Mathematics

The Solution of a Certain Inverse Problem

Authors: Khromov A. A.

The solution is given for the problem of findinging uniform approximations of a the right-hand side of a general linear ordinary differential equation in the case when approximations of the exact solution are known. The constructed method has a simple structure, produces approximations of the right-hand side on the whole interval of definition and does not employ boundary conditions.

Investigation Riemann – Hilbert Boundary Value Problem with Infinite Index on Circle

Authors: Fatykhov A. K., Shabalin P. L.

We consider the Riemann – Hilbert boundary value problem of analytic function theory with infinite index and the boundary condition on the circumference. The boundary condition coefficients are Holder’s continuous everywhere except one particular point where the coefficients have discontinuity of second kind (power order with the index is less than one). In this formulation the problem with infinite index is considered for the first time.

Expansion in Root Functions of Strongly Irregular Pencil of Differential Operators of the Second Order with Multiple Characteristics

Authors: Rykhlov V. S.

We consider the quadratic strongly irregular pencil of ordinary second order differential operators with constant coefficients and with a multiple root of the characteristic equation. The amounts of double expansions in biorthogonal Fourier series in the derived chains of such pencils and a necessary and sufficient condition for convergence of these expansions to the expanded vector-valued function are found. This necessary and sufficient condition is a differential equation relating the components of the expanded vector function.

A Case of an Explicit Solutions for the Three-element Problem of Carleman Type for Analytic Functions in a Circle

Authors: Perelman N. R.

The article investigates the three-element Carleman boundary value problem in the class of analytic functions, continuous extension to the contour in the Holder sense, when this problem can not be reduced to a two-element boundary value problems . The unit circle is considered as the contour .To be specific, we study a case of inverse shift. In this case, the solution of the problem is reduced to solving a system of two integral equations of Fredholm second kind; thus significantly used the theory of F. D. Gakhov about Riemann boundary value problem for analytic functions.

Solution of Cauchy Problem for Equation First Order Via Haar Functions

Authors: Lukomskii D. S., Lukomskii S. F., Terekhin P. A.

In this article we consider a Cauchy problem for the first order differential equation and are looking for its numerical solution. For this aim we represent the derivative of the solution as Haar decomposition. We also obtain estimates of approximate solution. The method is computationally simple and applications are demonstrated through illustrative examples. These examples show that in some cases the error of the proposed method is much less, than in second order Runge – Kutta method.

A Mixed Problem for a System of First Order Differential Equations with Continuous Potential

Authors: Burlutskaya M. S.

We study a mixed problem for a first order differential system with two independent variables and continuous potential when the initial condition is an arbitrary square summable vector-valued function. The corresponding spectral problem is the Dirac system. It sets the convergence almost everywhere of a formal decision, obtained by the Fourier method. It is shown that the sum of a formal decision is a generalized solution of a mixed problem, understood as the limit of classical solutions for the case of smooth approximation of the initial data of the problem.

Simulation of the Temperature and Electric Fields by High-current Pulse to the Electrode

Authors: Arutyunyan R. V.

The paper investigates the influence of nonlinearities thermophysical properties and phase transitions of melting and evaporation on the electrical and thermal processes at heating of a metallic electrode of high-current pulse. We formulate a mathematical model and develop a finite-difference method and a computer program, allowing effectively to carry out computer modeling of thermal and physical processes when exposed to high current pulse to metal electrodes. The article describes the results of the calculation of the fields based on cross-cutting enthalpy method.

Mazur Spaces and 4.3-intersection Property of (BM)-spaces

Authors: Alimov A. R.

The paper puts forward some combinatorial and geometric properties of finite-dimensional (BM)-spaces.

Well-posedness of the Dirichlet Problem for a Class of Multidimensional Elliptic-parabolic Equations

Authors: Aldashev S. A.

Correctness of boundary problems in the plane for elliptic equations is well analyzed by analitic function theory of complex variable. There appear principal difficulties in similar problems when the number of independent variables is more than two. An attractive and suitable method of singular integral equations is less strong because of lock of any complete theory of multidimensional singular integral equations.