Izvestiya of Saratov University.
ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

formal solution

Mixed Problem for a Homogeneous Wave Equation with a Nonzero Initial Velocity and a Summable Potential

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity u′t(x, 0) = ψ(x). The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem.

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

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.

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

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.

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.