Abstract:
The article gives a new short proof the V. A. Chernyatin theorem about the classical solution of the Fourier method of the mixed problem for the wave equation with fixed ends with minimum requirements on the initial data. Next, a similar problem for the simplest functional differential equation of the first order with involution in the case of the fixed end is considered, and also obtained definitive results. These results are due to a significant use of ideas A. N. Krylova to accelerate the convergence of series, like Fourier series. The results for other similar mixed problems given without proof.
Key words:
mixed problem, Fourier method, involution, classical solution, asymptotic form of eigenvalues and eigenfunctions, Dirac system.
Bibliographic databases:
Document Type:
Article
UDC:517.95+517.984
Language: Russian
Citation:
A. P. Khromov, M. Sh. Burlutskaya, “Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data”, Izv. Saratov Univ. Math. Mech. Inform., 14:2 (2014), 171–198
\Bibitem{KhrBur14}
\by A.~P.~Khromov, M.~Sh.~Burlutskaya
\paper Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2014
\vol 14
\issue 2
\pages 171--198
\mathnet{http://mi.mathnet.ru/isu501}
\crossref{https://doi.org/10.18500/1816-9791-2014-14-2-171-198}
\elib{https://elibrary.ru/item.asp?id=21719217}
Linking options:
https://www.mathnet.ru/eng/isu501
https://www.mathnet.ru/eng/isu/v14/i2/p171
This publication is cited in the following 7 articles:
K. Yu. Malyshev, “Predstavlenie funktsii Grina volnovogo uravneniya na otrezke v konechnom vide”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 22:4 (2022), 430–446
V. L. Leontiev, “Fourier method related with orthogonal splines in parabolic initial boundary value problem for domain with curvilinear boundary”, Ufa Math. J., 14:2 (2022), 56–66
Baskakov A.G., Polyakov D.M., “Fourier Method For a Mixed Problem With the Hill Operator”, Differ. Equ., 56:6 (2020), 679–684
M. Sh. Burlutskaya, “Classical and generalized solutions of a mixed problem for a system of first-order equations with a continuous potential”, Comput. Math. Math. Phys., 59:3 (2019), 355–365
I. S. Mokrousov, “Kriterii prinadlezhnosti klassu WlpWlp obobschennogo iz klassa LpLp resheniya volnovogo uravneniya”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 18:3 (2018), 297–304
M. Sh. Burlutskaya, “Fourier method in a mixed problem for the wave equation on a graph”, Dokl. Math., 92:3 (2015), 735
Sarsenbi A., Sadybekov M., “Eigenfunctions of a Fourth Order Operator Pencil”, International Conference on Analysis and Applied Mathematics (ICAAM 2014), AIP Conference Proceedings, 1611, eds. Ashyralyev A., Malkowsky E., Amer Inst Physics, 2014, 241–245
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