#### For citation:

Kurdyumov V. P., Khromov A. P., Khalova V. A. A Mixed Problem for a Wave Equation with a Nonzero Initial Velocity. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2018, vol. 18, iss. 2, pp. 157-171. DOI: 10.18500/1816-9791-2018-18-2-157-171, EDN: XQFNQT

# A Mixed Problem for a Wave Equation with a Nonzero Initial Velocity

We study a mixed problem for the wave equation with a continuous complex potential in the case of a nonzero initial velocity ut (x, 0) = ψ(x) and two types of two-point boundary conditions: the ends are ﬁxed and when each of the boundary boundary conditions contains a derivative with respect to x. A classical solution in the case ψ(x) ∈ W12[0, 1] is obtained by the Fourier method with respect to the acceleration of the convergence of Fourier series by the resolvent approach with the help of A. N. Krylov’s recommendations (the equation is satisﬁed almost everyw here). It is also shown that in the case when ψ(x) ∈ L[0, 1] the series of a formal solution for a problem with ﬁxed ends converges uniformly in any bounded domain, and for the second problem it converges only everywhere and for both problems is a generalized solution in the uniform metric.

- Burlutskaya M. Sh., Khromov A. P. Resolvent approach in the Fourier method. Dokl. Math., 2014, vol. 90, iss. 2, pp. 545–548. DOI: https://doi.org/10.1134/S1064562414060076
- Krylov A. N. On Some Differential Equations of Mathematical Physics Having Applications in Engineering. Leningrad, Gosudarstv. Izdat. Tehn.-Teor. Lit., 1950. 368 p. (in Russian).
- Kornev V. V., Khromov A. P. Resolvent approach to the Fourier method in a mixed problem for the wave equation. Comput. Math. Math. Phys., 2015, vol. 55, iss. 4, pp. 618–627. DOI: https://doi.org/10.1134/S0965542515040077
- Kurdyumov V. P., Khromov A. P. Obosnovanie metoda Fu r’e dlia volnovogo uravneniia priminimal’nykh trebovaniiakh na iskhodnye dannye [Justification of the Fourier method for the wave equation with minimum requirements for initial data]. Matematika. Mekhanika [Mathematics. Mechanics]. Saratov, Saratov Univ. Press, 2015, iss. 17, pp. 32–36 (in Russian).
- Gurevich A. P., Kurdyumov V. P., Khromov A. P. Justification of Fourier method in a mixed problem for wave equation with non-zero velocity. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2016, vol. 16, iss. 1, pp. 13–29 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2016-16-1-13-29
- Khromov A. P. Behavior of the formal solution to a mixed problem for the wave equation. Comput. Math. Math. Phys., 2016, vol. 56, iss. 2, pp. 243–255. DOI: https://doi.org/10.1134/S0965542516020135
- Steklov V. A. Fundamental Problems of Mathematical Physics. Moscow, Nauka, 1983. 432 p. (in Russian).
- Petrovsky I. G. Lectures on partial differential equations. New York, Interscience, 1954. 245 p. (Russ. ed.: Moscow, Gosudarstv. Izdat. Tehn.-Teor. Lit., 1953. 360 p.)
- Smirnov V. I. A Course of Higher Mathematics. Vol. 4. Reading, Mass., Addison-Wesley, 1964. (Russ. ed.: Moscow, Gostekhizdat, 1953. 804 p.)
- Ladyzhenskaya O. A. Mixed Problems for Hyperbolic Equations. Moscow, Gostekhizdat, 1953. 282 p. (in Russian).
- Il’in V. A. Selected Works. Vol. 1. Moscow, Maks-press, 2008. 727 p. (in Russian).
- Il’in V. A. The solvability of mixed problems for hyperbolic and parabolic equations. Russ. Math. Surv., 1960, vol. 15, iss. 1, pp. 85–142.
- Naimark M. A. Linear Differential Operators: Two Volumes Bound as One. Dover Publications, Inc., 2012. 528 p. (Russ. ed.: Moscow, Nauka, 1969. 528 p.)
- Rasulov M. L. Contour Integral Method and Its Application to Problems f or Differential Equations. Moscow, Nauka, 1964. 462 p. (in Russian).
- Vagabov A. I. Introduction to the Spectral Theory of Differential Operators. Rostov on Don, Izd-vo Rost. un-ta, 1994. 160 p. (in Russian).
- Marchenko V. A. Sturm – Liouville Operators and Their Applications. Kiev, Naukova Dumka, 1977. 392 p. (in Russian).
- Lavrentiev M. A., Shabat B. V. Methods of the Theory of Functions of a Complex Variable. Moscow, Nauka, 1965. 716 p. (in Russian).
- Carleson L. On convergence and growth of partial sums of Fourier series. Acta Math., 1966, vol. 116, no. 1, pp. 135–157.

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