An inverse problem for identification of a time- and space-dependent memory kernel of a special kind in heat conduction

Abstract

In this paper an inverse problem for identification of a memory kernel in heat conduction is dealt with where the kernel is represented by a finite sum of products of known spatiallydependent functions and unknown time-dependent functions. Using the Laplace transform method an existence and uniqueness theorem for the memory kernel is proved.

An identification problem arising in the theory of heat conduction for materials with memory

“…It should be stressed that the recovery of a memory kernel k depending on both time and space is a quite new problem, as far as first-order in time integro-differential equations are concerned. See, for instance, [2], [5] and [7] which, however, have to be considered one-dimensional in character. In [2] the kernel k depends on time and on only one space variable between the n variables of R n , n ≥ 2, whereas in [5] and [7] the kernel is assumed to be degenerate, i.e.…”

“…See, for instance, [2], [5] and [7] which, however, have to be considered one-dimensional in character. In [2] the kernel k depends on time and on only one space variable between the n variables of R n , n ≥ 2, whereas in [5] and [7] the kernel is assumed to be degenerate, i.e. of the form k(t, x) = N j=1 m j (t)µ j (x), but with the spacedependent functions µ j , j = 1, .…”

Determination of a time‐ and space‐dependent heat flux relaxation function by means of a restricted Dirichlet‐to‐Neumann operator

“…Let us denote byD the operator which assigns to a function ' the function h via the solution u of the problem (7), (8). By Corollary 1.1D transforms the set of functions ' satisfying (11) into the space C…”

“…In this section we will study the direct problem (7), (8 First we observe that the problem (7), (8) is formally equivalent to the following elliptic boundary value problem derived by means of the Laplace transform:…”

“…For instance, the papers [6,7] deal with the identiÿca-tion of the kernel depending on partial space variables or satisfying certain spherical symmetry conditions. The papers [8,9] treat the determination of the kernel representable as a ÿnite sum of products of known space-dependent and unknown time-dependent functions.…”