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Volumn 60, Issue 1, 2000, Pages 39-

Solving Second-Order Differential Equations with Lie Symmetries

Author keywords

Differential equation; Equivalence; Symmetry

Indexed keywords


EID: 18044404414     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1006321609161     Document Type: Article
Times cited : (8)

References (49)
  • 3
    • 85031697769 scopus 로고
    • Linear ordinary differential equations: Breaking through the order 2 barrier
    • P. S. Wang (ed.), ACM Press
    • Bronstein, M.: Linear ordinary differential equations: Breaking through the order 2 barrier, In: P. S. Wang (ed.), Proc. ISSAC 1992, Berkeley, California, ACM Press, 1992, pp. 42-48.
    • (1992) Proc. ISSAC 1992, Berkeley, California , pp. 42-48
    • Bronstein, M.1
  • 7
    • 0004121376 scopus 로고
    • Princeton Univ. Press, Princeton, Reprinted by Dover Publ., New York
    • Eisenhart, L. P.: Continuous Groups of Transformations, Princeton Univ. Press, Princeton, 1933 [Reprinted by Dover Publ., New York, 1961].
    • (1933) Continuous Groups of Transformations
    • Eisenhart, L.P.1
  • 9
    • 0000232659 scopus 로고
    • Symmetry and integrability by quadratures of ordinary differential equations
    • Gonzáles-López, A.: Symmetry and integrability by quadratures of ordinary differential equations, Phys. Lett. A 133 (1988), 190-194.
    • (1988) Phys. Lett. A , vol.133 , pp. 190-194
    • Gonzáles-López, A.1
  • 11
    • 0007127151 scopus 로고
    • Linearization of second order ordinary differential equations via Cartan's equivalence method
    • Grissom, C., Thomson, G. and Wilkens, G.: Linearization of second order ordinary differential equations via Cartan's equivalence method, J. Differential Equations 77 (1989), 1-15.
    • (1989) J. Differential Equations , vol.77 , pp. 1-15
    • Grissom, C.1    Thomson, G.2    Wilkens, G.3
  • 12
    • 0002247504 scopus 로고
    • Review of symbolic software for the computation of Lie symmetries of differential equations
    • Hereman, W.: Review of symbolic software for the computation of Lie symmetries of differential equations, Euromath Bull. 1 (1994), 45-97.
    • (1994) Euromath Bull. , vol.1 , pp. 45-97
    • Hereman, W.1
  • 14
    • 0031281466 scopus 로고    scopus 로고
    • Factorization of differential operators with rational function coefficients
    • van Hoeij, M.: Factorization of differential operators with rational function coefficients, J. Symbol. Comput. 24 (1997), 537-561.
    • (1997) J. Symbol. Comput. , vol.24 , pp. 537-561
    • Van Hoeij, M.1
  • 15
    • 0042397944 scopus 로고
    • Symmetries of second-order ordinary differential equations and Elie Cartan's method of equivalence
    • Hsu, L. and Kamran, N.: Symmetries of second-order ordinary differential equations and Elie Cartan's method of equivalence, Lett. Math. Phys. 15 (1988), 91-99.
    • (1988) Lett. Math. Phys. , vol.15 , pp. 91-99
    • Hsu, L.1    Kamran, N.2
  • 16
    • 0001577484 scopus 로고
    • Classification of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries
    • Hsu, L. and Kamran, N.: Classification of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries, Proc. London Math. Society (3) 58 (1989), 387-416.
    • (1989) Proc. London Math. Society (3) , vol.58 , pp. 387-416
    • Hsu, L.1    Kamran, N.2
  • 18
    • 0002856938 scopus 로고
    • Les systèmes d'équations aux dérivées partielles
    • Janet, M.: Les systèmes d'équations aux dérivées partielles, J. Math. 83 (1920), 65-123.
    • (1920) J. Math. , vol.83 , pp. 65-123
    • Janet, M.1
  • 19
    • 4244173093 scopus 로고
    • Die Zusammensetzung der stetigen endlichen Transformationsgruppen
    • Killing, W.: Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Math. Annal. 31 (1887), 252-290; 33 (1889), 1-48; 34 (1889), 57-122; 36 (1890), 161-189.
    • (1887) Math. Annal. , vol.31 , pp. 252-290
    • Killing, W.1
  • 20
    • 4244173093 scopus 로고
    • Killing, W.: Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Math. Annal. 31 (1887), 252-290; 33 (1889), 1-48; 34 (1889), 57-122; 36 (1890), 161-189.
    • (1889) Math. Annal. , vol.33 , pp. 1-48
  • 21
    • 4244173093 scopus 로고
    • Killing, W.: Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Math. Annal. 31 (1887), 252-290; 33 (1889), 1-48; 34 (1889), 57-122; 36 (1890), 161-189.
    • (1889) Math. Annal. , vol.34 , pp. 57-122
  • 22
    • 0041395769 scopus 로고
    • Killing, W.: Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Math. Annal. 31 (1887), 252-290; 33 (1889), 1-48; 34 (1889), 57-122; 36 (1890), 161-189.
    • (1890) Math. Annal. , vol.36 , pp. 161-189
  • 25
    • 0001149405 scopus 로고
    • Équations différentielle linéaires d'ordre n > 2 ayant une algèbre de Lie de symétrie de dimension n + 4
    • Krause, J. and Michel, L.: Équations différentielle linéaires d'ordre n > 2 ayant une algèbre de Lie de symétrie de dimension n + 4, C.R. Acad. Sci. 307 (1988), 905-910.
