On the solution of the convolution equation with two kernels

We suggest a constructive method for solving a nonsingular convolution equation with two kernels whose kernel functions are integrable on the entire line. The right-hand side is assumed to be integrable with power p ⩾ 1.     

Reconstruction of a convolution operator from the right-hand side on the real half-axis

We study a Volterra convolution integral equation of the first kind on a semi-infinite interval. Under some rather natural constraints on the kernel and the right-hand side of the Volterra integral equation (the kernel has bounded support, while the support of the right-hand side may be unbounded), it is possible to reconstruct the integral operator of the equation (i.e., the solution and the kernel of the integral operator) from the right-hand side of the equation. The uniqueness theorem is proved, the necessary and sufficient conditions for solvability are found, and the explicit formulas for the solution and the kernel are obtained.     

Optimal control of linear parabolic equation:the constrained right-hand side as control function

The quadratic cost control problem for a linear parabolic equation controlled by the constrained right-hand side of the equation is considered. Finite-dimensional approximation scheme for finding approximate solutions is proposed and its convergence rate is estimated. The set of optimal controls and its continuity properties with respect to the problem's data are studied. The convergence of the approximation scheme in the case when the control consists of both the constrained initial condition and the constrained right-hand side of the equation is proved.     

On the optimal reconstruction of a solution of the Poisson equation

We consider the problem of optimal reconstruction of a solution of the generalized Poisson equation in a bounded domain Q with homogeneous boundary conditions for the case in which the right-hand side of the equation is fuzzy. We assume that right-hand sides of the equations belong to generalized Sobolev classes and finitely many Fourier coefficients of the right-hand sides of the equations are known with some accuracy in the Euclidean metric. We find the optimal reconstruction error and construct a family of optimal reconstruction methods. The problem on the best choice of the coefficients to be measured is solved.     

Phragmén–Lindelöf type theorem for <Emphasis Type="Italic">m</Emphasis>-hessian equations

We derive an interior estimate for the gradient of a solution u to the m-Hessian equation with a certain right-hand side. The estimate depends on the oscillation of u and properties of the right-hand side of the equation. The proof is based on a modification of some ideas of Trudinger (1997). As a consequence of the main result, a theorem of Fragmén–Lindel?f type is obtained for solutions to the m-Hessian equations in the entire space . Bibliography: 4 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 147–155.     

Extremal solutions of boundary value problems

To prove the existence of a solution of a two-point boundary value problem for an nth-order operator equation by the a priori estimate method, we study extremal solutions of auxiliary boundary value problems for an nth-order differential equation with simplest right-hand side, which have a unique solution under certain restrictions on the boundary conditions.     

On the asymptotic stability of discontinuous systems analysed via the averaging method

The averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semi-continuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov’s first theorem and the Samoilenko-Stanzhitskii theorem to differential inclusions with an upper semi-continuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov’s theorem concerns finite time intervals, while the extension of the Samoilenko-Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required.     

Stability in the right-hand side of nonlocal difference schemes

Prior bounds expressing stability in the right-hand side are constructed for difference schemes approximating the heat equation with nonlocal boundary conditions. The constraints on the grid increment guaranteeing stability in the right-hand side are identical with the previously established criteria for stability in initial values. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 134–139, 2006.     

Stability in mean square of a linear stochastic differential equation

The stability of the zero solution of a first-order linear differential equation with a random right-hand side is investigated using moment equations. Transformations of moment equations are considered. Conditions for reducing the order of the moment equations are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 119–126, 1987.     

On the quasi-sohition of overdetermined til-conditioned systems of linear equations and its realization by means of parametric linear programming

The paper is concerned with the stability of the L _∞ -quasi-solutions In the sense of IVANOV for overdetermined, ill-conditioned linear equation systems. A stability theorem is proved, which considers perturbations in the coefficient-matrix of the equation system. Then an efficient way for the computation of L _∞-quasi-solutions is shown by means of parametric linear programming. The numerical investigations performed allow statements on the stability behaviour of such solutions when the right-hand side and the matrix of the equation systema reperturbed. It is shown that L _∞-quasi-solutions just provide good approximations if Euclidean-quasi-solutions or equivalent regularized solutions are very imprecisely.     

