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.
The paper is devoted to the analysis of linear ill-posed operator equations Ax = y with solution x0 in a Hilbert space setting. In an introductory part, we recall assertions on convergence rates based on general source conditions for wide classes of linear regularization methods. The source conditions are formulated by using index functions. Error estimates for the regularization methods are developed by exploiting the concept of Mathé and Pereverzev that assumes the qualification of such a method to be an index function. In the main part of the paper we show that convergence rates can also be obtained based on distance functions d(R) depending on radius R > 0 and expressing for x0 the violation of a benchmark source condition. This paper is focused on the moderate source condition x0 = A∗v. The case of distance functions with power type decay rates d(R) = (R–η/(1–η)) as R → ∞ for exponents 0 < η < 1 is especially discussed. For the integration operator in L2(0, 1) aimed at finding the antiderivative of a square-integrable function the distance function can be verified in a rather explicit way by using the Lagrange multiplier method and by solving the occurring Fredholm integral equations of the second kind. The developed theory is illustrated by an example, where the optimal decay order of d(R) → 0 for some specific solution x0 can be derived directly from explicit solutions of associated integral equations.
- 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.