    • (1988) C.R. Acad. Sci. , vol.307 , pp. 905-910
    • Krause, J.1    Michel, L.2
  • 26
    • 0041088002 scopus 로고
    • Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten I, II, III
    • Lie, S.: Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten I, II, III, Arch. Math. 8, pp. 187-224, 249-288 and 371-458 respectively (1883) [Gesammelte Abhandlungen, vol. V, pp. 240-281, 282- 310 and 362-427].
    • (1883) Arch. Math. , vol.8 , pp. 187-224
    • Lie, S.1
  • 27
    • 0041896713 scopus 로고    scopus 로고
    • Lie, S.: Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten I, II, III, Arch. Math. 8, pp. 187-224, 249- 288 and 371-458 respectively (1883) [Gesammelte Abhandlungen, vol. V, pp. 240-281, 282-310 and 362-427].
    • Gesammelte Abhandlungen , vol.V , pp. 240-281
  • 30
    • 19944408515 scopus 로고
    • Teubner, Leipzig, Reprinted by Chelsea Publ. Co., New York
    • Lie, S.: Vorlesungen über continuierliche Gruppen, Teubner, Leipzig, 1893 [Reprinted by Chelsea Publ. Co., New York, 1971].
    • (1893) Vorlesungen Über Continuierliche Gruppen
    • Lie, S.1
  • 31
    • 0042397951 scopus 로고
    • Sur une classe d'équations différentielle non linéaires
    • Liouville, R.: Sur une classe d'équations différentielle non linéaires, Comptes Rendus 103 (1886), 520-523.
    • (1886) Comptes Rendus , vol.103 , pp. 520-523
    • Liouville, R.1
  • 32
    • 0002790739 scopus 로고
    • Über vollständig reduzible lineare homogene Differentialgleichungen
    • Loewy, A.: Über vollständig reduzible lineare homogene Differentialgleichungen, Math. Annal. 56 (1906), 89-117.
    • (1906) Math. Annal. , vol.56 , pp. 89-117
    • Loewy, A.1
  • 33
    • 0025494888 scopus 로고
    • Symmetry Lie algebras of nth order ordinary differential equations
    • Mahomed, F. M. and Leach, P. G. L.: Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal. Appl. 151 (1990), 80-107.
    • (1990) J. Math. Anal. Appl. , vol.151 , pp. 80-107
    • Mahomed, F.M.1    Leach, P.G.L.2
  • 38
    • 21944451409 scopus 로고    scopus 로고
    • Symmetric algebras and normal forms of third order ordinary differential equations
    • Schmucker, A. and Czichowski, G.: Symmetric algebras and normal forms of third order ordinary differential equations, J. Lie Theory 8 (1998), 129-137.
    • (1998) J. Lie Theory , vol.8 , pp. 129-137
    • Schmucker, A.1    Czichowski, G.2
  • 39
    • 0041395768 scopus 로고
    • A factorization algorithm for linear homogeneous ODE's
    • G. Gonnet (ed.), ACM Press
    • Schwarz, F.: A factorization algorithm for linear homogeneous ODE's, In: G. Gonnet (ed.), Proc. ISSAC'89, ACM Press, 1989, pp. 17-25.
    • (1989) Proc. ISSAC'89 , pp. 17-25
    • Schwarz, F.1
  • 40
    • 0002150185 scopus 로고
    • An algorithm for determining the size of symmetry groups
    • Schwarz, F.: An algorithm for determining the size of symmetry groups, Computing 49 (1992), 95-115.
    • (1992) Computing , vol.49 , pp. 95-115
    • Schwarz, F.1
  • 41
    • 0029181650 scopus 로고
    • rd order ODE's
    • A. Levelt (ed.), ACM Press
    • rd order ODE's, In: A. Levelt (ed.), Proc. ISSAC'95, ACM Press, 1995, pp. 16-25.
    • (1995) Proc. ISSAC'95 , pp. 16-25
    • Schwarz, F.1
  • 42
    • 0030414170 scopus 로고
    • Janet bases of 2nd order ordinary differential equations
    • Lakshman (ed.), ACM Press
    • Schwarz F.: Janet bases of 2nd order ordinary differential equations, In: Lakshman (ed.), Proc. ISSAC'96, ACM Press, 1995, pp. 179-187.
    • (1995) Proc. ISSAC'96 , pp. 179-187
    • Schwarz, F.1
  • 44
    • 0005479703 scopus 로고    scopus 로고
    • Janet bases for symmetry groups
    • B. Buchberger and F. Winkler (eds), London Math. Soc. Lecture Notes Ser. 251, Cambridge Univ. Press
    • Schwarz, F.: Janet bases for symmetry groups, In: B. Buchberger and F. Winkler (eds), Groebner Bases and Applications, London Math. Soc. Lecture Notes Ser. 251, Cambridge Univ. Press, 1998, pp. 221-234.
    • (1998) Groebner Bases and Applications , pp. 221-234
    • Schwarz, F.1
  • 45
    • 84961387717 scopus 로고    scopus 로고
    • ALLTYPES: An ALgebraic Language and TYPESystem
    • J. Calmet and J. Plaza (eds), Artificial Intelligence and Symbolic Computation, Springer
    • Schwarz, F: ALLTYPES: An ALgebraic Language and TYPESystem, In: J. Calmet and J. Plaza (eds), Artificial Intelligence and Symbolic Computation, Lecture Notes in Artificial Intelligence 1476, Springer, 1998, pp. 270-283.
    • (1998) Lecture Notes in Artificial Intelligence , vol.1476 , pp. 270-283
    • Schwarz, F.1
  • 46
    • 0041896714 scopus 로고    scopus 로고
    • Equivalence classes and symmetries ob Abel's equation
    • Schwarz, F.: Equivalence classes and symmetries ob Abel's equation, to appear in J. Symbol. Comput.
    • J. Symbol. Comput.
    • Schwarz, F.1


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