Geodesic
Generalized solution of the initial-boundary-value problem for the wave equation with a mixed derivative and a general potential
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 99-121

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We study the initial-boundary-value problem in a half-strip for a second-order inhomogeneous hyperbolic equation with constant coefficients and a nonzero potential containing a mixed derivative. The equation considered is the equation of transverse vibrations of a moving finite string. The problems with general initial conditions (nonzero string profile and nonzero initial velocity of string points) and fixed ends (Dirichlet conditions) are examined. Theorems on the existence and uniqueness of a solution are formulated and formulas for the solution are obtained.
Keywords: partial differential equation, wave equation, hyperbolic equation, mixed derivative, generalized solution
Mots-clés : nonzero potential 
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     title = {Generalized solution of the initial-boundary-value problem for the wave equation with a mixed derivative and a general potential},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {99--121},
     publisher = {mathdoc},
     volume = {232},
     year = {2024},
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     url = {http://geodesic.mathdoc.fr/item/INTO_2024_232_a8/}
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V. S. Rykhlov. Generalized solution of the initial-boundary-value problem for the wave equation with a mixed derivative and a general potential. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 99-121. http://geodesic.mathdoc.fr/item/INTO_2024_232_a8/