Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T07:19:43.721Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

Published online by Cambridge University Press:  26 September 2008

Gregory J. Reid ,
Allan D. Wittkopf  and
Alan Boulton
Gregory J. Reid
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Allan D. Wittkopf
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Alan Boulton
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 6 , December 1996 , pp. 635 - 666
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Kovalevskaya, S. 1875 Zur Theorie der Partiellen Differentialgleichungen. J. Reine Agnew. Math. 80, 1–32.Google Scholar
[2] Tresse, A. 1894 Sur les invariants différentiels des groupes continus de transformations. Acta Mathematica 18, 1–88. (English translation I. Lisle 1989, available from the author.)CrossRefGoogle Scholar
[3] Riquier, C. 1910 Les Systèmes d'Équations aux Dérivées Partielles. Gauthier-Villars, Paris.Google Scholar
[4] Janet, M. 1920 Sur les systèmes d'équations aux dérivées partielles. J. de Math. 3, 65–151.Google Scholar
[5] Cartan, E. 1946 Les Systèmes Différentiels Extérieur et leurs Applications Géometrique. Hermann, Paris.Google Scholar
[6] Kuranishi, M. 1957 On E. Cartan's prolongation theorem of exterior differential systems. Am. J. Math. 79, 1–47.CrossRefGoogle Scholar
[7] Goldschmidt, H. 1967 Integrability criteria for systems of partial differential equations. J. Diff. Geom. 1, 269–307.Google Scholar
[8] Spencer, D. 1969 Overdetermined systems of linear differential equations. Bull. A.M.S. 75, 179–239.CrossRefGoogle Scholar
[9] Pommaret, J. F. 1978 Systems of Partial Differential Equations and Lie Pseudogroups. Gordon and Breach.Google Scholar
[10] Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L. & Griffiths, P. A. 1991 Exterior Differential Systems, Mathematical Sciences Research Institute publications 18, Springer Verlag.Google Scholar
[11] Reid, G. J. 1991 Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution. Eur. J. Appl. Math. 2, 293–318.CrossRefGoogle Scholar
[12] Brenan, K., Campbell, S. & Petzold, L. 1989 Numerical solution of initial-value problems in Differential-Algebraic Equations. Elsevier Science.Google Scholar
[13] Clarkson, P. A. & Mansfield, E. L. 1994 On a shallow water wave equation. Nonlinearity 7. 975–1000.CrossRefGoogle Scholar
[14] Bluman, G. W. & Cole, J. D. 1969 The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042.Google Scholar
[15] Bluman, G. W. & Kumei, S. 1989 Symmetries and Differential Equations. Springer Verlag.CrossRefGoogle Scholar
[16] Olver, P. J. 1986 Application of Lie Groups to Differential Equations. Springer Verlag.CrossRefGoogle Scholar
[17] Cartan, É. J. 1937 Les problèmes d'équivalence. Oeuvres Complètes, Part II 2, 1311–1334. Gauthier-Villars, Paris.Google Scholar
[18] Kramer, D., Stephani, H., MacCallum, H. & Herlt, E. 1980 Exact Solutions of Einstein's Field Equations. Deutscher Verl. d. Wiss, Berlin.Google Scholar
[19] Carminati, J. & McLenaghan, R. G. 1987 An explicit determination of the space-times on which the conformally invariant scalar wave equation satisfies Huygens' principle. Part II: Petrov type D space-times. Ann. Inst. Henri Poincaré, 47 (4), 337–354.Google Scholar
[20] Šurygin, V. A. & Janenko, N. N. 1961 On the realization on electronic computing machines of algebraico-differential algorithms. Problemy Kibernetika 6, 33–43 (in Russian).Google Scholar
