|
|
Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method |
Liu Xi-Zhong(刘希忠)† |
Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China |
|
|
Abstract This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders, showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method. The homotopy series solutions to the generalized Kawahara equation are consequently derived.
|
Received: 04 January 2010
Revised: 13 January 2010
Accepted manuscript online:
|
PACS:
|
02.30.Mv
|
(Approximations and expansions)
|
|
02.20.Hj
|
(Classical groups)
|
|
02.20.Qs
|
(General properties, structure, and representation of Lie groups)
|
|
02.30.Jr
|
(Partial differential equations)
|
|
Fund: Project supported by the National Natural Science Foundations of China (Grant Nos. 10735030, 10475055, 10675065 and 90503006), the National Basic Research Program of China (Grant No. 2007CB814800). |
Cite this article:
Liu Xi-Zhong(刘希忠) Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method 2010 Chin. Phys. B 19 080202
|
[1] |
Bluman G W and Kumei S 1989 Symmetries and Differential Equations Appl. Math. Sci. (Berlin: Springer-Verlag) p. 81
|
[2] |
Rogers C and Ames W F 1989 Nonlinear Boundary Value Problems in Science and Engineering (Boston: Academic)
|
[3] |
Olver P J 1993 Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 2nd ed. (New York: Springer-Verlag) p. 107
|
[4] |
Li J H and Lou S Y 2008 Chin. Phys. B 17 747
|
[5] |
Lian Z J, Chen L L and Lou S Y 2005 Chin. Phys. 14 1486
|
[6] |
Bluman G W and Cole J D 1969 J. Math. Mech. 18 1025
|
[7] |
Clarkson P A and Kruskal M D 1989 J. Math. Phys. 30 2201
|
[8] |
Jiao X Y and Lou S Y 2009 Chin. Phys. B 18 3611
|
[9] |
Tang X Y, Gao Y, Huang F and Lou S Y 2009 Chin. Phys. B 18 4622
|
[10] |
Zhang Y F and Zhang H Q 2002 Chin. Phys. 11 319
|
[11] |
Liao S J 1992 The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, PhD thesis, Shanghai Jiao Tong University
|
[12] |
Hayat T and Sajid M 2007 Phys. Lett. A 361 316
|
[13] |
Liao S J 2005 Appl. Math. Comput. 169 1186 par
|
[14] |
Liao S J 1999 Int. J. Nonlinear Mech. 34 759
|
[15] |
Liao S J 2003 Beyond Perturbation: Introduction to Homotopy Analysis Method (Boca Raton: Chapman and Hall/CRC)
|
[16] |
Liao S J 2004 Appl. Math. Comput. 147 499
|
[17] |
Jiao X Y, Yao R X and Lou S Y 2009 Chin. Phys. Lett. 26 040202
|
[18] |
Jiao X Y, Gao Y and Lou S Y 2009 Sci. Chin. Ser. G 52 1169
|
[19] |
Kakutani T and Ono H 1969 J. Phys. Soc. Jpn. 26 1305
|
[20] |
Kawahara T 1972 J. Phys. Soc. Jpn. 33 260
|
[21] |
Marchenko A B 1988 Prikl. Mat. Mekh. 52 230
|
[22] |
Il'ichev A T 1989 Current Mathematical Problems of Mechanics and Their Applications (Moscow: Nauka) p. 186
|
[23] |
Pomeau Y, Ramani A and Grammaticos B 1988 Physica D 31 127
|
[24] |
Boyd J P 1991 Phys. D 48 129
|
No Suggested Reading articles found! |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
Altmetric
|
blogs
Facebook pages
Wikipedia page
Google+ users
|
Online attention
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
View more on Altmetrics
|
|
|