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The computer calculation of Lie point symmetries of large systems of differential equations. (English) Zbl 0875.65079


MSC:

65L99 Numerical methods for ordinary differential equations
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
68W30 Symbolic computation and algebraic computation
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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[1] Olver, P. J., Applications of Lie Groups to Differential Equations (1988), Springer: Springer New York
[2] MACSYMA Reference Manual, Version 13, Computer Aided Mathematics Group (1989), Symbolics: Symbolics Burlington, MA
[3] MACSYMA User’s Guide, Computer Aided Mathematics Group (1988), Symbolics: Symbolics Burlington, MA
[4] Ames, W. F., Nonlinear Partial Differential Equations in Engineering (1972), Academic: Academic New York · Zbl 0255.35001
[5] Bluman, G. W.; Cole, J. D., Similarity Methods for Differential Equations (1974), Springer: Springer New York · Zbl 0292.35001
[6] Miller, W., Symmetry and Separation of Variables (1977), Addison-Wesley: Addison-Wesley Reading, MA
[7] Anderson, R. L.; Ibragimov, N. H., Lie-Bäcklund Transformations in Applications, (Studies in Applied Mathematics, vol. 1 (1979), SIAM: SIAM Philadelphia) · Zbl 0424.53004
[8] Ovsiannikov, L. V., Group Analysis of Differential Equations (1982), Academic: Academic New York · Zbl 0485.58002
[9] Winternitz, P., Nonlinear Phenomena, (Wolf, K. B., Lecture Notes in Physics, vol. 189 (1983), Springer: Springer New York), 263
[10] Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics (1985), Reidel: Reidel Boston
[11] Kalnins, E. G., Separation of Variables for Riemannian Spaces of Constant Curvature (1986), Longman: Longman Essex · Zbl 0658.53041
[12] Sattinger, D. H.; Weaver, O. L., Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics (1986), Springer: Springer New York · Zbl 0595.22017
[13] Fushchich, V. I.; Nikitin, A. G., Symmetries of Maxwell Equations (1987), Reidel: Reidel Dordrecht · Zbl 0644.35004
[14] (Levi, D.; Winternitz, P., Symmetries and Nonlinear Phenomena (1988), World Scientific: World Scientific Singapore) · Zbl 0747.00041
[15] Schwarz, F., SIAM Rev., 30, 450 (1988)
[16] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations (1989), Springer: Springer New York · Zbl 0718.35003
[17] Rogers, C.; Ames, W. F., Nonlinear Boundary Value Problems in Science and Engineering (1989), Academic: Academic New York · Zbl 0699.35004
[18] Stephani, H., Differential Equations: Their Solution using Symmetries (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0704.34001
[19] Winternitz, P., Partially Integrable Nonlinear Evolution Equations and their Physical Applications, (Conte, R.; Boccara, N. (1990), Kluwer: Kluwer Dordrecht), 515
[20] Vinogradov, A. M., Symmetries of partial differential equations, Part III. Symmetries of partial differential equations, Part III, Acta Appl. Math., 16 (1989), 2 · Zbl 0678.00015
[21] Schwarz, F., Comput. Phys. Commun., 27, 179 (1982)
[22] Computing, 36, 279 (1986), Addendum
[23] Schwarz, F., Comput. Phys. Commun., 39, 285 (1986)
[24] Schwarz, F., The Package SPDE for Determining Symmetries of Partial Differential Equations, User’s Manual (1987), Rand Corp: Rand Corp Santa Monica, CA, distributed with REDUCE 3.3
[25] Schwarz, F., Trends in Computer Algebra, (JanBen, R., Lecture Notes in Computing Science, vol. 296 (1988), Springer: Springer New York), 167
[26] Edelen, D. G.B., Isovector Methods for Equationa of Balance (1981), Sijthoff & Noordhoff: Sijthoff & Noordhoff Alphen aan den Rijn
[27] Gragert, P.; Kersten, P. H.M.; Martini, A., Acta Appl. Math., 1, 43 (1983)
[28] Kersten, P. H.M., Infinitesimal symmetries: a computational approach, (CWI Tract 34 (1987), Center for Mathematics and Computer Science: Center for Mathematics and Computer Science Amsterdam) · Zbl 0648.68052
