Time-periodic Poiseuille-type solution with minimally regular flow rate

The nonstationary Navier–Stokes equations are studied in the infinite cylinder Π = {x = (x', xn) ∈ Rn: x' ∈ σ ∈ R n – 1: – ∞ < xn < ∞, n = 2, 3} under the additional condition of the prescribed time-periodic flow-rate (flux) F(t). It is assumed that the flow-rate F belongs to the space L2(0, 2π), only. The time-periodic Poiseuille solution has the form u(x, t) = (0, ... , 0, U(x', t)), p(x,t) = –q(t)xn + p0(t), where (U(x', t), q(t)) is a solution of an inverse problem for the time-periodic heat equation with a specific over-determination condition. The existence and uniqueness of a solution to this problem is proved.

ON SMOOTHNESS OF SOLUTION OF ONE NONLOCAL PROBLEM FOR HYPERBOLIC EQUATION

In this paper we consider a nonlocal problem with integral boundary condition for hyperbolic equation. The conditions of the problem contain derivatives of the first order with respect to both x and t,, which can be interpreted as an elastic fixation of the right end rod in the presence of a certain damper, and since the conditions also contain integral of the desired solution, this condition is nonlocal. It is known that problems with nonlocal integral conditions are non-self-adjoint and, therefore, the study of solvability encounters difficulties that are not characteristic of self-adjoint problems. Additional difficulties arise also due to the fact that one of the conditions is dynamic. The attention of the article is focused on studying thesmoothness of the solution of the nonlocal problem. The concept of a generalized solution is introduced, and the existence of second-order derivatives and their belonging to the space L2 are proved. The proof is basedon apriori estimates obtained in this work.

On the resolvent existence and the separability of a hyperbolic operator with fast growing coefficients in L2(R2)

This paper studies the question of the resolvent existence, as well as, the smoothness of elements from the definition domain (separability) of a class of hyperbolic differential operators defined in an unbounded domain with greatly increasing coefficients after a closure in the space L2(R2). Such a problem was previously put forward by I.M. Gelfand for elliptic operators. Here, we note that a detailed analysis shows that when studying the spectral properties of differential operators specified in an unbounded domain, the behavior of the coefficients at infinity plays an important role.

Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation

It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L^{−1} have the following property (*) λn → 0, when n → ∞, moreover, than the faster convergence to zero so the operator L^{−1} is best approximated by finite rank operators. The following question: - Is it possible for a given nonlinear operator to indicate a decreasing numerical sequence characterized by the property (*)? naturally arises for nonlinear operators. In this paper, we study the above question for the nonlinear Sturm-Liouville operator. To solve the above problem the theorem on the maximum regularity of the solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space L2(R) (R = (−∞, ∞)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear SturmLiouville equation are also obtained. As is known, the obtained estimates of Kolmogorov widths give the opportunity to choose approximation apparatus that guarantees the minimum possible error.

Polyharmonic Decomposition of a Digital Image

A decomposition of the space L2(Q) into a direct sum of polyharmonic subspaces is obtained. The completeness of shift systems of the fundamental solution of the polyharmonic equation is proved. A convergent algorithm for solving the problem of separating the polyharmonic component of a function from L2(Q) is developed. The resulting decomposition is applied to some digital image processing problems; the results of computational experiments are presented.

Оптимальные формулы типа Эйлера-Маклорена для численного интегрирования в пространстве Соболева

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L2(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N ≥ m-3 and for any m ≥ 4 using S. L. Sobolev method which is based on the discrete analogue of the differential operator d2m/dx2m. In particular, for m = 4 and m = 5 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=6 new optimal quadrature formulas are obtained. At the end of this work some numerical results are presented. В настоящей статье рассматривается задача построения оптимальных квадратурных формул в смысле Сарда в пространстве L2(m)(0,1). Здесь квадратурная сумма состоить из значений подынтегральной функции в узлах и значений первой и третьей производных подынтегральной функции на концах интервала интегрирования. Найдены коэффициенты оптимальных квадратурных формул и вычислена норма оптимального функционала погрешности для любого натурального числа N ≥ m-3 и для любого m ≥ 4, используя метод С. Л. Соболева который основывается на дискретный аналог дифференциального оператора d2m/dx2m. В частности, при m = 4 и m = 5 получен оптимальность классической формулы Эйлера-Маклорена. Начиная с m = 6 получены новые оптимальные квадратурные формулы. В конце работы приведаны некоторые численные результаты.