Efficiently implementing two methods of the geometrical theory of differential equations: An experience in algorithm and software design. (English) Zbl 0693.65087
Authors’ summary: “This paper contains an analysis of two computational schemes for finding the infinitesimal symmetries and conservation laws of arbitrary systems of differential equations. The design of efficient algorithms implementing these computational schemes is described together with the design of a portable software system, SCoLAr, supporting these algorithms. A test run is listed in the Appendix and the prospect of using the SCoLAr program on middle class personal computers is discussed.”
Reviewer: P.Onumanyi
MSC:
| 65Z05 | Applications to the sciences |
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
symmetry; conservation law; computer algebra; portability; efficiency; PASCAL; personal computer; geometrical theorySoftware:
REDUCEReferences:
| [1] | Edelen, D. G. B.: Isovector Methods for Equations of Balance, Sijthoff & Nordhoff, Alphen a/d Rijn, 1981. |
| [2] | Gragert, P. K. H., Kersten, P. H. M., and Martini, R.: Symbolic computations in applied differential geometry, Acta Appl. Math. 1 (1983), 43-77. · Zbl 0539.68026 · doi:10.1007/BF02433841 |
| [3] | Hearn, A. C.: REDUCE User’s Manual Rand Corp., Santa Monica, 1983. |
| [4] | Vinogradov, A. M.: A spectral sequence associated with a nonlinear differential equation and algebro-geometric principia of the Lagrange restricted field theory, Doklady Acad. Nauk SSSR, 238(5) (1978), 1028-1031 (Russian). |
| [5] | Vinogradov, A. M.: Higher symmetries and conservation laws, in Group-Theoretical Methods in Physics, vol. 2, Nauka, Moscow, 1983, pp. 414-420 (Russian). |
| [6] | Vinogradov, A. M.: Local symmetries and conservation laws, Acta Appl. Math. 2 (1984), 21-78. · Zbl 0534.58005 · doi:10.1007/BF01405491 |
| [7] | Zharkov, A. Yu. et al.: Investigating the integrability of nonlinear evolution equations using symbolic manipulation system REDUCE-2, JINR, Dubna, 1983, Report R11-83?914 (Russian). |
| [8] | Gerdt, V. P. and Zharkov, A. Yu.: On application of computer algebra for classification of the integrable systems of nonlinear evolution equations, in Analytic Computations with Application to Theoretical Physics, JINR, Dubna, 1985, p. 225 (Russian). · Zbl 0577.35099 |
| [9] | Eliseev, V. P., Fedorova, R. N., and Kornyak, V. V.: A REDUCE program for determination of the Lie Symmetries of differential equations, JINR, Dubna, 1984, Report R11-84?238 (Russian). · Zbl 0567.34001 |
| [10] | Fedorova, R. N. and Kornyak, V. V.: Applying computer algebra systems for determination of the Lie and Lie-Bäcklund symmetries of differential equations, in Analytic Computations with Application to Theoretical Physics, JINR, Dubna, 1985, p. 248 (Russian). |
| [11] | Fedorova, R. N. and Kornyak, V. V.: Determination of the Lie-Bäcklund symmetries of differential equations using FORMAC, JINR, Dubna, 1985, Preprint E11-85?164. · Zbl 0567.34001 |
| [12] | Svinolupov, S. I. and Sokolov, V. V.: Func. Analiz 16 (1982), 86 (Russian). |
| [13] | Schwartz, F.: Automatically determining symmetries of ordinary differential equations, in Lecture Notes Comp. Sci., 162 (1983), 45-54. |
| [14] | Popov, M. D.: Automatized computing of defining equations for a Lie group, Izvestiya AN Belor. SSR, Ser. Phys.-Math., No. 2 (1985), 33-37 (Russian). · Zbl 0571.35010 |
| [15] | Vinogradov, A. M. and Krasilschik, I. S.: A method for computing higher symmetries of nonlinear evolution equations and the nonlocal symmetries, Doklady Acad. Nauk SSSR, 253(6) (1980), 1289-1293 (Russian). |
| [16] | Bocharov, A. V.: On a class of algorithmic computations in differential algebra with indefinite terms, in Object-oriented Computer Systems, ser. ?Voprosy kibernetiki?, vol. 125, Moscow, 1987 (Russian). |
| [17] | Cartan, E.: Sur la structure des gruppes infinies des transformationes, Ann. Ec. Norm. Super. (3), 21, 1904. |
| [18] | Finikov, S. P.: Cartan’s method of the exterior differential systems, Moscow-Leningrad, 1948 (Russian). |
| [19] | Pommaret, J. F.: Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, 1978. · Zbl 0418.35028 |
| [20] | Ganzha, V. G. and Yanenko, N. N.: Compatibility analysis for systems of differential equations on a computer, ITPM, Novosibirsk, 1982, ITPM Preprint #20-82 (Russian). |
| [21] | Vinogradov, A. M.: Symmetries and conservation laws of partial differential equations: Basic notions and results, Acta. Appl. Math. 15 (1989), 3-21. · Zbl 0692.35002 · doi:10.1007/BF00131928 |
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.