[21] Arǎis, E. A., Šapeev, V. P. & Janenko, N. N. 1974 Realization of Cartan's method of exterior differential forms on an electronic computer. Sov. Math. Dokl. 15 (1), 203–205.Google Scholar
[22] Ganǎa, V. G., Melesško, S. V., Murzin, F. A., Šapeev, V. P. & Janenko, N.N. 1981 Realization on a computer of an algorithm for studying the consistency of systems of partial differential equations. Sov. Math. Dokl. 24 (1), 638–640.Google Scholar
[23] Schwarz, F. 1984 The Riquier–Janet theory and its applications to nonlinear evolution equations. Physica Dll, 243–251.Google Scholar
[24] Hereman, W. 1994 Review of symbolic software for the computation of Lie symmetries of differential equations. Euromath Bull. 1, 45–79.Google Scholar
[25] Schwarz, F. 1992 Reduction and completion algorithms for partial differential equations. In Proc. ISSAC'92, 49–56. ACM Press, Berkeley.CrossRefGoogle Scholar
[26] Topunov, L. 1989 Reducing systems of linear differential equations to a passive form. Acta Appl. Math. 16, 191–206.CrossRefGoogle Scholar
[27] Bocharov, A. V. & Bronstein, M. L. 1989 Efficiently implementing two methods of the geometrical theory of differential equations: an experience in algorithm and software design. Acta Appl. Math. 16, 143–166.CrossRefGoogle Scholar
[28] Sherring, J. & Prince, G. 1992 D1MSYM – Symmetry Determination and Linear Differential Equations Package. Preprint, Department of Mathematics, LaTrobe University, Bundoora, Australia.Google Scholar
[29] Schwarz, F. 1992 An algorithm for determining the size of symmetry groups. Computing 49, 95–115.CrossRefGoogle Scholar
[30] Reid, G. J. & McKinnon, D. K. 1992 Solving systems of linear PDEs in their coefficient field by recursively decoupling and solving ODEs. Preprint, Department of Mathematics, University of British Columbia, Vancouver, Canada.Google Scholar
[31] Czichowski, G. & Thiede, M. 1992 Gröbner Bases, Standard Forms of Differential Equations and Symmetry Computation. Seminar Sophus Lie, Darmstadt-Erlangen-Greifswald-Leipzig.Google Scholar
[32] Reid, G.J. 1991 Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. Euro. J. Appl. Math. 2, 319–340.CrossRefGoogle Scholar
[33] Reid, G. J., Lisle, I. G., Boulton, A. & Wittkopf, A. D. 1992 Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs. In Proc. ISSAC '92, Berkeley, California, P. S. Wang. ACM Press.Google Scholar
[34] Oaku, T. 1994 Algorithms for finding the structure of solutions of linear partial differential equations. In Proc. ISSAC '94.CrossRefGoogle Scholar
[35] Ritt, J. F. 1950 Differential algebra. Amer. Math. Soc. Colloq. Publns. 13. A.M.S., New York.Google Scholar
[36] Geddes, K. O., Czapor, S. R. & Labahn, G. 1992 Algorithms for Computer Algebra. Kluwer Academic.CrossRefGoogle Scholar
[37] Kolchin, E. R. 1973 Differential Algebra and Algebraic Groups. Academic Press.Google Scholar
[38] CarrÀ-Ferro, G. 1987 Gröbner Bases and Differential Algebra. Lecture Notes in Comp. Sci. 356, 128–140. Springer-Verlag.Google Scholar
[39] Ollivier, F. 1991 Standard Bases of Differential Ideals. Lecture Notes in Comp. Sci. 508. 304–321. Springer-Verlag.Google Scholar
[40] MANSFIELD, E. 1991 Differential Gröbner Bases. PhD Thesis, University of Sydney.Google Scholar
[41] Mansfield, E. L. & Fackerell, E. D. 1993 Differential Gröbner Bases. Preprint 92–108, School of Mathematics, Physics, Computer Science, and Electronics, Macquarie University, Sydney, Australia.Google Scholar
