Infinite system of the ordinary differential equations necessary for the solution of one initial and regional task, by means of trigonometrical ranks

About the authors:

Sergei P. Bautin, Dr. Sci. (Phys.-Math), Professor, Department of Higher and Applied Mathematics, Snezhinsk Physical-Technical Institute, National Research Nuclear University MEPhI (Snezhinsk); eLibrary AuthorID, sbautin@usurt.ru

Olga D. Zorina, Postgraduate, Department of the higher and applied mathematics, Ural state university of railway transport

Abstract:

Non-stationary solutions of full system of the equations of Navier−Stokes in a onedimensional case are under construction by means of infi nite trigonometrical series and thereby currents of the compressed viscous heat-conducting gas are modelled. The case when entry conditions for the full system of the equations of Navier−Stokes transfer the uniform based gas is considered. And as a total state it is required to come into a fortune of non-uniform rest with a linear profi le of the temperature. For creation of the solution of objective special representations of trigonometrical ranks which coeffi cients are required functions from time are used. For required coeffi cients by means of procedure of projection the infi nite system of the ordinary differential equations in a normal form is written out. Thanks to concrete identical transformations the system is signifi cantly simplified: in the right parts of the equations there are no double sums.

References:

1. Bautin, S. P. Predostavlenie reshenij sistemy uravnenij Navie-Stoksa v okrestnosti kontaktnoj kharakteristiki // Prikladnaia matematika i mekhanika. 1987. 51, vip.4. Pp. 574584.

2. Bautin, S. P. Analiticheskoe postroenie techenij viazkogo gaza s pomoschyu posledovatelnosti linerizovannykh sistem Navie-Stoksa//Prikladnaja matematika i mekhanika.1988.T. 52, vip. 4. Рр. 579-589.

 3. Bautin, S. P. Kharakteristicheskie poverkhnosti v techeniakh gaza// Prikladnaja matematika i mekhanika. 2001. T. 65, vip. 5. Рр. 862-875.

4.  Bautin, S. P. Characteristic Cauchy problem and its appendices in gas dynamics. Novosibirsk: Science, 2009. 368 p.

5.  Bautin, S. P., Zamyslov, V. E. Submission of approximate decisions of full system of the equations of Navier-Stokes in a one-dimensional case // Computing technologies, 2012. T. 17. № 3. Pр. 3-12.

6. Zamislov, V. E., Skachkov, P. P. Sravnenie dvukh priblizhennikh metodov reshenija odnoj nachalno-kraevoj zadachi gazovoj dinamiki s uchetom viazkosti i teploprovodnosti// Vestnik UrGUPS. 2012. № 4 (16). Рр. 29-38.

7.  Bautin, S. P., Zamyslov, V. E. One-dimensional periodic currents of viscous heatconducting gas // Messenger of URGUPS. 2013. № 1(17). Pр. 4-13.

8.  Zamyslov, V. E. Standing waves of full system of an uravnenya of Navier-Stokes in a one-dimensional case // Computing technologies. 2013. T. 18. № 2. Pр. 33-45.

 9.  Bautin, S. P., Zamislov, V. E., Skachkov, P. P. Matematicheskoe modelirovanie trigonometricheskimi rjadami odnomernykh techenij vjazkogo teploprovodnogo gaza. Novosibirsk: Nauka, 2014. 90 р.

10.  Bautin, S. P. Odno predstavlenie periodicheskikh trekhmernykh nestatsionarnykh reshenij polnoj sistemy uravnenij Navie-Stoksa. Preprint. Ekaterinburg: Izd-vo UrGUPS, 2015. 46 р.