In this paper we recover two unknown kernels related to a thermal body Ω with memory consisting of two different sub-bodies Ω1 and Ω2, when the boundaries of Ω1 and Ω2 have a common (closed) surface Γ intersecting the boundary ∂Ω of Ω. The additional measurements are performed on two (accessible) subsets of ∂Ω1 and ∂Ω2. For this problem we prove existence, uniqueness and continuous dependence on the data in the framework of Sobolev spaces of L2-type in space.
A generalized method of Lavrent'ev regularization involving an unbounded operator is studied. Assuming source condition in terms of the inverse of the unbounded operator, error estimates of scale type for the regularized solution are derived. The method is applied to ill-posed problems containing selfadjoint operators and Volterra equations of the first kind.
By means of the Laplace transform method sufficient conditions for the existence of exponentially decaying memory kernels in heat flow and viscoelasticity are derived solving corresponding inverse problems. The observation functionals of the inverse problems are built up by n eigenfunctions of the related elliptic equation or the data of the direct problems possess n non-vanishing Fourier coefficients, only. In the special cases n = 1 and n = 2 the Laplace transforms of the memory kernel are given in explicit form.
- We consider the inverse problem of identification of memory kernels in one-dimensional heat flow are dealt with where the kernel is represented by a finite sum of products of known spatially-dependent functions and unknown time-dependent functions. As additional conditions for the inverse problems observations of both heat flux and temperature are prescribed. Using the Laplace transform method we prove an existence and uniqueness theorem for the memory kernel.