resolvent

On Classic Solution of the Problem for a Homogeneous Wave Equation with Fixed End-Points and Zero Initial Velocity

Authors: Khromov A. P.

The paper gives necessary and sufficient conditions of classic solution for a homogeneous wave equation with a summable potential, fixed end-point, and zero initial velocity. With the use of Fourier method and Krylov method of improving series rate convergence an analogue of d’Alembert formula is derived in the form of exponentially convergent series. The paper essentially supports and extends the results of our work carried out in 2016. The suggested new method, based on the use of divergent (in Euler’s sense) series, is very economical in using well-known mathematical facts.

Equiconvergence Theorem for Integral Operator with Involution

Authors: Nazarova E. V., Khalova V. A.

In the paper, the integral operator with kernel having discontinuities of the first kind at the lines t = x and t = 1 − x is studied. The equiconvergence of Fourier expansions for arbitrary integrable function f(x) in eigenfunctions and associated functions of the considered operator and expansions of linear combination of functions f(x) and f(1 − x) in trigonometric system is proved. The equiconvergence is studied using the method based on integration of the resolvent using spectral value. Methods, developed by A. P.

Approximating Properties of the Powers of the Differentiation Operator Resolvent

Authors: Khromov A. A.

The families of operators are constructed and their approximating properties are investigated in the problems of approximating the derivative of function sand the smooth solutions of integralequations.

Approximate Solution of an Optimal Control Problem with Linear Nonhomogeneous Control System in Hilbert Space

Authors: Kurdyumov V. P.

For an optimal control problem with a linear differential equation in Hilbert space and quadratic criteria necessary and sufficient conditions of control functions optimality and approximate formulas of the expansions of these functions in eigenfunctions of the control system operator have been obtained.

On Analogue of Jordan – Dirichlet Theorem about the Convergence of the Expansions in Eigenfunctions of a Certain Class of Differential-Difference Operators

Authors: Khalova V. A.

An analogue of Jordan – Dirichlet theorem is established of convergence of the expansions in eigen functions of the operator Ly = αy′(x) − y′(1 − x) with the boundary condition U(y) = ay(0) + by(1) − (y,ϕ) = 0.

The Approached Calculation of Eigenvalues of the Discrete Operator by Means of Spectral Traces of Resolvent Degree

Authors: Maleko Е. М.

Letadiscreteself-adjointoperatorT actsinaseparableHilbertspace and have the kernel resolvent, and eigenvalues and eigenfunctions of the operator T be known. In the paper the method of calculation of eigenvalues of the perturbed operator T + P is considered. Resolvent of this operator is presented as convergent Neumann series on eigenfunctions of the operator T. The point of the method is that at first is found a set of numbers which approximate traces of the resolvent degrees of the operator T + P.

Solution of Integral Equations via Resolvents of Simplest Differential Operators

Authors: Khromov A. A.

Families of operators for approximate solution of integral equations with unbounded inverse operators and right parts of equations specified by the root-mean-square approximations are constructed.

Resolvent Approach to Fourier Method in a Mixed Problem for Non-homogeneous Wave Equation

Authors: Kornev V. V., Khromov A. P.

Fourier method of obtaining classic solution is being justified in a mixed problem for non-homogeneous wave equation with a complex potential and fixed boundary conditions under minimal conditions on initial data. The proof is based on resolvent approach which does not need any information on eigen and associated functions of the corresponding spectral problem. 

Justification of Fourier Method in a Mixed Problem for Wave Equation with Non-zero Velocity

Authors: Gurevich A. P., Kurdyumov V. P., Khromov A. P.

In the paper, using contour integration of the resolvent of the corresponding spectral problem operator, justification of Fourier method in two mixed problems for wave equation with trivial initial function and non-zero velocity is given. The boundary conditions of these problems, together with fixed endpoint conditions, embrace all cases of mixed problems with the same initial conditions for which the corresponding spectral operators in Fourier method have regular boundary conditions. The problems are considered under minimal requirements on initial data. A. N.

Pages