×

Symmetry and solution of neutron transport equations in nonhomogeneous media. (English) Zbl 1474.35044

Summary: We propose the group-theoretical approach which enables one to generate solutions of equations of mathematical physics in nonhomogeneous media from solutions of the same problem in a homogeneous medium. The efficiency of this method is illustrated with examples of thermal neutron diffusion problems. Such problems appear in neutron physics and nuclear geophysics. The method is also applicable to nonstationary and nonintegrable in quadratures differential equations.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds

References:

[1] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations. Symmetries and Differential Equations, Applied Mathematical Sciences 81 (1989), Berlin, Germany: Springer, Berlin, Germany · Zbl 0698.35001
[2] Ovsiannikov, L. V., Group Analysis of Differential Equations (1982), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0485.58002
[3] Olver, P. J., Applications of Lie Groups to Differential Equations (1986), New York, NY, USA: Springer, New York, NY, USA · Zbl 0588.22001
[4] Olver, P. J., Direct reduction and differential constraints, Proceedings of the Royal Society A: Mathematical and Physical Sciences, 444, 1922, 509-523 (1994) · Zbl 0814.35003 · doi:10.1098/rspa.1994.0035
[5] Olver, P. J.; Rosenau, P., The construction of special solutions to partial differential equations, Physics Letters A, 114, 3, 107-112 (1986) · Zbl 0937.35501 · doi:10.1016/0375-9601(86)90534-7
[6] Fushchich, W. I.; Tsyfra, I. M., On a reduction and solutions of nonlinear wave equations with broken symmetry, Journal of Physics A: Mathematical and General, 20, 2, L45-L48 (1987) · Zbl 0663.35045 · doi:10.1088/0305-4470/20/2/001
[7] Levi, D.; Winternitz, P. W., Nonclassical symmetry reduction: example of the Boussinesq equation, Journal of Physics A: Mathematical and General, 22, 15, 2915-2924 (1989) · Zbl 0694.35159 · doi:10.1088/0305-4470/22/15/010
[8] Clarkson, P. A., New similarity solutions for the modified Boussinesq equation, Journal of Physics A: Mathematical and General, 22, 13, 2355-2367 (1989) · Zbl 0704.35116 · doi:10.1088/0305-4470/22/13/029
[9] Clarkson, P. A.; Kruskal, M. D., New similarity reductions of the Boussinesq equation, Journal of Mathematical Physics, 30, 10, 2201-2213 (1989) · Zbl 0698.35137 · doi:10.1063/1.528613
[10] Tsyfra, I. M., Non-local ansätze for nonlinear heat and wave equations, Journal of Physics A: Mathematical and General, 30, 6, 2251-2262 (1997) · Zbl 0932.35008 · doi:10.1088/0305-4470/30/6/042
[11] Zhdanov, R. Z.; Tsyfra, I. M.; Popovych, R. O. A., A precise definition of reduction of partial differential equations, Journal of Mathematical Analysis and Applications, 238, 1, 101-123 (1999) · Zbl 0936.35012 · doi:10.1006/jmaa.1999.6511
[12] Megrabov, A. G., On group approach to inverse problems for differential equations, Soviet Physics Doklady, 308, 3, 583-586 (1984) · Zbl 0599.35132
[13] Kozachok, I. A., Direct problem of the neutron-neutron method: statement and solution using the \(P_2\)-approximation, Geophysical Journal, 5, 3, 3-10 (1983)
[14] Beckurts, K. H.; Wirtz, K., Neutron Physics (1964), New York, NY, USA: Springer, New York, NY, USA · Zbl 0129.22801
[15] Boyer, C., The maximal “kinematical” invariance group for an arbitrary potential, Helvetica Physica Acta, 47, 589-605 (1974)
[16] Fushchich, W.; Symenoh, Z.; Tsyfra, I., Symmetry of the Schrödinger equation with variable potential, Journal of Nonlinear Mathematical Physics, 5, 1, 13-22 (1998) · Zbl 0946.35076 · doi:10.2991/jnmp.1998.5.1.3
[17] Symenoh, Z.; Tsyfra, I., Equivalence transformations and symmetry of the Schrödinger equation with variable potential, Proceedings of the 2nd International Conference on Symmetry in Nonlinear Mathematical Physics (SNMP ’97) · Zbl 0947.35146
[18] Kaplansky, I., An Introduction to Differential Algebra (1957), Paris, France: Hermann, Paris, France · Zbl 0083.03301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.