zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The first integral method for solving some important nonlinear partial differential equations. (English) Zbl 1176.35149
Summary: Exact solutions of some important nonlinear partial differential equations are obtained by using the first integral method. The efficiency of the method is demonstrated by applying it for two selected equations.
MSC:
35Q51Soliton-like equations
35C05Solutions of PDE in closed form
References:
[1]Mitchell, A.R., Griffiths, D.F.: The Finite Difference Method in Partial Equations. Wiley, New York (1980)
[2]Adomian, G.: Solving Frontier Problem of Physics: The Decomposition Method. Kluwer Academic, Boston (1994)
[3]Parkes, E.J., Duffy, B.R.: An Automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations. Comput. Phys. Commun. 98, 288–300 (1998) · Zbl 0948.76595 · doi:10.1016/0010-4655(96)00104-X
[4]Khater, A.H., Malfiet, W., Callebaut, D.K., Kamel, E.S.: The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction–diffusion equations. Chaos Solitons Fractals 14, 513–522 (2002) · Zbl 1002.35066 · doi:10.1016/S0960-0779(01)00247-8
[5]Evans, D.J., Raslan, K.R.: The tanh function method for solving some important nonlinear partial differential equation. Int. J. Comput. Math. 82(7), 897–905 (2005) · Zbl 1075.65125 · doi:10.1080/00207160412331336026
[6]Fan, E.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000) · Zbl 1167.35331 · doi:10.1016/S0375-9601(00)00725-8
[7]Fan, E.: Traveling wave solutions for generalized Hirota–Satsuma coupled KdV systems. Z. Naturforsch. A 56, 312–318 (2001)
[8]Elwakil, S.A., El-Labany, S.K., Zahran, M.A., Sabry, R.: Modified extended tanh-function method for solving nonlinear partial differential equations. Phys. Lett. A 299, 179–188 (2002) · Zbl 0996.35043 · doi:10.1016/S0375-9601(02)00669-2
[9]Gao, Y.T., Tian, B.: Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics. Comput. Phys. Commun. 133, 158–164 (2001) · Zbl 0976.65092 · doi:10.1016/S0010-4655(00)00168-5
[10]Tian, B., Gao, Y.T.: Observable solitonic features of the generalized reaction diffusion model. Z. Naturforsch. A 57, 39–44 (2002)
[11]Tang, X.-Y., Lou, S.-Y.: Abundant structures of the dispersive long-wave equation in (2+1)-dimensional spaces. Chaos Solitons Fractals 14, 1451–1456 (2002) · Zbl 1037.35062 · doi:10.1016/S0960-0779(02)00077-2
[12]Tang, X.-Y., Lou, S.-Y.: Localized excitations in (2+1)-dimensional systems. Phys. Rev. E 66, 46601 (2002) · doi:10.1103/PhysRevE.66.046601
[13]Feng, Z.S.: On explicit exact solutions to the compound Burgers–KdV equation. Phys. Lett. A 293, 57–66 (2002) · Zbl 0984.35138 · doi:10.1016/S0375-9601(01)00825-8
[14]Feng, Z.S.: The first integer method to study the Burgers–Korteweg–de Vries equation. Phys. Lett. A: Math. Gen. 35, 343–349 (2002) · Zbl 1040.35096 · doi:10.1088/0305-4470/35/2/312
[15]Feng, Z.S.: Exact solution to an approximate sine-Gordon equation in (n+1)-dimensional space. Phys. Lett. A 302, 64–76 (2002) · