Identification of Weakly Singular Memory Kernels in Heat Conduction

Abstract

Inverse problems of identification of memory kernels in linear heat conduction are dealt with in case of weakly singular kernels in the space Lp and of continuous kernels with power singularity. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence and stability of solutions are proved.

A class of inverse problems for viscoelastic material with dominating Newtonian viscosity

“…By Corollary 1 in [10], Eq. (52) has a unique solution for any x G A if G is an operator in E satisfying the condition…”

“…The proof of the applied Corollary 1 in [10] uses the contraction principle in a scale of norms ||m||CT for sufficiently large a. Therefore the method of successive approximation can be used for calculation of the solution m to Eq.…”

“…In contrast to the "hyperbolic" cases dealt with in [8,9] in the present "parabolic" case the needed assumptions are as weak as in the corresponding inverse problems for heat flow (cf. [10]). …”

“…For simplicity, in the following we only deal with continuous memory kernels in the generic "main case". Corresponding problems with weakly singular memory kernels can be dealt with as in [9] and [10] and additional cases to the main case as in [8,9]. Further, as in [8]- [10] inverse problems with observation functionals containing the traction can be treated (see also Grasselli [3]).…”

The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods

“…Coupled systems of Volterra equations of the first and second kind arise for example in a slightly different form in problems of identification of memory kernels in heat conduction and viscoelasticity (see, e.g., [5] and [6]). As a starting point for our investigations we have chosen the form (1.1), (1.2).…”