Mixed problem for the inhomogeneous wave equation with a summable potential

The resolvent approach to the Fourier method is used to examine the behavior of the formal solution to the mixed problem for an inhomogeneous wave equation with a summable potential. A classical solution is obtained under minimum conditions imposed on the initial function. It is shown that, in the case of square summable initial and driving functions, the series sum of the formal solution is a weak solution of the mixed problem.     

A mixed problem for an inhomogeneous wave equation with a summable potential

A mixed problem for an inhomogeneous wave equation with fixed ends in the case of a summable potential is studied. Using the Krylov method for acceleration of the convergence of Fourier series, a classical solution under minimal conditions on the initial data and a generalized solution in the case of quadratic summable initial data and perturbing function are obtained.     

Behavior of the formal Fourier solution of the wave equation with a summable potential

The convergence of the formal Fourier solution to a mixed problem for the wave equation with a summable potential is analyzed under weaker assumptions imposed on the initial position u(x, 0) = φ(x) than those required for a classical solution up to the case φ(x)∈ L_p[0,1] for p > 1. It is shown that the formal solution series always converges and represents a weak solution of the mixed problem.     

On the convergence of the formal Fourier solution of the wave equation with a summable potential

The convergence of the formal Fourier solution to a mixed problem for the wave equation with a summable potential is analyzed under weaker assumptions imposed on the initial position u(x, 0) = φ(x) than those required for a classical solution up to the case φ(x) ∈ L_p[0,1] for p > 1. It is shown that the formal solution series always converges and represents a weak solution of the mixed problem.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

The initial boundary value problem for quasi-linear wave equation with viscous damping

In this paper, the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi-linear wave equation with viscous damping are proved. The nonexistence of the global solution for this problem is discussed by an ordinary differential inequality. Finally, an example is given.     

On the characteristic initial value problem for the wave equation in odd spatial dimensions with radial initial data

Summary The paper obtains an explicit solution of the characteristic initial value problem for the wave equation in odd spatial dimensions with radial initial data via solution of a characteristic boundary value problem involving a singular differential equation. The solution of the latter problem is obtained by a modified Riemann method. It is shown that on the time axis the solution of the original problem reduces to the solution that is obtainable by the use of Asgeirsson’s mean value theorem. Entrata in Redazione il 29 agosto 1971.     

Inverse scattering problem for a wave equation with absorption

We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1580–1584, November, 2007.