Nonsmooth eigenfunctions in problems of mathematical physics

We study whether V.A. Il’in’s method for proving the uniqueness of the solution of a mixed problem for a hyperbolic equation applies to a problem with transmission conditions in the interior of the interval. We show that the system of eigenfunctions corresponding to this problem is complete in the space L ²(0, l) and is a Riesz basis in this space.     

Some applications of mathematical programming to problems in mathematical physics

Siberian Mathematical Journal -     

Attractors for certain nonlinear problems of mathematical physics

One proves the existence of a compact, connected global attractor for the Navier-Stokes-Voigt equations, for pseudoparabolic equations, and for quasilinear wave equations with a strongly dissipative term.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 50–54, 1986.     

Group analysis of boundary-value problems of mathematical physics

We obtain conditions for invariance and invariant solvability of boundary-value problems of mathematical physics. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 140–144, January, 1999.     

Pseudospherical surfaces and some problems of mathematical physics

In the paper, some aspects of the interrelation of Lobachevsky geometry and nonlinear differential equations are discussed. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.     

Applications of Tauberian theorems in some problems in mathematical physics

We give some multidimensional Tauberian theorems for generalized functions and show examples of their application in mathematical physics. In particular, we consider the problems of stabilizing the solutions of the Cauchy problem for the heat kernel equation, multicomponent gas diffusion, and the asymptotic Cauchy problem for a free Schrödinger equation in the norms of different Banach spaces among others.     

Some problems of mathematical physics in the whole Euclidean space

Doklady Mathematics -     

Inverse problems for a mathematical model of redox sorption

We consider a mathematical model of sorption that allows for external diffusion kinetics and a redox reaction. Two inverse problems are considered for this model, uniqueness is proved, and numerical solution methods are proposed. The efficiency of the numerical methods is investigated by computer experiments. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 15–23, 2006.     

Generalized potentials and their application in solving problems of mathematical physics

We introduce the concept of potential for bounded bodies. We construct the potentials for a two-dimensional semi-infinite region. We solve the problems of determining the nonsteady state temperature fields in a semi-infinite plate with a straight-line cut situated perpendicularly to the boundary of the region under boundary conditions of the first or second kind on the boundary of the region and the edges of the cut.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 52–58.     

The basis for an operational method of solving certain problems in mathematical physics

In this paper we obtain sufficient conditions for the applicability of an operational method for solving a mixed problem for an equation of parabolic type with discontinuous coefficients.Translated from Matematicheskie Zametki, Vol. 13, No. 1, pp. 125–134, January, 1973.     

On some nonlinear problems in mathematical physics with exponents close to critical ones

The paper deals with several problems having the following common feature. The differential equation under consideration contains a power function of the solution or of its gradient, with the exponent depending on a small parameter. This dependence is such that a certain property of the solutions corresponding to the positive values of the parameter may disappear as the latter tends to zero. This phenomenon is shown to be lacking in the case where not only the said exponent, but also the coefficients of the equation depend on the small parameter in a suitable way. Several theorems are proved on sharp conditions, ensuring such uniformity of various properties exhibited by the solutions. Bibliography: 44 titles. Dedicated to Olga Arsenievna Oleinik This paper has been supported in part by the Russian Foundation for Basic Research (Project 93-013-1744) and by the International Science Foundation (Grant MIE000). Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

Solving boundary value problems of mathematical physics using radial basis function networks

A neural network method for solving boundary value problems of mathematical physics is developed. In particular, based on the trust region method, a method for learning radial basis function networks is proposed that significantly reduces the time needed for tuning their parameters. A method for solving coefficient inverse problems that does not require the construction and solution of adjoint problems is proposed.     

On p-adic mathematical physics

A brief review of some selected topics in p-adic mathematical physics is presented. The text was submitted by the authors in English.