[29] Kersten, P. H.M., Acta Appl. Math., 16, 207 (1989)
[30] Eliseev, V. P.; Fedorova, R. N.; Kornyak, V. V., Comput. Phys. Commun., 36, 383 (1985)
[31] Fedorova, R. N.; Kornyak, V. V., A REDUCE program for computing determining equations of Lie-Bäcklund symmetries of differential equations, (Report R11-87-19 (1987), JINR: JINR Dubna) · Zbl 0567.34001
[32] Fedorova, R. N.; Kornyak, V. V., Comput. Phys. Commun., 39, 93 (1986)
[33] Fushchich, W. I.; Kornyak, V. V., J. Symb. Comput., 7, 611 (1989)
[34] Wolf, T., Analytic solutions of differential equations with computer algebra systems, (Preprint 87/5 (1987), Universität Jena)
[35] Head, A. K., LIE: A muMATH Program for the calculation of the LIE algebra of differential equations (1990), CSIRO Division of Material Sciences: CSIRO Division of Material Sciences Clayton
[36] Vafeades, P.; Park, E. K., Proc. ISMM Int. Symp. Computer Applications in Design, Simulation and Analysis, 310 (1990), New Orleans, Louisiana
[37] P. Vafeades, SYMCON: a MACSYMA package for the determination of symmetries and conservation laws of PDEs, J. Symb. Comput., submitted.; P. Vafeades, SYMCON: a MACSYMA package for the determination of symmetries and conservation laws of PDEs, J. Symb. Comput., submitted.
[38] Popov, M. D., Izv. Akad. Nauk B. SSR Ser. Fiz. Mat., 2, 33 (1985)
[39] Bocharov, A. V., DEliA: a system of exact analysis of differential equations using S. Lie approach, (Report OWIMEX (1989), Program Systems Institute, Pereslavl-Zalessky: Program Systems Institute, Pereslavl-Zalessky USSR)
[40] Bocharov, A. V.; Bronstein, M. L., Acta Appl. Math., 16, 143 (1989)
[41] Davison, D. K., MANDO: a computer program for symbolic manipulation of differential operators generating continuous transformations, (M.Sc. Thesis (1973), University of the Pacific: University of the Pacific Stockton, CA)
[42] Nagao, G. G., DETERMININGEQS: a computer program for approximating Lie generators admitted by dynamical systems, (M.Sc. Thesis (1980), University of the Pacific: University of the Pacific Stockton, CA)
[43] Rosenau, P.; Schwarzmeier, J. L., Similarity solutions of systems of partial differential equations using MACSYMA, (Report COO-3077-160 MF-94 (1979), Courant Institute of Mathematical Sciences, New York University)
[44] Schwarzmeier, J. L.; Rosenau, P., Using MACSYMA to calculate similarity transformations of partial differential equations, (Report LAUR 88-4157 (1988), Los Alamos National Laboratory)
[45] Steinberg, S., Proc. 1979 MACSYMA User’s Conf., (Lewis, V. E. (1979), MIT Press: MIT Press Boston), 408
[46] Steinberg, S., MACSYMA Newsletter, 7, 3 (1990)
[47] Champagne, B.; Winternitz, P., A MACSYMA program for calculating the symmetry group of a system of differential equations, (Report CRM-1278 (1985), Centre de Recherches Mathématiques: Centre de Recherches Mathématiques Montréal) · Zbl 0643.35097
[48] Rosencrans, S. I., Comput. Phys. Commun., 38, 347 (1985)
[49] Schwarz, F., An algorithm for determining the size of symmetry groups (1989), private communication
[50] Reid, G. J., Differential Equations and Representation Theory, (Hussin, V., Proc. Annual Seminar of the Canadian Math. Soc. (1990), Les Publications de Centre de Recherches Mathématiques: Les Publications de Centre de Recherches Mathématiques Montréal), 363
[51] Reid, G. J., Finding abstract Lie symmetry algebras of differential equations without integrating determining equations, (Technical Report 90-4 (1990), Inst. of Applied Mathematics, The University of British Columbia: Inst. of Applied Mathematics, The University of British Columbia Vancouver) · Zbl 0768.35002
[52] Reid, G. J., J. Phys. A, 23, L853 (1990) · Zbl 0724.35001
[53] Karpman, V. I., Phys. Lett. A, 136, 216 (1989)
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