ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

# смешанная задача

## Divergent series and generalized mixed problem for a wave equation of the simplest type

With the use of the operation of integrating the divergent series of a formal solution in the separating variables method, there are obtained the results concerning a generalized mixed problem (homogeneous and non-homogeneous) for the wave equation. The key moment consists in finding the sum of the divergent series  which corresponds to the simplest mixed problem with a summable initial function.

## Mixed problem for simplest hyperbolic first order equations with involution

In this paper investigates the mixed problem for the first order differential equation with involution at the potential and with periodic boundary conditions. Using the received refined asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to avoid investigation of the uniform convergence of the series, obtained by term by term differentiation of formal solution on method of Fourier.

## Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data

The article gives a new short proof the V. A. Chernyatin theorem about the classical solution of the Fourier method of the mixed problem for the wave equation with fixed ends with minimum requirements on the initial data. Next, a similar problem for the simplest functional differential equation of the first order with involution in the case of the fixed end is considered, and also obtained definitive results. These results are due to a significant use of ideas A. N. Krylova to accelerate the convergence of series, like Fourier series.

## 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.

## Structure of Mixed Problem Solution for Wave Equation on Compact Geometrical Graph in Nonzero Initial Velocity Case

A D’Alambert formula analogue for wave equation on the compact geometrical graph with generalized smooth transmission conditions is being proved.

## The Mixed Problem for the Differential Equation with Involution and Potential of the Special Kind

For the solution of some mixed problem with involution and real symmetrical potential, explicit analytical formula has been found with the use of the Fourier method. Techniques allowing to avoid term-byterm differentiation of the functional series and impose the minimum conditions for initial problem data, are used.

## Substantiation of Fourier Method in Mixed Problem with Involution

In this paper the mixed problem for the first order differential equation with involution is investigated. Using the received specified asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to transform a series representing the formal solution on Fourier method, and to prove the possibility of its term by term differentiation. At the same time on the initial problem data minimum requirements are imposed.