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In early works the author studied the Dirichlet and Poincaré problems for multidimensional hyperbolic equations, which shows the well-posedness of these problems in cylindrical domains, significantly dependent on the height of the considered cylindrical domain. Here a multidimensional region inside a characteristic cone is considered, in which the Dirichlet and Poincaré problems have unique solutions for one class of hyperbolic equations.

The paper is devoted to the development of the theory of inverse problems for evolution equations with summands rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints (overdetermination conditions) are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet spectral problem with the homogeneous boundary conditions in a cylindrical domain of Euclidean space for multidimensional hyperbolic equation with wave operator. We construct the solution as an expansion in multidimensional spherical functions; prove the existence and uniqueness theorems. The obtained conditions of the problem unique solvability essentially depend on the “height” of the cylinder.

In the present work a fourth order accuracy decomposition scheme is constructed for a multidimensional nonhomogeneous hyperbolic equation on the basis of rational splitting of cosine operator-function. The main operator of the equation is self-adjoint and positive definite. The stability of the constructed scheme is shown and the error of the approximate solution is estimated. On the basis of the constructed scheme, numerical calculations of test problems for three-dimensional case are carried out. The numerical results obtained verify that the suggested scheme has a fourth order accuracy. [ABSTRACT FROM PUBLISHER]