732
Views
13
CrossRef citations to date
0
Altmetric
Original Articles

Symmetries and Integrating Factors

&
Pages 73-91 | Received 01 Jan 2002, Published online: 21 Jan 2013

References

  • Abraham-Shrauner , B and Guo , A . 1993 . “ Hidden and Nonlocal Symmetries of Nonlinear Differential Equations ” . In Modern Group Analysis: Advanced Analytic and Computational Methods in Mathematical Physics , Edited by: Ibragimov , N H , Torrisi , M and Valenti , A . 1 – 5 . Dordrecht : Kluwer .
  • Abraham-Shrauner B Leach P G L Hidden Symmetries of Nonlinear Ordinary Differential Equations, in Exploiting Symmetry in Applied and Numerical Analysis, Editors: Allgower E, George K and Miranda R, Lecture Notes in Applied Mathematics, Vol. 29, American Mathematical Society, Providence, 1993, 1–10
  • Abraham-Shrauner , B , Govinder , K S and Leach , P G L . 1995 . Integration of Second Order Equations not Possessing Point Symmetries . Phys. Lett. , A203 : 169 – 174 .
  • Andriopoulos , K , Leach , P G L and Flessas , G P . 2000 . Complete Symmetry Groups of Ordinary Differential Equations and Their Integrals: Some Basic Considerations . J. Math. Anal. Appl. , 262 : 256 – 273 .
  • Bianchi L Lezione sulla teoria dei gruppi ciontinui finiti di transformazioni, Enrico Spoerri, Pisa, 1918
  • Braude , S-J . 1935 . The Motion of Electrons in Electric and Magnetic Fields Taking into Consideration the Action of the Space Charge . Phys. Z. Sowjetunion , VIII : 667 – 674 .
  • Cheb-Terrab , E S and Roche , A D . 1999 . Integrating Factors for Second Order ODEs . J. Sym. Comp. , 27 : 501 – 519 .
  • Dickson , L E . 1925 . Differential Equations from the Group Standpoint . Ann. Math. , 25 : 287 – 378 .
  • Eisenhart , L P . 1961 . Continuous Groups of Transformations , New York : Dover .
  • Géronimi , C , Feix , M R and Leach , P G L . 2001 . Exponential Nonlocal Symmetries and Nonnormal Reduction of Order . J. Phys. A: Math. Gen. , 34 : 10109 – 10117 .
  • Govinder , K S and Leach , P G L . 1995 . On the Determination of Nonlocal Symmetries . J. Phys. A: Math. Gen. , 28 : 5349 – 5359 .
  • Govinder , K S and Leach , P G L . 1997 . A Group Theoretical Approach to a Class of Second Order Ordinary Differential Equations not Possessing Lie Point Symmetries . J. Phys. A: Math. Gen. , 30 : 2055 – 2068 .
  • Head , A K . 1993 . LIE, a PC Program for Lie Analysis of Differential Equations . Comp. Phys. Commun. , 77 : 241 – 248 .
  • Ince , E L . 1927 . Ordinary Differential Equations , London : Longmans, Green and Co .
  • Ince , E L . 1956 . Integration of Ordinary Differential Equations , Edinburgh : Oliver and Boyd .
  • Kamke E Differentialgleichungen: Lösungsmethoden und Lösungen, Band I, 10 th Aufl., B G Teubner, Stuttgart, 1983
  • Krause , J . 1994 . On the Complete Symmetry Group of the Classical Kepler Problem . J. Math. Phys. , 35 : 5734 – 5748 .
  • Leach , P G L , Nucci , M C and Cotsakis , S . 2001 . Symmetry, Singularities and Integrability in Complex Dynamics V: Complete Symmetry Groups of Nonintegrable Ordinary Differential Equations . J. Nonlin. Math. Phys. , 8 : 475 – 490 .
  • Lie , S . 1967 . Differentialgleichungen , New York : Chelsea .
  • Lie , S . 1971 . Continuierliche Gruppen , New York : Chelsea .
  • Lie , S . 1977 . Berührungstransformationen , New York : Chelsea .
  • Muller , J-J . 1937 . Oscillations électroniques dans le magnétron . Rev. Gén. Élect. , XLII : 389 – 406 .
  • Muller , J-J . 1937 . Oscillations électroniques dans le magnétron . Rev. Gén. Élect. , XLII : 419 – 434 .
  • Noether , E . 1918 . Invariante Variationsprobleme . König. Gesell. Wissen Göttingen, Math.- Phys. Kl. , Heft 2 : 235 – 237 .
  • Nucci M C Interactive REDUCE Programs for Calculating Classical, Nonclassical and Lie–Bäcklund Symmetries of Differential Equations, Preprint, Georgia Tech. Math. 062090-051, Georgia Institute of Technology, 1990
  • Nucci , M C . 1996 . The Complete Kepler Group Can Be Derived by Lie Group Analysis . J. Math. Phys. , 37 : 1772 – 1775 .
  • Nucci M C Interactive REDUCE Programs for Calculating Lie Point, Non-Classical, Lie–Bäcklund, and Approximate Symmetries of Differential Equations: Manual and Floppy Disk, in CRC Handbook of Lie Group Analysis of Differential Equations. Vol. III: New Trends, Editor: Ibragimov N H, CRC press, Boca Raton, 1996, 415–481
  • Nucci , M C and Leach , P G L . 2001 . The Harmony in the Kepler Problem . J. Math. Phys. , 42 : 746 – 764 .
  • Pillay , T and Leach , P G L . 1997 . A General Approach to Symmetries of Differential Equations . Problems of Nonlinear Analysis in Engineering Systems , 2 : 33 – 39 .
  • Pillay , T and Leach , P G L . 2000 . Chaos, Integrability and Symmetry . S. Afr. J. Sci. , 96 : 371 – 376 .
  • Sherring , J , Head , A K and Prince , G E . 1997 . Dimsym and LIE: Symmetry Determine Packages . Math. Comp. Model. , 25 : 153 – 164 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.