We consider the three-particle discrete Schrodinger operator H_µ,γ(К), К ϵТ³ associated to a system of three particles (two particle are fermions with mass 1 and third one is an another particle with mass m = 1/y < 1) interacting through zero range pairwise potential µ> 0 on the three-dimensional lattice Z³. It is proved that for γϵ(1, γ₀) (γ₀≈4,7655) the operator H_µ,γ(π), π=(π,π,π), has no eigenvalue and has only unique eigenvalue with multiplicity three for γ>γ₀ lying right of the essential spectrum for sufficiently large µ.

The paper considers a two-point boundary value problem with homogeneous boundary conditions for a single nonlinear ordinary differential equation of order 4n. Using the well-known Krasnoselsky theorem on the expansion (compression) of a cone, sufficient conditions for the existence of a positive solution to the problem under consideration are obtained. To prove the uniqueness of a positive solution, the principle of compressed operators was invoked. In conclusion, an example is given that illustrates the fulfillment of the obtained sufficient conditions for the unique solvability of the problem under study.

We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables: the function is smooth outside the discontinuity lines, and at each point on the line it has a discontinuity of the first kind. We construct averaging procedures and study global discrete regularizing algorithms for approximating discontinuity lines. Lipschitz conditions are imposed on the discontinuity lines. A parametric family of fractal lines is constructed, for which all conditions can be checked analytically. A fractal is indicated that has a large fractal dimension, for which the efficiency of the constructed methods can be guaranteed.

The second initial-boundary value problem in a bounded domain for a fractional-diffusion equation with the Bessel operator and the Gerasimov-Caputo derivative is investigated. Theorems of existence and uniqueness of the solution of the inverse problem of determining the lowest coefficient in a one-dimensional fractional diffusion equation under the condition of integral observation are obtained. The Schauder principle was used to prove the existence of the solution.

Using the Schwartz function method, we have obtained a new exact solution for the problem of stationary capillary waves of finite amplitude on the surface of a liquid that has a finite depth. The reliability of the solution was confirmed by the results of numerical verification of the main boundary equation. The obtained solution of the problem is general in the sense that for any Weber number one can find the corresponding wave configuration. Parametric analysis showed a nonmonotonic dependence of the wave-length and its amplitude on the Weber number. The fact that the problem has one more branch of the solution (the trivial solution) indicates the possibility of the existence of other branches. The Schwartz function method cannot guarantee finding all solutions of the problem even from the specified class of functions. Therefore, the question of reproducing the known exact solution of W. Kimmersley for this problem and its reliability remains open. Note that for the parameter ß included in the main boundary equation, W. Kimmersley preliminarily laid down assumption ß = 1. The found exact solution has the property that ß > 1 and cannot coincide with W. Kimmersley’s solution.

In this paper, we consider the one-dimensional Schrodinger equation on the semiaxis with an additional exponential potential. Using transformation operators with the asymptotics at infinity, a triangular representation of a special solution of this equation is found. An estimate is obtained with respect to the kernel of the representation

In this paper, numerical simulation of a compressible gas in plane channels of constant and variable cross-section is performed within the framework of two-dimensional parabolized Navier-Stokes equations. The "narrow channel approximation" model is used for the numerical solution of the uranation system.

A number of transformations are described in detail, such as de-dimensioning the equation system, reducing the considered area to a square, as well as thickening the calculated points in large gradients of gas dynamic parameters. Flow conservation conditions are used to determine the pressure gradient.

An effective method is given for simultaneously determining the pressure gradient and the longitudinal velocity, then other gas dynamic parameters stable for subsonic and supersonic flows, as well as a method for determining the critical flow rate for solving Laval nozzle problems.

The results of methodical calculations are presented, with the aim of verifying the effectiveness of the developed calculation methodology, as well as confirming the reliability of the results obtained by comparing them with data from other authors