernel. One of the integral equations is well-posed while another is ill-posed. A regularization approach for constructing an approximate solution to the ill-posed integral equation is proposed. Theoretical analysis and numerical experiment are provided to support the method.

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   A.G.Ramm , Inverse problems for parabolic equations 2, Communications in Nonlinear Science and Numerical simulation, Volume 12, Issue 6, September 2007, Pages 865-868
   

Let u_t − u_xx = h(t) in 0 ⩽ x ⩽ π, t ⩾ 0. Assume that u(0, t) = v(t), u(π, t) = 0, and u(x, 0) = g(t). The problem is: what extra data determine the three unknown functions {h, v, g} uniquely? This question is answered and an analytical method for recovery of the above three functions is proposed.

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