Abstract
A method for computing symmetries and conservation laws of integro-differential equations is proposed. It resides in reducing an integro-differential equation to a system of boundary differential equations and in computing symmetries and conservation laws of this system. A geometry of boundary differential equations is constructed like the differential case. Results of the computation for the Smoluchowski’s coagulation equation are given.
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References
Ovsiannikov, L. V.: Group Analysis of Differential Equations, Academic Press, New York, 1982.
Ibragimov, N. H.: Group Transformations in the Mathematical Physics, Reidel, Dordrecht, 1985.
Krasil’shchik, I. S., Lychagin, V. V., and Vinogradov, A. M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.
Olver, P.: Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
Vinogradov, A. M.: Local symmetries and conservation laws, Acta Appi Math. 2 (1) (1984), 21–78.
Krasil’shchik, I. S. and Vinogradov, A. M.: Nonlocal symmetries and the theory of coverings, Acta Appl. Math. 2 (1) (1984), 79–96.
Taranov, V. B. : On symmetry of one-dimensional high-freguency motions of collisionless plasma, Zh. Tekh. Fiz. 46(6) (1976), 1271–1277 (in Russian).
Bounimovich, A. I. and Krasnoslobodsev, A. V: Group invariant solutions of kinetic equations, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 4 (1982), 135–140 (in Russian).
Bounimovich, A. I. and Krasnoslobodsev, AV: On some invariant transformations of kinetic equations, Vestnik Moskov. Univ., Ser. 1, Mat. Mekh. 4 (1983), 69–72 (in Russian).
Bobylev, AV : Exact solutions of the nonlinear Boltzmann equations and the theory of relaxation of the Maxwell gas, Teoret. Mat. Fiz. 60(2) (1984), 280–310 (in Russian).
Meleshko, SV: Group properties of equations of motions of a viscoelastic medium, Model. Mekh. 2 (19)(4) (1988), 114–126 (in Russian).
Grigoriev, Yu. N. and Meleshko, S. V: Group theoretical analysis of kinetic Boltzmann equation and its models, Arch. Mech. 42 (6) (1990), 693–701.
Senashov, SI : Group classification of an equation of a viscoelastic rod, Model. Mekh. 4 (21)(1) (1990), 69–72 (in Russian).
Kovalev, VF., Krivenko, SV., and Pustovalov, VV: Symmetry group for kinetic equations for collisionless plasma, Pisma Zh. Eksper. Teoret. Fiz. 55(4) (1992), 256–259 (in Russian).
Chetverikov, V. N. and Kudryavtsev, A. G.: Modelling integro-differential equations and a method for computing their symmetries and conservation laws, Adv. Sov. Math, (to appear in 1995 ).
Drake, R. L.: A general mathematical survey of the coagulation, in G. M. Hidy and J. R. Brock (eds), Topics in Current Aerosol Research, Part 2, International Reviews in Aerosol Physics and Chemistry, Vol. 3, Pergamon, New York, 1972.
Vinokurov, L. I. and Kac, AV Power solutions of the kinetic equation for the stationary coagulation of atmospheric aerosols, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmosfer. Okeana 16(6) (1980), 601 (in Russian).
Lushnikov, A. A.: Evolution of coagulating systems, J. Colloid Interface Sci. 45 (3) (1973), 549–556.
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© 1995 Kluwer Academic Publishers
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Chetverikov, V.N., Kudryavtsev, A.G. (1995). A Method for Computing Symmetries and Conservation Laws of Integro-Differential Equations. In: Kersten, P.H.M., Krasil’Shchik, I.S. (eds) Geometric and Algebraic Structures in Differential Equations. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0179-7_4
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DOI: https://doi.org/10.1007/978-94-009-0179-7_4
Publisher Name: Springer, Dordrecht
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