Generalized initial-boundary problem for the wave equation with mixed derivative

Abstract

We study an initial-boundary problem for a second-order inhomogeneous hyperbolic equation in a half-strip of the plane containing a mixed derivative with constant coefficients and zero or nonzero potential. This equation is the equation of transverse oscillations 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 opposite sides of the origin. The classical solution of the initial-boundary problem is determined. In the case of zero potential, a uniqueness theorem for the classical solution is formulated and a formula for the solution is given in the form of a series consisting of 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. The main theorems on finite formulas for the generalized solution in the case of homogeneous and inhomogeneous problems are formulated. To prove these theorems, we apply an approach that uses the theory of divergent series in the sense of Euler, proposed by A.P. Khromov (axiomatic approach). Using this approach, on the basis of formulas for solutions in the form of a series, the formulated main theorems are proved. Further, as an application of the main theorems obtained, we prove a theorem on the existence and uniqueness of a generalized solution of the initial-boundary problem in the presence of a nonzero summable potential and give a formula for the solution in the form of an exponentially convergent series.