Geodesic
On the solution of the initial-boundary problem in a half-strip for a hyperbolic equation with a mixed derivative
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 226 (2023), pp. 89-107

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An initial-boundary problem for an inhomogeneous second-order hyperbolic equation in a half-strip of a plane with constant coefficients and a mixed derivative is studied. This problem describes transverse oscillations of a finite string with fixed ends. We introduce the notion of a classical solution of the initial-boundary problem, prove a uniqueness theorem for the classical solution, and obtain a formula for the solution in the form of a series whose terms are contour integrals containing the initial data of the problem. A definition of a generalized solution is given and finite formulas for this generalized solution are found.
Mots-clés : oscillation equation
Keywords: hyperbolic equation, mixed derivative, initial boundary value problem, classical solution, generalized solution. 
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     author = {V. S. Rykhlov},
     title = {On the solution of the initial-boundary problem in a half-strip for a hyperbolic equation with a mixed derivative},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {89--107},
     publisher = {mathdoc},
     volume = {226},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2023_226_a9/}
}
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V. S. Rykhlov. On the solution of the initial-boundary problem in a half-strip for a hyperbolic equation with a mixed derivative. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 226 (2023), pp. 89-107. http://geodesic.mathdoc.fr/item/INTO_2023_226_a9/