Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source

Physica D: Nonlinear Phenomena

Volumn 144, Issue 1-2, 2000, Pages 97-123

  Qu, Changzheng ^a   Zhang, Shunli ^a   Liu, Ruochen ^a  

^a NORTHWEST UNIVERSITY   (China)

Author keywords

Generalized conditional symmetry; Quasilinear diffusion equations; Separation of variables; Symmetry group

Indexed keywords

EID: 0039592610     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(00)00069-5     Document Type: Article

References (62)

1. W.F. Ames, Nonlinear Partial Differential Equations in Engineering, Vol. I, Academic Press, New York, 1967.
2. W.F. Ames, Nonlinear Partial Differential Equations in Engineering, Vol. II, Academic Press, New York, 1972.
3. G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989.
4. Dordnitsyn V.A. Comput. Maths. Math. Phys. 22:1982;115-122.
5. Oron A., Rosenau P. Phys. Lett. A. 118:1986;172-176.
6. Galaktionov V.A., Dorodnitsyn V.A., Elenin G.G., Kurdyumov S.P., Samarskii A.A. J. Soviet Math. 41:1988;1222-1292.
7. Bluman G.W., Cole J.D. J. Math. Mech. 18:1969;1025-1042.
8. Nucci M.C., Clarkson P.A. Phys. Lett. A. 164:1992;49-56.
9. Clarkson P.A., Mansfield E.L. Physica D. 70:1993;250-288.
10. Arrigo D.J., Broadbridge P., Hill J.M. IMA J. Appl. Math. 52:1994;1-24.
11. Arrigo D.J., Hill J.M. Stud. Appl. Math. 94:1995;21-39.
12. Fokas A.S., Liu Q.M. Phys. Rev. Lett. 72:1994;3293-3296.
13. Zhdanov R.Z. J. Phys. A. 28:1995;3841-3850.
14. Qu C.Z. Stud. Appl. Math. 99:1997;107-136.
15. Qu C.Z. IMA J. Appl. Math. 62:1999;283-302.
16. Qu C.Z. J. Aust. Math. Soc. B. 41:1999;1-40.
17. Chou K.S., Qu C.Z. J. Phys. A. 32:1999;6271-6286.
18. C.Z. Qu, Nonlinear Anal. Theory Meth. Appl., in press.
19. Bluman G.W., Reid J.D., Kumei S. J. Math. Phys. 29:1988;806-812.
20. Akhatov I.Sh., Gazizov R.K., Ibragimov N.H. Soviet Math. Dokl. 35:1987;384-388.
21. Galaktionov V.A. Nonlinear Anal. Theory Meth. Appl. 23:1994;1595-1621.
22. Galaktionov V.A., Posashkov S.A. Physica D. 99:1996;217-236.
23. Galaktionov V.A. Proc. Roy. Soc. Edinburgh. 125:1995;225-246.
24. Galaktionov V.A., Posashkov S.A. Comput. Maths. Math. Phys. 34:1994;313-321.
25. Svirshchevskii S.R. Phys. Lett. A. 199:1995;344-348.
26. King J.R. J. Phys. A. 23:1990;5441-5464.
27. King J.R. J. Phys. A. 23:1990;3681-3697.
28. King J.R. J. Phys. A. 24:1991;5721-5745.
29. King J.R. J. Aust. Math. Soc. B. 33:1992;321-349.
30. Burgan J.R., Munier A., Feix M.R., Fijalkow E. SIAM J. Appl. Math. 44:1984;11-18.
31. King J.R. Physica D. 63:1993;35-65.
32. Fuschych W.I., Zhdanov R.Z. J. Nonlinear Math. Phys. 1:1994;60-64.
33. Galaktionov V.A. Differential Integral Equations. 3:1990;863-874.
34. Cherniha R.M. J. Phys. A. 31:1998;8179-8198.
35. Newell A.C., Tabor M., Zeng Y.B. Physica D. 29:1987;1-68.
36. Cariello F., Tabor M. Physica D. 39:1989;77-94.
37. Cariello F., Tabor M. Physica D. 53:1991;59-70.
38. Dolye P.W., Vassiliou P.J. Int. J. Nonlinear Mech. 33:1998;315-326.
39. Dolye P.W. J. Phys. A. 29:1996;7581-7595.
40. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
41. A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, A.P. Mikhailov, Blow-up in Problems for Quasilinear Parabolic Equations, Nauka, Moscow, 1987.
42. Levine H.A. SIAM Rev. 32:1990;262-288.
43. W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Reading, MA, 1977.
44. Kalnins E.G., Miller W. J. Math. Phys. 26:1985;1560-1565.
45. Kalnins E.G., Miller W. J. Math. Phys. 26:1985;2168-2173.
46. Zhdanov R.Z. J. Phys. A. 27:1994;L291-L297.
47. Zhdanov R.Z., Revenko I.V., Fushchych W.I. J. Math. Phys. 36:1995;5506-5521.
48. Zhdanov R.Z. J. Math. Phys. 38:1997;1197-1217.
49. P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
50. Storm M.L. J. Appl. Phys. 22:1951;940-951.
51. Bluman G.W., Kumei S. J. Math. Phys. 21:1980;1019-1023.
52. Fokas A.S., Yortsos Y.C. SIAM J. Appl. Math. 42:1982;318-332.
53. Clarkson P.A., Fokas A.S., Ablowitz M. SIAM J. Appl. Math. 49:1989;1188-1209.
54. Mullins W.W. J. Appl. Phys. 28:1957;333-339.
55. Broadbridge P. J. Math. Phys. 30:1989;1645-1651.
56. Gage M., Hamilton R. J. Differential Geom. 23:1986;69-96.
57. Grayson M. J. Differential Geom. 26:1987;285-314.
58. Angenent S. Ann. Math. 132:1990;451-483.
59. Chou K.S., Zhu X.P. J. Differential Geom. 50:1998;471-504.
60. Gage M. Duke Math. J. 50:1983;1225-1229.
61. S. Angenent, in: Gregynog (Ed.), Shrinking Doughnuts. Nonlinear Diffusion Equations and their Equilibrium States, 3 (Gregynog, 1989, pp. 21-38), Progr. Nonlinear Differential Equations Appl., 7 Birkhäuser, Boston, MA (1992).
62. S. Angenent, in: Gregynog (Ed.), Shrinking Doughnuts. Nonlinear Diffusion Equations and their Equilibrium States, 3 (Gregynog, 1989, pp. 21-38), Progr. Nonlinear Differential Equations Appl., 7 Birkhäuser, Boston, MA (1992).