[42] Kähler, E. 1934 Einführung in die Theorie der Systeme von Differentialgleichungen. B. G. Teubner, Leipzig.Google Scholar
[43] Vessiot, E. 1924 Sur une théorie nouvelle des problèmes généraux d'integration. Bull. Soc. Math. Fr. 52, 336–395.CrossRefGoogle Scholar
[44] Schü, J., Seiler, W. M. & Calmet, J. 1992 Algorithmic methods for Lie Pseudogroups. In Proc. Modern Group Analysis, Acireale.Google Scholar
[45] Hartley, D. & Tucker, R. W. 1991 A constructive implementation of the Cartan–Kähler theory of exterior differential systems. J. Symb. Comp. 12, 655–667.CrossRefGoogle Scholar
[46] Hudson, A. 1987 Symbolic computation of involutivity of PDES. Masters Thesis, University of Sydney.Google Scholar
[47] Lewy, H. 1957 An example of a smooth linear partial differential equation without solution. Ann. Math. 66, 155–158.CrossRefGoogle Scholar
[48] Carrà Ferro, G. & Sit, W. Y. 1993 On Term-Orderings and Rankings. Lecture Notes in Pure and Applied Math., Computational Algebra, 151, Dekker.Google Scholar
[49] Weispfenning, V. 1993 Differential-term orders. In Proc. of ISAAC '93. Kiev. ACM Press.Google Scholar
[50] Rust, C. 1993 On The Classification of Rankings of Partial Derivatives. Preprint.Google Scholar
[51] Auslander, L. & Mackenzie, R. E. 1963 Introduction to Differentiable Manifolds. McGraw-Hill, New York.Google Scholar
[52] Gunning, R. C. & Rossi, H. 1965 Analytic Functions of Several Complex Variables. Prentice-Hall.Google Scholar
[53] Gianni, P., Trager, B. & Zacharias, G. 1989 Grobner Bases and Primary Decomposition of Polynomial Ideals. In Computational Aspects of Commutative Algebra, Robbiano, L., editor, pp. 15–33. Academic Press.Google Scholar
[54] Clarkson, P. A. & Kruskal, M. D. 1989 New similarity solutions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213.CrossRefGoogle Scholar
[55] Reid, G. J. & Wittkopf, A. D. 1993 The long guide to the Standard Form package. Programs and documentation available by anonymous ftp to math.ubc.ca in directory /pub/reid/standardform.Google Scholar
[56] WOLF, T. 1987 A package for the analytic investigation and exact solutions of differential equations. In Proc. EUROCAL'87, Leipzig, GDR, J. H., Davenport, , editor. Lecture Notes in Computer Science, 378, 479–491. Springer-Verlag.CrossRefGoogle Scholar
[57] Becker, T. & Weispfennino, V. 1993 Gröbner Bases: A computational appraoch to commutative algebra. Springer-Verlag.CrossRefGoogle Scholar
[58] Mansfield, E. 1992 A simple criterion for involutivity. Preprint M94/16.Google Scholar
[59] Reid, G. J., Weih, D. T. & Wittkopf, A. D. 1993 A Point symmetry group of a differential equation which cannot be found using infinitesimal methods. In Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics. Ibragimov, N. H., Torrisi, M. and Valenti, A., editors. 93–99. Kluwer.Google Scholar
[60] Thomas, J. M. 1929 Riquier's existence theorems. Annals of Math. 30, 285–310.CrossRefGoogle Scholar
[61] Carrà-Ferro, G. & Duzhin, S. V. 1993 Differential-algebraic and differential-geometric approach to the study of involutive symbols. In Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, Ibragimov, N. H., Torrisi, M. and Valenti, A., editors, pp. 93–99. Kluwer.Google Scholar
[62] Seiler, W. M., Vassiliou, P. J. & Rogers, C. 1995 Formal Analysis of the general Cauchy problem for a system associated with the (2 + 1)-dimensional Krichever–Novitou equation. Preprint.CrossRefGoogle Scholar