Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations

Journal of Symbolic Computation

Volumn 23, Issue 5-6, 1997, Pages 517-533

  Mansfield, Elizabeth L ^a   Clarkson, Peter A ^a  

^a UNIVERSITY OF KENT   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031142055     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsco.1996.0105     Document Type: Article

References (38)

1. Ablowitz, M.J., Clarkson, P.A. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering. L.M.S. Lect. Notes Math., 149, Cambridge, Cambridge University Press.
2. Bluman, G.W., Cole, J.D. (1969). The general similarity solution of the heat equation. J. Math. Mech. 18:1025-1042.
3. Bluman, G.W., Kumei, S. (1989) Symmetries and Differential Equations. Appl. Math. Sci., 81, Berlin, Springer-Verlag.
4. Buchberger, B. (1985) Gröbner bases: an algorithmic method in polynomial ideal theory. In Bose, N.K., ed., Recent Trends in Multidimensional Systems Theory, Dordrecht, Reidel, pp. 184-232.
5. Champagne, B., Hereman, W., Winternitz, P. (1991). The computer calculation of Lie point symmetries of large systems of differential equations. Comp. Phys. Comm. 66:319-340
6. Char, B.W., Geddes, K.O., Gonnet, G.H., Leong, B.L., Monagan, M.B., Watt, S.M. (1991). Maple Library Reference Manual. New York, Springer-Verlag
7. Clarkson, P.A., (1996). Nonclassical symmetry reductions for the Boussinesq equation. Chaos, Solitons and Fractals, 5:2261-2301.
8. Clarkson, P.A., Kruskal, M.D. (1989). New similarity solutions of the Boussinesq equation. J. Math. Phys. 30:2201-2213.
9. Clarkson, P.A., Mansfield, E.L. (1993). Nonclassical symmetry reductions and exact solutions of nonlinear reaction-diffusion equations. In Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, Clarkson, P.A., ed., Dordrecht, Kluwer, 375-389
10. Clarkson, P.A., Mansfield, E.L. (1994a). Symmetry reductions and exact solutions of a class of nonlinear heat equations. Physica 70D:250-288.
11. Clarkson, P.A., Mansfield, E.L. (1994b). On a shallow water wave equation. Nonlinearity 7:975-1000.
12. Clarkson, P.A., Mansfield, E.L. (1994c). Algorithms for the nonclassical method of symmetry reduction. SIAM J. Appl. Math. (to appear).
13. Clarkson, P.A., Olver, P.J. (1994). Symmetries and the Chazy Equation. J. Diff. Eqns. (to appear).
14. Cox, D., Little, J., O'Shea, D.O. (1992). Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics, New York, Springer-Verlag.
15. Hawkins, T. (1994). The birth of Lie's theory of groups. Math. Intell. 16:6-17.
16. Head, A.K. (1993). LIE: A PC Program for Lie Analysis of Differential Equations. Comp. Phys. Comm. 77:241-248.
17. Hereman, W. (1994). Review of symbolic software for the computation of Lie symmetries of differential equations. Euromath Bull. 1:45-79.
18. Ibragimov, N.Kh. (1992). Group analysis of ordinary differential equations and the invariance principle in mathematical physics. Russian Math. Surveys 47:89-156.
19. Ibragimov, N.Kh. (1994). CRC Handbook of Lie Group Analysis of Differential Equations. Volume I: Symmetries, Exact Solutions and Conservation Laws. Boca Raton, CRC Press.
20. Janet, M. (1929) Leçons sur les Systèmes d'Équations. Gauthier-Villers, Paris.
21. Kandri-Rody, A., Weispfenning, V. (1990). Noncommutative Gröbner bases in algebras of solvable type. J. Symbolic Computation 9:1-26.
22. Lisle, I.G. (1992). Equivalence Transformations for Classes of Differential Equations. Ph.D. Thesis, University of British Columbia, Vancouver, Canada.
23. Mansfield, E.L. (1993). diffgrob2: A symbolic algebra package for analysing systems of PDES using MAPLE, ftp euclid.exeter.ac.uk, login: anonymous, password: your email address, directory: pub/liz
24. Mansfield, E.L., Fackerell, E.D. (1992). Differential Gröbner bases. Macquarie University preprint 92/108.
25. Mora, T. (1994). An introduction to commutative and noncommutative Gröbner bases. Theoretical Computer Science 134:131-173.
26. Olver, P.J. (1993) Applications of Lie Groups to Differential Equations. Second Edition, New York, Springer-Verlag.
27. Ovsiannikov, L.V. (1982). Group Analysis of Differential Equations. Ames W.F. trans., New York, Academic Press.
28. Pankrat'ev, E.V. (1989). Computations in differential and difference modules. Acta Appl. Math. 16:167-189.
29. Reid, G.J. (1991a). Algorithms for reducing a system of PDES to standard form, determining the dimension of its solution space and calculating its Taylor series solution. Europ. J. Appl. Math. 2:293-318.
30. Reid, G.J. (1991b). Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. Europ. J. Appl. Math. 2:319-340.
31. Reid, G.J., Wittkopf, A. (1993). A Differential Algebra Package for MAPLE, ftp 137.82.36.23 login: anonymous, password: your email address, directory: pub/standardform
32. Riquier, C. (1910). Les Systèmes d'Equations aux Dérivées Partielles. Gauthier-Villars, Paris.
33. Schwarz, F. (1985). Automatically determining symmetries of partial-differential equations. Computing 34:91-106.
34. Schwarz, F. (1992). An algorithm for determining the size of symmetry groups. Computing 49:95-115.
35. Sherring, J., Prince, G. (1993). DIMSYM-Symmetry Determination and Linear Differential Equations Package. LaTrobe University preprint.
36. Topunov, V.L. (1989). Reducing systems of linear-differential equations to a passive form. Acta Appl. Math. 16:191-206.
37. Wolf, T., Brand, A. (1993). Heuristics for overdetermined systems of PDES. In Johnson, J.H., McKee, S., Vella, A., eds, Proc. Meeting on Artificial Intelligence in Mathematics, Oxford, Clarendon Press, in press.
38. Wolf, T. (1995). Private communication.