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Kurdyumov V. P. Classic and generalized solutions of the mixed problem for wave equation with a summable potential. Part I. Classic solution of the mixed problem. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2023, vol. 23, iss. 3, pp. 311-319. DOI: 10.18500/1816-9791-2023-23-3-311-319, EDN: GUFKKJ

# Classic and generalized solutions of the mixed problem for wave equation with a summable potential. Part I. Classic solution of the mixed problem

The resolvent approach and the using of the idea of A. N. Krylov on the acceleration of convergence of Fourier series, the properties of a formal solution of a mixed problem for a homogeneous wave equation with a summable potential and a zero initial function are studied. This method makes it possible to obtain deep results on the convergence of a formal series with arbitrary boundary conditions and without overestimating the requirements for the smoothness of the initial data. The different-order boundary conditions considered in the article are such that the operator corresponding to the spectral problem may have an infinite set of multiple eigenvalues and their associated functions. A classical solution is obtained without overstating the requirements for the initial velocity $u'_t(x,0) = \psi(x)$. It is shown that for $\psi(x) \in L[0,1]$ the formal solution, being the uniform limit of the classical ones, is a generalized solution, and when $\psi(x) \in L_p[0,1], ~ 1 <p\leqslant 2$, the formal solution has much smoother properties than the case $\psi(x) \in L[0,1]$.

- Naymark M. A.
*Lineynye differentsial'nye operatory*[Linear Differential Operators]. Moscow, Nauka, 1969. 528 p. (in Russian). - Khromov A. P. Mixed problem for homogeneous wave equation with non-zero initial velocity.
*Computational Mathematics and Mathematical Physics*, 2018, vol. 58, iss. 9, pp. 1531–1543. https://doi.org/10.1134/S0965542518090099 - Kurdyumov V. P., Khromov A. P., Khalova V. A. Mixed problem for a homogeneous wave equation with a nonzero initial velocity with 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, EDN: BEUDSC - Burlutskaya M. S., Khromov A. P. Resolvent approach in the Fourier method.
*Doklady Mathematics*, 2014, vol. 90, iss. 2, pp. 545–548. https://doi.org/10.1134/S1064562414060076, EDN: UFVTOF - 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, EDN: WSPZGX - Burlutskaya M. S., Khromov A. P. Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders.
*Differential Equations*, 2017, vol. 53, iss. 4, pp. 497–508. https://doi.org/10.1134/S0012266117040085, EDN: XNEPDT - Carleson L. On convergence and growth of partial sums of Fourier series.
*Acta Mathematica*, 1966, vol. 116, iss. 1, pp. 135–157. https://doi.org/10.1007/BF02392815 - Hunt R. On the convergence of Fourier series. In:
*Orthogonal Expansious and Their Continuous Analogues.*Proceedings of the Conference Held at Southern Illinois University, Edwardsville, April 27–29, 1967. Carbondale, JL, Southern Illinois University Press, 1968, pp. 235–255. - Il'in V. A. Existence of a reduced system of eigen- and associated functions for a nonselfadjoint ordinary differential operator.
*Proceedings of the Steklov Institute of Mathematics*, 1979, vol. 142, pp. 157–164. https://www.mathnet.ru/eng/tm2564 - Gurevich A. P., Kurdyumov V. P., Khromov A. P. Justification of the Fourier method in a mixed problem for a wave equation with a nonzero initial velocity.
*Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2016, vol. 16, iss. 1, pp. 13–29 (in Russian). https://doi.org/10.18500/1816-9791-2016-16-1-13-29, EDN: VUSODD - Rasulov M. L.
*Metod konturnogo integrala*[Contour Integral Method]. Moscow, Nauka, 1964. 462 p. (in Russian). - Vagabov A. I.
*Vvedeniye v spektral'nuyu chuvstvitel'nost' differentsial'nykh reaktsiy*[Introduction to the Spectral Theory of Differential Operators]. Rostov-on-Don, Rostov University Publ., 1994. 160 p. (in Russian).

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