Jaan Janno

Solitary waves in nonlinear microstructured materials

Dispersive effects due to microstructure of materials combined with nonlinearities give rise to solitary waves. In this paper the existence of solitary wave solutions is proved for a rather general hierarchical governing equation which accounts for nonlinearities on both macro- and microscales. Properties of the waves are established. Waves are asymmetric in the case of the nonlinearity in the microscale. Dispersive effects are due to the scale dependence.

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.

An inverse solitary wave problem related to microstructured materials

An inverse problem for determining coefficients of a one-dimensional wave equation of nonlinear microstructured material is considered. The solution of the problem is based on measurements gathered from two independent solitary waves. Uniqueness for the inverse problem is proved and a stability estimate is derived.

On Lavrentiev Regularization for Ill-Posed Problems in Hilbert Scales

Abstract In this article we study the problem of identifying the solution x † of linear ill-posed problems Ax = y in a Hilbert space X where instead of exact data y noisy data y δ ∈ X are given satisfying with known noise level δ. Regularized approximations are obtained by the method of Lavrentiev regularization in Hilbert scales, that is, is the solution of the singularly perturbed operator equation where B is an unbounded self-adjoint strictly positive definite operator satisfying . Assuming the smoothness condition we prove that the regularized approximation provides order optimal error bounds (i) in case of a priori parameter choice for and (ii) in case of Morozovs discrepancy principle for s ≥ p. In addition, we provide generalizations, extend our study to the case of infinitely smoothing operators A as well as to nonlinear ill-posed problems and discuss some applications.

Microstructured materials: inverse problems

Introduction.- 1 Inverse problems and non-destructive evaluation.- 2 Mathematical models of microstructured solids.- 3 Linear waves.- 4 Inverse problems for linear waves.- 5 Solitary waves in nonlinear models.- 6 Inverse problems for solitary waves.- 7 Summary.- References.- Index

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.

A Parabolic Integro-Differential Identification Problem in a Barrelled Smooth Domain

We consider the problem of recovering a spaceand time-dependent kernel in a parabolic integro-differential equation. The related domain is assumed to be smooth and provided with two bases. Global existence and uniqueness results are proved.

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.