REPRESENTATION OF SOLUTIONS TO THE BURGERS EQUATION TRIGONOMETRIC SERIES
https://doi.org/10.26583/vestnik.2022.238
Abstract
The paper describes a technique for representing solutions of a nonlinear partial differential equation – the Burgers equation – in the form of an infinite trigonometric series from a spatial variable. The coefficients of the series are the desired functions of time. The procedure for obtaining an infinite system of ordinary differential equations, the solutions of which set the desired coefficients of the series, is described. Due to the specific properties of the solutions of the considered infinite systems of ordinary differential equations, the theorems on multiple frequencies are proved and the convergence of an infinite trigonometric series in some neighborhood of the point t = 0 and for all values of the independent variable x is investigated. With the help of finite sums, concrete approximate solutions of the Burgers equation are constructed. In particular, it is established that the solution has large values of derivatives in the spatial variable at a finite time under given smooth initial conditions. Which, nevertheless, does not lead to the occurrence of unreasonable oscillations or to the destruction of the solution.
About the Authors
S. P. Bautin
Snezhinsk Institute of Physics and Technology, National Research Nuclear University MEPhI
Russian Federation
Russian Federation
V. E. Zamyslov
Ural State Transport University
Russian Federation
Russian Federation
References
1. Bautin S.P., Zamyslov V.E., Skashkov P.P. Ma-tematicheskoe modelirovanie trigonometricheskimi raj-dami odnomernyh techenii vsajzkogo teploprovodnogo gaza [Mathematical modeling of trigonometric series of one-dimensional flows of a viscous heat-conducting gas]. Novosibirsk: Nauka Publ.; Ekaterinburg: UrGUPS Publ. 2014. 91 p.
2. Mendousse J.S. Nonlinear discipativte distortion of progressive sound waves at moderate amplitudes // J. A coust. Soc. Am. 1953. V. 25. Р. 51–54.
3. Hohlov R.V. K teorii udarnyh radiovoln v nelinein-yh liniajh [On the theory of shock radio waves in non-linear lines] // Radiotehnika I etltktronika. 1961. V. 6. № 6. Р. 917–925 (in Russian).
4. Bautin S.P. Characteristicheskaj zadacha Koshy i ee prilozeniaj v gazovoi dinamike [The characteristic Cauchy problem and its applications in gas dynamics]. Novosibirsk: Nauka, 2009. 368 p. (in Russian).
5. Zamyslov V.E. Stoajchii volny kak resheniaj polnoi sistemy ueavnenii Navie-Stoksa v odnomernom sluchae [Standing waves as solutions of the complete system of Navier-Stokes equations in the one-dimensional case] // Vychislitelnye tehnologii. 2013. V. 18. № 2. Р. 33–45 (in Russian).
6. Kurmaeva K.V., Titov S.S. Specyalnye rajdy s kratnymi chastotami dlaj odnomernyh techenii szy-maemogo gaza v obobchenii teorem V.E. Zamyslova // [Special series with multiple frequencies for one-dimensional flows of compressible gas in generalization of V.E. Zamyslov's theorems] // Vestnik Uralskogo gosudarstvennogo universiteta putei soobsheniaj. 2016. № 3. Р. 18–28 (in Russian).
7. Landau L.D., Lifcsych E.M. Teoreticheskaja fizika. vol.VI. Gidrodinamika [Theoretical physics. Vol. VI. Hydrodynamics]. M.: Nauka Publ, 1988. 736 p.
8. Aldoshyna I., Pritts R, Приттс Р. Muzykalnaaj akustika [Musical acoustics]. SPb.: Kompozitor publ., 2006. 260 p.
9. Kovalevskaj S.V. Nauchnye Trudy. K teorii differ-encyalnyh uravnenii s chastnymi proizvodnyvi [Scien-tific works. On the theory of partial differential equa-tions]. M.: Izd-vo AN SSSR Publ., 1948. 368 p.
10. Bautin S.P. Analiticheskaja teplovaja volna [Analytical heat wave]. M.: Fizmatlit Publ., 2003. 88 p.
Review
For citations:
Bautin S.P., Zamyslov V.E. REPRESENTATION OF SOLUTIONS TO THE BURGERS EQUATION TRIGONOMETRIC SERIES. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2022;11(4):305-318. (In Russ.) https://doi.org/10.26583/vestnik.2022.238
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