Band 13, Heft 4

New formulae for solutions and coefficients of multidimensional kinetic equations are presented in the paper.

In the paper we consider inverse problems for evolutionary linear and non-linear equations. We introduce constructive methods for their investigation, in particular, in some cases of the formula.

We consider the problem of identifying the potential in the one-dimensional Schrödinger equation with input Dirichlet data, from measured Neumann data. Knowledge of the Dirichlet to Neumann map together with spectral controllability results for the Schrödinger equation obtained using properties of exponential Riesz bases allow recovery of the spectral data. Once the the spectral data is recovered, we use the Boundary Control method to solve the identification problem.

Two-phase model for compressible viscous dusty gas is considered. Global-in-time unique solvability of the Cauchy problem and decay of solutions are proved for this model provided sufficiently small initial data.

This paper deals with the construction of effective heuristic methods for investigating the effect of corrosion on the inaccessible side of a thin metallic plate. In particular, we derive explicit formulas for the Thin Plate Approximation of the positive functions γ and θ that, in our model, describe energy dispersion and material loss respectively.

Hyperbolic equations and inequalities in octants with the lateral Cauchy data at coordinate planes are considered. Lipschitz stability estimate is established in the case when both the inhomogeneous right hand side and (unknown) initial conditions at { t = 0} have a finite support. This is the first stability estimate for such a Cauchy problem in an infinite domain. Refocusing of time reversal fields in octants follows. It is shown that the modified Quasi-Reversibility Method can be applied for the numerical solution of such a Cauchy problem including computational time reversal.

In the present work, we treat a coated-optics problem in which it is required to choose an optimal refractive index of intermediate layer in a three-layer medium providing for maximum transmission of radiation through the medium. The extremum problem is considered as a formulation of the inverse problem for the stationary radiation transfer equation with generalized matching conditions for the Fresnel reflection-refraction at the interfaces between the media. Some analytical solutions are obtained.

Qualitative and quantitative analysis of optimal-control problems and inverse problems in mathematical physics largely implies the calculation of functional gradients giving optimality criteria or closing errors. As a rule, this procedure is used in calculating necessary extremum conditions and in gradient methods. If a system under study is linear, then calculation of a functional gradient presents no serious difficulties. Yet, in nonlinear problems of mathematical physics there arise substantial difficulties stemming from the fact that, as the control (parameter to be identified) is put to variation, the state function of the system also undergo changes. Thus, for the functional gradient to be found, differentiability of the state function with respect to control must be established. Substantiation of this property for nonlinear infinite-dimensional systems is a point far from being trivial. Moreover, the example given below shows that, although the system of interest contains no nonsmooth expressions, the required property can be violated. This circumstance does not allow one to calculate functional gradients by standard methods and use traditional optimization methods. Nonetheless, the system under study displays a property that can be considered as a weak form of differentiability of the state function with respect to control. As a result, we arrive at the notion of extended derivative of an operator, presenting generalization of the classical types of operator derivatives. With the help of this notion, one can find a functional gradient even if its direct determination is impossible. In this way, we can enlarge the area of applicability of optimization methods. It is shown that, for extended derivatives, an analogue to the inverse function theorem can be formulated which makes it possible to give a general scheme permitting calculation of functional gradients in nonlinear problems of mathematical physics even in those cases in which the state function turns out to be a nondifferentiable function of the control variable.