Generalized solution of the simplest initial boundary value problem for a homogeneous hyperbolic equation with a mixed derivative

V. S. Rykhlov

Taurida Journal of Computer Science Theory and Mathematics, no. 2 (2023), pp. 72-88

Résumé

An initial boundary value problem for a homogeneous second-order hyperbolic equation with constant coefficients and a mixed derivative is investigated in a half-strip of the plane. The equation in question is the equation of transverse vibrations of a moving finite string. The case of zero initial velocity and fixed ends (Dirichlet conditions) is considered. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on different sides of the origin. The main result of the article is formulated, namely, the theorem on the finite formula for the generalized solution and the method of obtaining this formula is briefly described. The main advantage of this formula is that it does not require any preliminary continuation of the initial function beyond the segment of its definition. The method is based on the idea of A. P. Khromov to use for this the theory of divergent series in the understanding of L. Euler (axiomatic approach). In the special case of the simplest string oscillation equation this formula for generalized solution has a different kind if compared with the formula, obtained earlier by A. P. Khromov. Next, it is determined the classical solution of the initial boundary value problem under consideration. The uniqueness theorem of the classical solution is formulated in the case of its existence and a formula is given for solving it in the form of a series whose members are contour integrals containing the initial data of the problem. Based on this formula, the concepts of a generalized initial boundary value problem and a generalized solution are introduced. Next, a detailed proof of the previously formulated main theorem of the article is given. The resulting formula for the generalized solution compared with the corresponding result for the classical solution. At the end, a brief history of the problem is given.

Détail

Keywords: hyperbolic equation, second order, mixed derivative in the equation, half-bands, initial boundary value problem, zero initial velocity, fixed ends, classical solution, generalized initial boundary value problem, generalized solution, uniqueness classical solution, finite formulas for the generalized solution
Mots-clés : constant coefficients, divergent series.