Geodesic
Generalized d'Alembert formula for the telegraph equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 1, Tome 199 (2021), pp. 66-79.

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We examine the mixed problem for the telegraph equation with periodic boundary conditions. Using A. P. Khromov's method, we construct a series, which represents the generalized d'Alembert formula for the equation considered. Under minimal conditions for input data, this series gives a generalized solution of the problem. If the criterion of the existence of a (unique) classical solution is fulfilled, then this series also gives a classical solution. The case of a summable potential of the equation is considered. In the case of zero potential, the series mentioned becomes the ordinary d'Alembert formula.
Keywords: Fourier series, contour integral method, hyperbolic equation, generalized solution, classical solution. 
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I. S. Lomov. Generalized d'Alembert formula for the telegraph equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 1, Tome 199 (2021), pp. 66-79. http://geodesic.mathdoc.fr/item/INTO_2021_199_a7/

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