Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution

Abstract

We present an algorithm for the identification of the relaxation kernel in the theory of diffusion systems with memory (or of viscoelasticity), which is linear, in the sense that we propose a linear Volterra integral equation of convolution type whose solution is the relaxation kernel. The algorithm is based on the observation of the flux through a part of the boundary of a body. The identification of the relaxation kernel is ill posed, as we should expect from an inverse problem. In fact, we shall see that it is mildly ill posed, precisely as the deconvolution problem which has to be solved in the algorithm we propose. Copyright © 2016 John Wiley & Sons, Ltd.

A linear algorithm for the identification of a weakly singular relaxation kernel using two boundary measurements

“…1. the algorithm extends to the case of the singular kernel N(t) in (1.3) the algorithm introduced in [19,20] when N(t) ∈ C 3 ([0, T ]). The proofs given in these papers heavily depend on smoothness of the relaxation kernel even for t = 0.…”

“…When the system is "stable", i.e. the free evolution tends to a stationary temperature as usually happens in applications, an initial condition in the form of a ramp is easily achievable as the equilibrium temperature when the ends of the bar are kept at constant temperatures for a time long enough, see [20].…”

Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory

“…The positive answer was given in [17]. The temperature ξ(x) is the solution of the following problem:…”

Reconstruction of kernel depending also on space variable

“…2 Just considering the hyperbolic case, we can mention other works. [3][4][5][6][7][8][9][10] Much less has been done in the case of kernels depending also on space variables. Nevertheless, this occurrence is very important in applications where anisotropic linear viscoelastic solids are involved (see, for instance, Skrzypek and Ganczarski 11 ).…”

“…We consider first the purely hyperbolic problem (4), which is nothing but (1) in case k ≡ 0. To this aim, we introduce the elliptic operator with homogeneous boundary conditions A 0 (precisely defined in (10) and show (Theorem 2.1) that its domain is precisely the class of elements in the Sobolev space H 2 ((0,π) × Ω) with vanishing boundary conditions. This implies (by essentially well known facts) that sufficient conditions assuring that (4) has a unique solution in ∩ 2 j¼0 C 2− j ð½0; T; H j ðð0; πÞ×ΩÞÞ are that u 0 ∈D(A 0 ) u 1 ∈ DðA 1=2 0 Þ (which is a closed subspace of H 1 ((0, π) × Ω) and f∈W 1,1 (0, T; L 2 ((0,π)×Ω).…”