Geodesic
About the classical solution of the mixed problem for the wave equation
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 1, pp. 56-66.

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The classic solution of the mixed problem for a wave equation with a complex potential and minimal smoothness of initial data is established by the Fourier method. The resolvent approach consists of constructing formal solution with the help of the Cauchy–Poincaré method of integrating the resolvent of the corresponding spectral problem over spectral parameter. The method requires no information about eigen and associated functions and uses only the main part of eigenvalues asymptotics. Krylov's idea of accelerating the convergence of Fourier series is essentially employed. The boundary conditions of the mixed problem can produce multiple spectrum and infinite number of associated functions in the spectral problem, thus making more difficult the analysis of the formal solution. 
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A. P. Khromov. About the classical solution of the mixed problem for the wave equation. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 1, pp. 56-66. http://geodesic.mathdoc.fr/item/ISU_2015_15_1_a8/

[1] Steklov V. A., The main tasks of mathematical physics, Nauka, M., 1983, 432 pp. (in Russian)

[2] Petrovsky I. G., Lectures on partial differential equations, Dover Publ. Inc., 1992, 245 pp.

[3] Smirnov V. I., A Course of Higher Mathematics, in 5 vol., v. 4, Gostekhizdat, M., 1953, 204 pp. (in Russian)

[4] Ladyzhenskaya O. A., Mixed problem for a hyperbolic equation, Gostekhizdat, M., 1953, 282 pp. (in Russian)

[5] Il'in V. A., Selected works, in 2 vol., v. 1, OOO «Maks-press», M., 2008, 727 pp. (in Russian)

[6] Il'in V. A., “The solvability of mixed problems for hyperbolic and parabolic equations”, Rus. Math. Surv., 15:1 (1960), 85–142 | DOI

[7] Chernyatin V. A., Justification of the Fourier method in a mixed problem for partial differential equations, Moscow Univ. Press, M., 1991, 112 pp. (in Russian)

[8] Krylov A. N., On some differential equations of mathematical physics with applications in technical matters, GITTL, L., 1950, 368 pp. (in Russian)

[9] Burlutskaya M. Sh., Khromov A. P., “Initial-boundary value problems for first-order hyperbolic equations with involution”, Doklady Math., 84:3 (2011), 783–786 | DOI

[10] Naymark M. A., Linear differential operators, Nauka, M., 1969, 528 pp. (in Russian)

[11] Rasulov M. L., The method of the contour integral, Nauka, M., 1964, 462 pp. (in Russian)

[12] Vagabov A. I., Introduction to the spectral theory of differential operators, Rostov Univ. Press, Rostov-on-Don, 1994, 106 pp. (in Russian)

[13] Marchenko V. A., Sturm–Liouville Operators and Applications, Naukova Dumka, Kiev, 1977, 332 pp. (in Russian)