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Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 2, Pages 258–275 (Mi izv1240)  

This article is cited in 3 scientific papers (total in 5 papers)

On the asymptotics of the solution of a problem with a small parameter

A. M. Il'in


Abstract: The problem $\partial_tu+\partial_x\varphi(u)=\varepsilon\partial_x^2u$, $u(x,t_0)=\psi(x)$, is considered, where $\varphi,\psi\in C^\infty$, $\varphi"(u)>0$, $0\leqslant\varepsilon\ll1$. It is assumed that for $\varepsilon=0$ the problem has a generalized solution with one smooth line of discontinuity, so that this line, modeling a shock wave, appears within the strip $\Omega=\{t_0\leqslant t\leqslant T\}$. The asymptotics of a solution, uniform in $\Omega$ up to any degree in $\varepsilon$, is constructed and justified.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 34:2, 261–279

Bibliographic databases:

UDC: 517.956
MSC: Primary 35K55, 35B25, 35C20; Secondary 76L05
Received: 24.03.1986
Revised: 17.01.1988

Citation: A. M. Il'in, “On the asymptotics of the solution of a problem with a small parameter”, Izv. Akad. Nauk SSSR Ser. Mat., 53:2 (1989), 258–275; Math. USSR-Izv., 34:2 (1990), 261–279

Citation in format AMSBIB
\Bibitem{Ili89}
\by A.~M.~Il'in
\paper On the asymptotics of the solution of a~problem with a~small parameter
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1989
\vol 53
\issue 2
\pages 258--275
\mathnet{http://mi.mathnet.ru/izv1240}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=998296}
\zmath{https://zbmath.org/?q=an:0703.35012}
\transl
\jour Math. USSR-Izv.
\yr 1990
\vol 34
\issue 2
\pages 261--279
\crossref{https://doi.org/10.1070/IM1990v034n02ABEH000629}


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    This publication is cited in the following articles:
    1. V. M. Babich, L. A. Kalyakin, M. D. Ramazanov, N. Kh. Rozov, “Arlen Mikhailovich Il'in (on the occasion of the 70th anniversary)”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S1–S7  mathnet  mathscinet  zmath  elib
    2. Allaberen Ashyralyev, Yaşar Sözen, “A note on the parabolic equation with an arbitrary parameter at the derivative”, Mathematical and Computer Modelling, 54:11-12 (2011), 2565  crossref
    3. “Arlen Mikhailovich Ilin (k vosmidesyatiletiyu so dnya rozhdeniya)”, Ufimsk. matem. zhurn., 4:2 (2012), 3–12  mathnet  mathscinet
    4. S. V. Zakharov, “Dvukhparametricheskie asimptotiki v bisingulyarnoi zadache Koshi dlya parabolicheskogo uravneniya”, Tr. IMM UrO RAN, 23, no. 2, 2017, 94–103  mathnet  crossref  elib
    5. S. F. Dolbeeva, V. N. Pavlenko, S. V. Matveev, O. N. Dementev, A. V. Melnikov, E. A. Sbrodova, A. A. Solovev, V. I. Ukhobotov, V. E. Fedorov, E. A. Fominykh, A. A. Ershov, “Arlen Mikhailovich Ilin. 85 let so dnya rozhdeniya”, Chelyab. fiz.-matem. zhurn., 2:1 (2017), 5–9  mathnet  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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