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Approximate symmetries and solutions of the Kompaneets equation. (English. Russian original) Zbl 1297.35183

J. Appl. Mech. Tech. Phys. 55, No. 2, 220-224 (2014); translation from Prikl. Mekh. Tekh. Fiz. 55, No. 2, 38-42 (2014).
Summary: Different approximations of the Kompaneets equation are studied using approximate symmetries, which allows consideration of the contributions of all terms of this equation previously neglected in the analysis of the limiting cases.

MSC:

35Q35 PDEs in connection with fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35B06 Symmetries, invariants, etc. in context of PDEs
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
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[1] A. S. Kompaneets, ”Establishment of Thermal Equilibrium between Quanta and Electrons,” Zh. Eksp. Teor. Fiz. 31(5), 876–885 (1956) [Sov. Phys. JETP 4, 730 (1957)]. · Zbl 0108.42903
[2] Ya. B. Zel’dovich, ”Interaction of Free Electrons with Electromagnetic Radiation,” Usp. Fiz. Nauk 115(5), 161–197 (1975) [Sov. Phys. Uspekhi 18 (2), 79–98 (1975)]. · doi:10.3367/UFNr.0115.197502a.0161
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[4] V. A. Baikov, R. K. Gazizov, and N. Kh. Ibragimov, ”Approximate Symmetries,” Mat. Sb. 136(4), 435–450 (1988) [Math. USSR Sbornik 64 (2), 427–441 (1989)].
[5] Yu. Yu. Bagderina and R. K. Gazizov, ”Invariant Representation and Symmetry Reduction for Differential Equations with a Small Parameter,” Comm. Nonlinear Sci. Numer. Simulat. 9(1), 3–11 (2004). · Zbl 1041.35003 · doi:10.1016/S1007-5704(03)00010-8
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