Infinite-modal approximate solutions of the Bryan-Pidduck equation

Author
O. O. Hukalov1, V. D. Gordevskyy2
1) B. Verkin Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine; 2) V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
Abstract
The nonlinear integro-differential Bryan-Pidduck equation for a model of rough spheres is considered. An approximate solution is constructed in the form of an infinite linear combination of some Maxwellian modes with coefficient functions that depend on time and spatial coordinate. Sufficient conditions for the infinitesimality of the uniformly-integral error between the parts of the Bryan-Pidduck equation are obtained.
Keywords
Bryan-Pidduck equation; rough spheres; uniform-integral error; infinite-modal approximate solutions; global Maxwellian; screws
DOI
doi:10.15330/ms.49.1.95-108
Reference
1. S. Chapman, T.G.Cowling, The mathematical theory of non-uniform gases, Cambridge Univ. Press, Cambridge, 1952.

2. C. Cercignani, M. Lampis, On the kinetic theory of a dense gas of rough spheres, J. Statist. Phys., 53 (1988), 655–672.

3. V.D. Gordevskyy, Explicit approximate solutions of the Boltzmann equation for the model of rough spheres, Reports of the National Academy of Sciences of Ukraine, 4 (2000), 10–13. (in Ukrainian)

4. V.D. Gordevskyy, Approximate billow solutions of the kinetic Bryan-Pidduck equation, Math. Meth. Appl. Sci., 23 (2000), 1121–1137.

5. V.D. Gordevskyy, A.A. Gukalov, Maxwell distributions in a model of rough spheres, Ukr. Mat. Zh., 63 (2011), ¹5, 629–639. (in Russian)

6. C. Cercignani, The Boltzman Equation and its Applications, Springer, New York, 1988.

7. T. Carleman, Problems Mathematiques dans la Theorie Cinetique des Gas, Almqvist & Wiksells, Uppsala, 1957.

8. H. Grad, On the kinetic theory of rarefield gases, Comm. Pure and Appl. Math., 2 (1949), ¹4, 331–407.

9. O.G. Fridlender, Local Maxwellian Solutions of the Boltzmann Equation, J. Appl. Math. Mech., 29 (1965), ¹5, 973–977. (in Russian)

10. V.D. Gordevskyy, On the non-stationary Maxwellians, Math. Meth. Appl. Sci., 27 (2004), ¹2, 231–247.

11. V.D. Gordevskyy, Biflow distributions with screw modes, Theor. Math. Phys., 126 (2001), ¹2, 234–249.

12. A.A. Gukalov, Interaction between ”Accelarating-Packing” Flows for the Bryan-Pidduck Model, J. of Math. Phys., Anal., Geometry, 9 (2013), ¹3, 316–331.

13. V.D. Gordevskyy, Approximate Biflow Solutions of the Kinetic Bryan-Pidduck Equation, Math. Meth. Appl. Sci., 23 (2003), 1121–1137.

14. V.D. Gordevskyy, A.A. Gukalov, Interaction of the Eddy Flows in the Bryan–Pidduck Model, Vysnik Kharkiv Univ., Ser. Mat. Prykl. Mat. Mech., 64 (2011), ¹2, 27–41. (in Russian)

Pages
95-108
Volume
49
Issue
1
Year
2018
Journal
Matematychni Studii
Full text of paper
pdf
Table of content of issue