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Rykhlov V. S. The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2023, vol. 23, iss. 2, pp. 183-194. DOI: 10.18500/1816-9791-2023-23-2-183-194, EDN: VJGXBX

# The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution

An initial boundary value problem for an inhomogeneous second-order hyperbolic equation on a finite segment with constant coefficients and a mixed derivative is investigated. The case of fixed ends 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 classical solution of the initial boundary value problem is determined. The uniqueness theorem of the classical solution is formulated and proved. A formula is given for the solution in the form of a series whose members are contour integrals containing the initial data of the problem. The corresponding spectral problem for a quadratic beam is constructed and a theorem is formulated on the expansion of the first component of a vector-function with respect to the derivative chains corresponding to the eigenfunctions of the beam. This theorem is essentially used in proving the uniqueness theorem for the classical solution of the initial boundary value problem.

- Tolstov G. P. On the mixed second derivative.
*Matematicheskii Sbornik. Novaya Seriya*, 1949, vol. 24 (66), iss. 1, pp. 27–51 (in Russian). - Naimark M. A.
*Linear Differential Operators*. New York, Ungar Publ. Co. Part I, 1967. 144 p.; Part 2, 1968. 352 p. (Russ. ed.: Moscow, Nauka, 1969. 528 p.). - Khromov A. P. Behavior of the formal solution to a mixed problem for the wave equation.
*Computational Mathematics and Mathematical Physics*, 2016, vol. 56, iss. 2, pp. 243–255. https://doi.org/10.1134/S0965542516020135 - Khromov A. P. On classic solution of the problem for a homogeneous wave equation with fixed end-points and zero initial velocity.
*Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2019, vol. 19, iss. 3, pp. 280–288 (in Russian). https://doi.org/10.18500/1816-9791-2019-19-3-280-288 - Khromov A. P. Divergent series and functional equations related to geometric progression analogues.
*Sovremennye metody teorii kraevykh zadach: materialy mezhdun. konf. : Voronezhskaya vesennyaya matematicheskaya shkola “Pontryaginskie chteniya – XXX”*[Modern methods of the Theory of boundary value problems: Materials of the International conference Voronezh Spring Mathematical School “Pontryagin Readings – XXX”]. Voronezh, Voronezh State University Publ., 2019, pp. 291–300 (in Russian). EDN: TXWNBY - Khromov A. P. Divergent series and Fourier method for wave equation.
*Contemporary Problems of Function Theory and Their Applications*: Materials of the 20th International Saratov Winter School. Saratov, Nauchnaya kniga, 2020, pp. 433–439 (in Russian). EDN: IFLQGK - Khromov A. P. Divergent series and generalized mixed problem for wave equation.
*Contemporary Problems of Function Theory and Their Applications*: Materials of the 21st International Saratov Winter School (Saratov, January 31 – February 4, 2022). Saratov, Saratov State University Publ., 2022, iss. 21, pp. 319–324 (in Russian). EDN: JPPSUX - Krylov A. N.
*O nekotorykh differentsial’nykh uravneniyakh matematicheskoy fiziki, imeyushchikh prilozheniya v tekhnicheskikh voprosakh*[On Some Differential Equations of Mathematical Physics that Have Applications to Technical Problems]. Moscow, Leningrad, GITTL, 1950. 368 p. (in Russian). - Euler L.
*Differentsial’noe ischislenie*[Differential Calculus]. Moscow, Leningrad, GITTL, 1949. 580 p. (in Russian). - Khromov A. P., Kornev V. V. Classical and generalized solutions of a mixed problem for a nonhomogeneous wave equation.
*Computational Mathematics and Mathematical Physics*, 2019, vol. 59, iss. 2, pp. 275–289. https://doi.org/10.1134/S096554251902009X - Kurdyumov V. P., Khromov A. P., Khalova V. A. Mixed problem for a homogeneous wave equation with a nonzero initial velocity and a summable potential.
*Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2020, vol. 20, iss. 4, pp, 444–456 (in Russian). https://doi.org/10.18500/1816-9791-2020-20-4-444-456 - Khromov A. P., Kornev V. V. Divergent series in the Fourier method for the wave equation.
*Trudy Instituta Matematiki i Mekhaniki UrO RAN*, 2021, vol. 27, iss. 4, pp. 215–238 (in Russian). https://doi.org/10.21538/0134-4889-2021-27-4-215-238 - Lomov I. S. Effective application of the Fourier technique for constructing a solution to a mixed problem for a telegraph equation.
*Moscow University Computational Mathematics and Cybernetics*, 2021, vol. 45, iss. 4, pp. 168–173. https://doi.org/10.3103/S0278641921040038, EDN: IUPUAQ - Lomov I. S. Effectiv application of the Fourier method to solving a mixed problem for the telegraph equation.
*Contemporary Problems of Function Theory and Their Applications*: Materials of the 21st International Saratov Winter School (Saratov, January 31 – February 4, 2022). Saratov, Saratov State University Publ., 2022, iss. 21, pp. 178–180 (in Russian). EDN: TZKVJG - Rykhlov V. S. Divergent series method for solving a mixed problem for a hyperbolic equation.
*Sbornik materialov mezhdunarodnoj konferentsii “XXXII Krymskaya Osennyaya Matematicheskaya Shkola-simpozium po spektral’nym i evolyucionnym zadacham” (KROMSH-2021)*[Collection of Materials of the International Conference “XXXII Crimean Autumn Mathematical School-Symposium on Spectral and Evolutionary Problems” (KROMSH-2021)]. Simferopol, Polyprint, 2021, pp. 22 (in Russian). - Rykhlov V. S. The solution of the initial boundary value problem for a hyperbolic equation with a mixed derivative.
*Contemporary Problems of Function Theory and Their Applications*: Materials of the 21st International Saratov Winter School (Saratov, January 31 – February 4, 2022). Saratov, Saratov State University Publ., 2022, iss. 21, pp. 252–255 (in Russian). EDN: ICBZND - Shkalikov A. A. Boundary problems for ordinary differential equations with parameter in the boundary conditions.
*Journal of Soviet Mathematics*, 1986, vol. 33, iss. 6, pp. 1311–1342. https://doi.org/10.1007/BF01084754

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