Identification of a memory kernel in a semilinear integrodifferential parabolic problem with applications in heat conduction with memory

Abstract

a b s t r a c tIn this contribution, the reconstruction of a solely time-dependent convolution kernel is studied in an inverse problem arising in the theory of heat conduction for materials with memory. The missing kernel is recovered from a measurement of the average of temperature. The existence, uniqueness and regularity of a weak solution is addressed. More specific, a new numerical algorithm based on Rothe's method is designed. The convergence of iterates to the exact solution is shown.

Error Estimates for the Time Discretization of a Semilinear Integrodifferential Parabolic Problem with Unknown Memory Kernel

“…In this section, we firstly repeat how the authors from [18] have built up a numerical scheme for problem (1.1). Secondly, we describe why and how this numerical scheme needs to be changed to be able to prove higher stability results.…”

“…In [18], the development of a numerical algorithm for problems of type (1.1)-(1.2) has been provided under the condition that…”

“…The first goal of this paper is to slightly change the numerical scheme from [18], such that higher stability results can be obtained. These stability results are needed for the second goal, i.e., to perform an error analysis, from which the strong convergence of the numerical approximations to the kernel K follows.…”

“…The outline of this paper is as follows. First, the numerical scheme of [18] and some corresponding a priori estimates are adapted in Section 2. In the same section, also the convergence of the approximations towards the unique weak solution is proved.…”

Identification of a memory kernel in a nonlinear integrodifferential parabolic problem

“…The error analysis of the scheme is performed in [13]. The interested reader is also referred to [14], where the authors studied the same partial differential equation with a convolution of the form K * ∆u instead of K * u and source term f (u) instead of f (u, ∇u). More recently, in [15], using semigroup theory, the authors investigated the reconstruction of an unknown memory kernel from a more general linear integral overdetermination Φ(u(t)) = m(t) in an abstract linear (of convolution type) evolution equation of parabolic type:…”