classical solution

On Classical Solvability of One-Dimensional Mixed Problem for Fourth Order Semilinear Biparabolic Equations

Authors: Khudaverdiyev K. I., Heydarova M. N.

Existence and uniqueness of classical solution of one-dimensional mixed problem with Riquier type homogenous boundary conditions for one class of fourth order semilinear biparabolic equations are studied. A priori estimates method is used to prove the existence in large theorem for classical solution of mixed problem under consideration..

Substantiation of Fourier Method in Mixed Problem with Involution

Authors: Burlutskaya M. S., Khromov A. P.

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.

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

Authors: Khromov A. P.

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.

About the Classical Solution of the Mixed Problem for the Wave Equation

Authors: Khromov A. P.

The classic solution of the mixed problem for a wave equation with a complex potential and minimal smoothness of initial data is established by the Fourier method. The resolvent approach consists of constructing formal solution with the help of the Cauchy – Poincaré method of integrating the resolvent of the corresponding spectral problem over spectral parameter. The method requires no information about eigen and associated functions and uses only the main part of eigenvalues asymptotics. Krylov’s idea of accelerating the convergence of Fourier series is essentially employed.

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

Authors: Khromov A. P., Burlutskaya M. S.

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.

Mixed problem for simplest hyperbolic first order equations with involution

Authors: Burlutskaya M. S., Khromov A. P.

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.