L. Von Wolfersdorf

Inverse Problems for Identification of Memory Kernels in Viscoelasticity

Inverse problems for identification of the memory kernel in the linear constitutive stress-strain relation of Boltzmann type are reduced to a non-linear Volterra integral equation using Fouriers method for solving the direct problem. To this equation the contraction principle in weighted norms is applied. In this way global existence of a solution to the inverse problem is proved and stability estimates for it are derived.

On Tikhonov Regularization for Identifying Memory Kernels in Heat Conduction and Viscoelasticity

Inverse problems for identifying memory kernels in linear heat conduction and viscoelasticity are dealt with by the method of least squares with Tikhonov regularization. Based on well-known results of Engl, Kunisch, and Neubauer convergence and convergence rates of sequences of approximate solutions in spaces H1 and H2, respectively, are obtained. The convergence rates are confirmed by numerical computations. Es werden inverse Probleme der Identifikation von Memory-Kernen in der linearen Warmeleitung und Viskoelastizitat mit der Methode der kleinsten Quadrate und Tichonovscher Regularisierung behandelt. Unter Benutzung bekannter Resultate von Engl, Kunisch und Neubauer wird die Konvergenz von Folgen von Naherungslosungen in den Raumen H1 bzw. H2 gezeigt, und es werden Konvergenzraten fur diese Folgen erhalten. Numerische Rechnungen bestatigen diese Konvergenzraten.

Inverse problems for identification of memory kernels in thermo‐ and poro‐viscoelasticity

Inverse problems for identification of the four memory kernels in one-dimensional linear thermoviscoelasticity are reduced to a system of non-linear Volterra integral equations using Fouriers method for solving the direct problem. To this system of equations the contraction principle in weighted norms is applied. In this way global in time existence of a solution to the inverse problems is proved and stability estimates for it are derived. In analogous way inverse problems for the memory kernels in linear poroviscoelasticity can be handled.

A Class of Quadratic Integral-Differential Equations

A nonlinear integro-ordinary differential equation built up by a linear ordinary differential operator of n th order with constant coefficients and a quadratic integral term is dealt with. The integral term represents the so-called autocorrelation of the unknown function. Applying the Fourier cosine transformation, the integral-differential equation is reduced to a quadratic boundary value problem for the complex Fourier transform of the solution in the upper half-plane. This problem in turn is reduced to a linear boundary value problem which can be solved in closed form. There are infinitely many solutions of the integral-differential equation depending on the prescribed zeros of a function related to the complex Fourier transform.A nonlinear integro-ordinary differential equation built up by a linear ordinary differential operator of n th order with constant coefficients and a quadratic integral term is dealt with. The integral term represents the so-called autocorrelation of the unknown function. Applying the Fourier cosine transformation, the integral-differential equation is reduced to a quadratic boundary value problem for the complex Fourier transform of the solution in the upper half-plane. This problem in turn is reduced to a linear boundary value problem which can be solved in closed form. There are infinitely many solutions of the integral-differential equation depending on the prescribed zeros of a function related to the complex Fourier transform.