On the solution of the inhomogeneous polyharmonic equation and the inhomogeneous helmholtz equation

We present formulas that simplify finding the solutions of the Poisson equation, the inhomogeneous polyharmonic equation, and the inhomogeneous Helmholtz equation in the case of a polynomial right-hand side. They are based on the representation of an analytic function by harmonic functions. The resulting formulas remain valid for some analytic right-hand sides for which the corresponding operator series converge.     

Some controllability results for the 2D Kolmogorov equation

In this article, we prove the null controllability of the 2D Kolmogorov equation both in the whole space and in the square. The control is a source term in the right-hand side of the equation, located on a subdomain, that acts linearly on the state. In the first case, it is the complementary of a strip with axis x and in the second one, it is a strip with axis x.The proof relies on two ingredients. The first one is an explicit decay rate for the Fourier components of the solution in the free system. The second one is an explicit bound for the cost of the null controllability of the heat equation with potential that the Fourier components solve. This bound is derived by means of a new Carleman inequality.     

On determining the source function in heat and mass transfer problems under integral overdetermination conditions

We examine an inverse problem of determining the right-hand side (the source function) in a parabolic equation from integral overdetermination data. By a solution to a parabolic equation we mean a weak solution, and the right-hand side in this equation can be a distribution of a certain class. Under some conditions on the data of the problem, we demonstrate that this inverse problem is well posed and, in particular, some stability estimates hold.     

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

In this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary ∂Ω and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1).     

Localization of the discontinuity line of the right-hand side of a differential equation

We propose a new approach to studying the inverse problems for differential equations with constant coefficients. Its application is illustrated by an example of some partial differential equation with three independent variables. The right-hand side of the equation is assumed to be a function discontinuous in spatial variables. In the inverse problem, it is required to find some hull containing the discontinuity line of the right-hand side. An algorithm for constructing such a hull is obtained: It is a square whose sides are tangent to the discontinuity line.     

Asymptotics of the solution to a stationary piecewise-smooth reaction-diffusion equation with a multiple root of the degenerate equation

A piecewise-smooth second-order singularly perturbed differential equation whose right-hand side is a nonlinear function with a discontinuity on some curve is investigated. This is a new class of problems in the case where the degenerate equation has a multiple root on the left-hand side of the curve which separates the domain and an isolated root on the right-hand side of that curve. The asymptotics of a solution with an internal layer near a point on the discontinuous curve and the transition point is constructed. The method to construct the internal layer function is proposed. The behavior of the solution in the internal layer consisting of four zones essentially differs from the case of isolated roots. For sufficiently small parameter values, the existence of a smooth solution with an internal layer from the multiple root of the degenerate equation to the isolated root in the neighborhood of a point on the discontinuous curve is proved. The method can be shown to be effective in the given example.     

A priori properties of solutions of nonlinear equations with degenerate coercivity and L 1-data

A Dirichlet problem for a second-order nonlinear elliptic equation in the general divergent form with a right-hand side from L ¹ is considered. The high-order coefficients in the equation are assumed to satisfy the degenerate coercivity condition. The main results concern a priori properties of summability and some estimates of entropy solutions of this problem. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.     

Sign-definiteness of solution to inhomogeneous higher-order equation of mixed parabolic-hyperbolic type

We study solutions of a polycaloric equation and an equation of mixed parabolichyperbolic type of the second order. We prove the sign-definiteness of the solution in dependence of the right-hand side of the equation. Based on these results we study the sign-definiteness of a solution to a higher-order inhomogeneous equation of mixed parabolic-hyperbolic type in dependence on the right-hand side of the equation.     

Convergence of the Galerkin method for a wave equation with singular right-hand side

We consider analogs of the Galerkin method for a linear wave equation of the fifth order with generalized functions on the right-hand side. Theorems on the convergence of an approximate method, depending on the order of singularity of the right-hand side, are proved. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 778–786, June, 2006.     

Asymptotics of Solutions of Difference Equations with Variable Coefficients

Linear difference equations in a Hilbert space with coefficients depending on the number n of the equation are considered. It is assumed that the coefficients differ from constant ones by a finite sum of exponentially vanishing terms as n → ∞. An asymptotic formula for solutions as n → ∞ is obtained. The coefficients in the asymptotic expansion are expressed as linear functionals on the space of sequences in the terms on the right-hand side.