formal solution

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.

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.