Problem with generalized fractional differentiation operators for a hyperbolic equation degenerating in the interior of the domain

We study the unique solvability of a nonlocal problem with generalized fractional differentiation operators in the boundary condition for a hyperbolic equation degenerating in the interior of the domain.     

Estimates of deviations from exact solutions of initial boundary value problems for the wave equation

We derive computable upper bounds of the difference between an exact solution of the initial boundary value problem for a linear hyperbolic equation and any function in a space-time cylinder that belongs to the respective energy class. We prove that the bounds vanish if and only if the approximate solution coincides with the exact one. Bibliography: 13 titles. Dedicated to the jubilee of dear Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 93–106.     

Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems

We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates.     

Guaranteed Functional Error Estimates for the Reissner-Mindlin Plate Problem

We consider a class of explicitly computed majorants that provide accuracy estimates for a solution to the Reissner-Mindlin plate problem. The proof is based on a generalized statement of the problem. It does not require any extra analysis of properties of controlled approximate solutions is independent of features of the finite element method for obtaining approximate solutions. Bibliography: 13 titles. Illustrations: 1 figure. __________ Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 159–167.     

Bitsadze-Samarskii boundary-value problem for moisture transport equation

We prove existence and uniqueness of the solution of the Bitsadze— Samarskii boundary-value problem for the hyperbolic equation of moisture transport in a region that contains a parabolic degeneracy curve in its interior.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 45–49, 1988.     

A posteriori error estimates for viscous flow problems with rotation

A new functional type a posteriori error estimates for the Stokes problem with rotating term are presented. The estimates give guaranteed upper bounds for the energy norm of the error and provide reliable error indication. Computational properties of the estimates are demonstrated by a number of numerical examples. Bibliography: 37 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 23–34.     

Functional approach to a posteriori error estimation for elliptic optimal control problems with distributed control

A new approach to the a posteriori analysis of distributed optimal control problems is presented. The approach is based on functional type a posteriori estimates that provide computable and guaranteed bounds of errors for any conforming approximations of a boundary value problem. Computable two-sided a posteriori estimates for the cost functional and estimates for approximations of the state and control functions are derived. Numerical results illustrate the efficiency of the approach. Bibliography: 35 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 3–14     

On a class of two-dimensional adjoint integral equations of Volterra type

In a rectangular domain, we consider a two-dimensional integral equation of Volterra type with fixed singular kernels. For various values of the parameters occurring in this equation, we prove its unique solvability and establish the asymptotic behavior of the obtained solution.