Lothar von Wolfersdorf

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

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 spatially-dependent functions and unknown time-dependent functions. Using the Laplace transform method an existence and uniqueness theorem for the memory kernel is proved.

Some results and a conjecture on the degree of ill-posedness for integration operators with weights

In this paper, we are looking for answers to the question whether a noncompact linear operator with non-closed range applied to a compact linear operator mapping between Hilbert spaces can, in a specific situation, destroy the degree of ill-posedness determined by the singular value decay rate of the compact operator. We partially generalize a result of Vu Kim Tuan and Gorenflo (1994 Inverse Problems 10 949–55) concerning the non-changing degree of ill-posedness of linear operator equations with fractional integral operators in L 2 (0, 1) when weight functions appear. For power functions m(t) = t α (α > −1), we prove the asymptotics σn(A) ∼ 1 0 m(t) dt πn for the singular values of the composite operator [Ax](s) = m(s) s 0 x(t)dt in L

Identification of Weakly Singular Memory Kernels in Viscoelasticity

Inverse problems for the identification of memory kernels in the linear theory of viscoelasticity with constitutive stress-strain-relation of Boltzmann type are dealt with in the case of weakly singular kernels in the space L p , and of continuous kernels with power singularity at zero. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence, uniqueness, and stability of solutions are proved.

On the theory of convolution equations of the third kind

Existence results of Part I of the paper are generalized to two types of autoconvolution equations of the third kind having free terms with nonzero values at x = 0 like the well-known Bernstein–Doetsch equation for the Jacobian theta zero functions. Also uniqueness results for the linear convolution equations in Part I of the paper are extended to more general function spaces. Further, a special class of integro-differential equations with autoconvolution integral and two classes of the linear singular Abel–Volterra equations are dealt with.

Identification of a special class of memory kernels in one-dimensional heat flow

Abstract - 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.

An inverse problem for identification of a time- and space-dependent memory kernel in viscoelasticity

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

On a Class of Nonlinear Convolution Equations

By means of weighted norms existence and uniqueness theorems are proved for some classes of nonlinear convolution equations in Lebesgue spaces L and spaces C of continuous functions. The applicability of the theorems is shown by examples.

Regularization of a class of nonlinear Volterra equations of a convolution type

A method of regularization of a class of nonlinear Volterra equations of a convolution type is analysed. The equations arise when solving inverse problems of determining the memory kernels in a heat flow.

On the analysis of distance functions for linear ill-posed problems with an application to the integration operator in L 2

The paper is devoted to the analysis of linear ill-posed operator equations Ax = y with solution x 0 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 x 0 the violation of a benchmark source condition. This paper is focused on the moderate source condition x 0 = 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 L 2(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 x 0 can be derived directly from explicit solutions of associated integral equations.

On the Determination of a Density Function by Its Autoconvolution Coefficient

We deal with a modification of the well-known ill-posed autoconvolution equation x*x = y on a finite interval, e.g., analyzed in [8]. In this paper, we focus on solutions that are probability density functions and assume to have data of the autoconvolution coefficient k of the density function x, which we define as the quotient of the autoconvolution function x*x and x itself. The corresponding inverse problem leads to the nonlinear integral equation kx − x*x = 0 of the third kind. For this equation, we give results on existence and make notes on uniqueness and stability. We show the ill-posedness of the equation by an example and make assertions on its regularization by Tikhonovs method. In this context, we prove the weak closedness of the forward operator for some appropriate domain.