# резольвента

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

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.

## Solution of Integral Equations via Resolvents of Simplest Differential Operators

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.

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

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

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

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.

## Approximating Properties of the Powers of the Differentiation Operator Resolvent

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.

## Approximating Properties of Solutions of the Differential Equation with Integral Boundary Condition

With the use of the solution of the first-order differential equation the approximations to the continuous functions with integral boundary conditions are constructed.

## Equiconvergence Theorem for Integral Operator with Involution

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.

## A Mixed Problem for a Wave Equation with a Nonzero Initial Velocity

We study a mixed problem for the wave equation with a continuous complex potential in the case of a nonzero initial velocity ut (x, 0) = ψ(x) and two types of two-point boundary conditions: the ends are ﬁxed and when each of the boundary boundary conditions contains a derivative with respect to x. A classical solution in the case ψ(x) ∈ W12[0, 1] is obtained by the Fourier method with respect to the acceleration of the convergence of Fourier series by the resolvent approach with the help of A. N.

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

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.