N. I. Chernykh

Order of the best spline approximations of some classes of functions

The rate of decrease of the upper bounds of the best spline approximations Em,n(f)p with undetermined n nodes in the metric of the space Lp(0, 1) (1≤p≤∞) is studied in a class of functionsf(x) for which ∥fm+1 (x)∥Lq(0, 1)≤1(1≤q≤t8) or var {f(m) (x); 0, 1}≤1 (m=1, 2, ..., the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as n → ∞, and asymptotic formulas are obtained for p=∞ and 1≤q≤∞ in the case of an approximation by broken lines (m=1). The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.

On the mechanics of helical flows in an ideal incompressible nonviscous continuous medium

We find a general solution to the problem on the motion in an incompressible continuous medium occupying at any time a whole domain D ⊂ R3 under the conditions that D is an axially symmetric cylinder and the motion is described by the Euler equation together with the continuity equation for an incompressible medium and belongs to the class of helical flows (according to I.S. Gromeka’s terminology), in which sreamlines coincide with vortex lines. This class is constructed by the method of transformation of the geometric structure of a vector field. The solution is characterized in Theorem 2 in the end of the paper.

Transformation That Changes the Geometric Structure of a Vector Field

We propose a method of constructing vector fields with certain vortex properties by means of transformations that change the value of the field vector at every point, the form of the field lines, and their mutual position. We discuss and give concrete examples of the prospects of using the method in applications involving solution of partial differential equations, including nonlinear ones.

Harmonic wavelets in boundary value problems for harmonic and biharmonic functions

We consider boundary value problems in a disk and in a ring for homogeneous equations with the Laplace operator of the first and second orders. Solutions are represented in terms of bases of harmonic wavelets in Hardy spaces of harmonic functions in a disk and in a ring, which were constructed earlier.

Interpolating-Orthogonal Wavelet Systems

Based upon Meyer wavelets, new systems of periodic wavelets and wavelets on the whole axis are constructed; these systems are orthogonal and interpolating simultaneously. Estimates of the errors of approximation of different classes of smooth functions by these wavelets are obtained.

On the construction of unit longitudinal-vortex vector fields with the use of smooth mappings

A solution is given for the problem of constructing a unit vector field collinear to the field of its curl. The solution is based on the use of a suitably parametrized orthogonal transformation of a unit vector field that is potential in ℝ3. The result is stated in the theorem that contains the recipe for constructing the required field.

Interpolation Wavelets in Boundary Value Problems

We propose and validate a simple numerical method that finds an approximate solution with any given accuracy to the Dirichlet boundary value problem in a disk for a homogeneous equation with the Laplace operator. There are many known numerical methods that solve this problem, starting with the approximate calculation of the Poisson integral, which gives an exact representation of the solution inside the disk in terms of the given boundary values of the required functions. We employ the idea of approximating a given 2π-periodic boundary function by trigonometric polynomials, since it is easy to extend them to harmonic polynomials inside the disk so that the deviation from the required harmonic function does not exceed the error of approximation of the boundary function. The approximating trigonometric polynomials are constructed by means of an interpolation projection to subspaces of a multiresolution analysis (approximation) with basis 2π-periodic scaling functions (more exactly, their binary rational compressions and shifts). Such functions were constructed by the authors earlier on the basis of Meyer-type wavelets; they are either orthogonal and at the same time interpolating on uniform grids of the corresponding scale or only interpolating. The bounds on the rate of approximation of the solution to the boundary value problem are based on the property ofMeyer wavelets to preserve trigonometric polynomials of certain (large) orders; this property was used for other purposes in the first two papers listed in the references. Since a numerical bound of the approximation error is very important for the practical application of the method, a considerable portion of the paper is devoted to this issue, more exactly, to the explicit calculation of the constants in the order bounds of the error known earlier.

Description of a helical motion of an incompressible nonviscous fluid

We consider the problem of describing the motion of a fluid filling at any specific time t ≥ 0 a domain D ⊂ R3 in terms of velocity v and pressure p. We assume that the pair of variables (v, p) satisfies a system of equations that includes Euler’s equation and the incompressible fluid continuity equation. For the case of an axially symmetric cylindrical layer D, we find a general solution of this system of equations in the class of vector fields v whose lines for any t ≥ 0 coincide everywhere in D with their vortex lines and lie on axially symmetric cylindrical surfaces nested in D. The general solution is characterized in a theorem. As an example, we specify a family of solutions expressed in terms of cylindrical functions, which, for D = R3, includes a particular solution obtained for the first time by I.S. Gromeka in the case of steady-state helical cylindrical motions.

A solution class of the Euler equation in a torus with solenoidal velocity field

A system of equations with respect to a pair (V, p) of a scalar field and a vector field in a torus D is considered. The system consists of the Euler equation with a given vector field f and the solenoidality equation for the field V. We seek for solutions (V, p) of this system such that the lines of the vector field V inside D coincide with meridians of tori embedded in D with the same circular axis. Conditions on the vector field f under which the problem is solvable are established, and the whole class of such solutions is described.

Some solutions of continuum equations for an incompressible viscous medium

We consider the Navier-Stokes equations for an incompressible medium that fills at any time t ≥ 0 an open axially symmetric cylindric layer D. We find solutions of these equations in the class of motions described by velocity fields whose lines for t ≥ 0 coincide with their vortex lines and lie on axially symmetric cylindrical surfaces in D.