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.     

Classical solution of the first mixed problem for a third-order hyperbolic equation with the wave operator

We study the classical solution of the boundary value problem for a third-order hyperbolic equation defined on the plane in a half-strip with the Cauchy conditions on the lower base of the domain and with the Dirichlet conditions on the lateral boundary. By the method of characteristics, in the case of two independent variables, we find the solution of the problem in closed form and prove its uniqueness.     

Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilizationof the wave equation

We deal here with a second order elliptic mixed problem which is posed in a regular open bounded domain of . We study the regularity of its solution. We apply our results to the boundary stabilization of the wave equation.     

Numerical solution of a time-like Cauchy problem for the wave equation

Let D ? ?ⁿ be a bounded domain with piecewise-smooth boundary, and q(x,t) a smooth function on D × [0, T]. Consider the time-like Cauchy problem Given g, h for which the equation has a solution, we show how to approximate u(x,t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.     

A mixed Dirichlet-Robin problem for a nonlinear Kirchhoff-Carrier wave equation

In this paper, a mixed Dirichlet-Robin problem for a nonlinear Kirchhoff-Carrier wave equation is studied. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, we prove the existence and uniqueness of a weak solution of the above problem. An asymptotic expansion of high order in many small parameters of solutions is also discussed.     

Approximate solution to a time optimal boundary control problem for the wave equation

The paper provides some examples of mutually dual unconstrained optimization problems originating from regularization problems for systems of linear equations and/or inequalities. The solution of each of these mutually dual problems can be found from the solution of the other problem by means of simple formulas. Since mutually dual problems have different dimensions, it is natural to solve the unconstrained optimization problem of the smaller dimension.     

Numerical solution of the inverse problem for the wave equation

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostiki, pp. 6–9, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Fourier method in a mixed problem for the wave equation on a graph

A mixed problem for the wave equation on the simplest geometric graph consisting of two ring edges that touch at a point is considered. The approach used is based on the contour integration of the operator’s resolvent. With the help of a special transformation of a formal series, a classical solution of the problem is obtained under minimum conditions imposed on the initial data. This approach makes it possible to do without an expensive analysis of improved asymptotics for the eigenvalues and eigenfunctions of the operator and to avoid the difficulties associated with the possible multiplicity of the operator’s spectrum.     

The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation

The paper deals with the problem of solvability of the mixed problem for a linear second-order hyperbolic partial differential equation. The minimal necessary and sufficient conditions for the existence of a unique classical solution to this problem are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1232–1238, September, 1993.     

Solvability of the mixed problem for the wave equation with a dynamic boundary condition

We study the solvability of a mixed problem for the equation of vibrations of a bounded string with a given oblique derivative at one end in the sense of a generalized solution in the Sobolev space.     

On the existence of a solution of a quasilinear mixed problem for a quasiwave equation

We establish conditions for the existence and uniqueness of a solution of a quasilinear mixed problem for a quasiwave equation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1182–1185, September, 1997.     

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.     

Classical solution of the first mixed problem for the Klein-Gordon-Fock equation in a half-strip

We consider a classical solution of the first boundary value problem for the Klein-Gordon-Fock equation in a half-strip in the one-dimensional case. We prove the existence and uniqueness of the classical solution under certain smoothness conditions and matching conditions for the given functions. To solve the problem, one should solve Volterra integral equations of the second kind.