Classical solution of the initial-boundary value problem for the wave equation with mixed derivative

V. S. Rykhlov

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 70 (2024) no. 3, pp. 451-486

Résumé

In this paper, we study the initial-boundary value problem for a second-order nonhomogeneous hyperbolic equation in a half-strip of the plane with constant coefficients, containing a mixed derivative, with zero and nonzero potentials. This equation is the equation of transverse oscillations of a moving finite string. We consider the case of fixed ends (Dirichlet conditions). We assume that the roots of the characteristic equation are simple and lie on the real axis on different sides of the origin. We formulate our previously proven theorems on finite formulas for a generalized solution in the case of homogeneous and nonhomogeneous problems. Then, based on these formulas, we prove theorems on finite formulas for a classical solution or, in other words, a solution almost everywhere. In the second part of the paper, we formulate theorems on generalized solution of the initial-boundary value problem with ordinary potential and potential of general type, which we had proved earlier. These results are based on the idea of treating an equation with a potential as an inhomogeneity in an equation without a potential. This idea was previously used by A. P. Khromov and V. V. Kornev in the case of equation without mixed derivative. Further, on the basis of formulas for generalized solution to the problem with potentials, we prove theorems on the corresponding formulas for classical solutions for these two types of potentials.

Détail

Keywords: nonhomogeneous hyperbolic equation, initial-boundary value problem, mixed derivative, generalized solution, classical solution.