Certain nonlocal problem for a degenerate hyperbolic equation

Translated from Matematicheskie Zametki, Vol. 51, No. 3, pp. 91–96, March, 1992.     

A nonlocal problem for degenerate hyperbolic equation

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.     

Mixed nonlocal problem for a nonlinear singular hyperbolic equation

In this paper, we study a nonlocal mixed problem for a nonlinear hyperbolic equation. Based on some a priori estimates and some density arguments, we prove the well posedness of the associated linear problem. The existence and uniqueness of the weak solution of the nonlinear problem are then established by applying an iterative process based on the obtained results for the linear problem. Copyright © 2009 John Wiley & Sons, Ltd.     

On the solvability of a nonlocal problem for a hyperbolic equation of the second kind

In the characteristic triangle for a hyperbolic equation of the second kind we study a nonlocal problem, where the boundary value condition contains a linear combination of Riemann–Liouville fractional integro-differentiation operators. We establish variation intervals of orders of fractional integro-differentiation operators, taking into account parameters of the considered equation with which the mentioned problem has either a unique solution or more than one solution.     

On a nonlocal boundary value problem for a degenerating second-order hyperbolic equation with a spectral parameter

We find necessary and sufficient conditions for the unique solvability of the generalized Darboux problem for a degenerating second-order linear hyperbolic equation of the first kind with two independent variables and with a spectral parameter.     

On a nonlocal problem for a parabolic equation

We study a nonlocal boundary-value problem for a parabolic equation in a two-dimensional domain, establish ana priori estimate in the energy norm, prove the existence and uniqueness of a generalized solution from the classW ₂ ^1,0 (Q _T ), and construct a difference scheme for the second-order approximation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 790–800, June, 1995.     

A problem with nonlocal initial data for one-dimensional hyperbolic equation

In this paper we consider a boundary-value problem for one-dimensional hyperbolic equation with nonlocal initial data in integral form. We prove the existence and uniqueness of the generalized solution.     

Solution of a nonlocal problem of the first kind for a hyperbolic equation

The spectral method is applied to establish solvability in the class of finite-energy generalized solutions of the mixed problem for a hyperbolic equation with a nonlocal boundary condition of the first kind. The solution is representable as a biorthogonal series in the root functions of the corresponding spectral problem. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 167–178, 2004.     

Initial-boundary value problem with a nonlocal boundary condition for a multidimensional hyperbolic equation

We consider a mixed initial-boundary value problem for a multidimensional (with respect to the space variables) hyperbolic equation with a nonlocal boundary condition containing an integral of the desired solution. We prove the unique solvability of the problem in the space W ₂ ¹ .     

On a nonlocal problem for a degenerating parabolic-hyperbolic equation

For an equation of mixed type in a rectangular domain, we use spectral analysis to establish a uniqueness criterion for the solution of a problem with a nonlocal condition relating the values of the unknown solution that belong to different types of the considered equation. We prove the stability of the solution with respect to the nonlocal condition.     

On a certain nonlinear nonlocal Sobolev-type wave equation

The initial boundary-value problem for the nonlinear nonlocal Sobolev equation is studied. Sufficient conditions for local and for global (with respect to time) solvability are obtained. For the case of local (not global) solvability, upper and lower bounds for the lifespan of the solution are obtained in the form of explicit and implicit formulas.     

Inverse problem for a hyperbolic equation with nonlocal boundary condition containing a delay argument

We investigate a multitype Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Assuming that the mean of the increment X of the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices is negative and the random variable Xe ^X has zero mean, we find the asymptotics of the survival probability at time n as n → ∞.     

Question of well-posedness of a certain inverse problem for a hyperbolic integrodifferential equation

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 3, pp. 69–77, May–June, 1992.     

A numerical method for solving one nonlocal boundary value problem for a third-order hyperbolic equation

A nonlocal boundary value problem for a third-order hyperbolic equation with variable coefficients is considered in the one- and multidimensional cases. A priori estimates for the nonlocal problem are obtained in the differential and difference formulations. The estimates imply the stability of the solution with respect to the initial data and the right-hand side on a layer and the convergence of the difference solution to the solution of the differential problem.     

Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument

We consider a problem for a quasilinear hyperbolic equation with a nonlocal condition that contains a retarded argument. By reducing this problem to a nonlinear integrofunctional equation, we prove the existence and uniqueness theorem for its solution. We pose an inverse problem of finding a solution-dependent coefficient of the equation on the basis of additional information on the solution; the information is given at a fixed point in space and is a function of time. We prove the uniqueness theorem for the solution of the inverse problem. The proof is based on the derivation and analysis of an integro-functional equation for the difference of two solutions of the inverse problem.     

Dirichlet problem for a nonlocal wave equation

We study the Dirichlet problem for a nonlocal wave equation in a rectangular domain. We prove the existence and uniqueness of a solution of the problem and show that determining whether the solution is unique can be reduced to determining whether a function of Mittag-Leffler type has real zeros. The obtained uniqueness condition turns into the uniqueness condition for the solution of the Dirichlet problem for the wave equation as the order of the fractional derivative in the equation tends to 2.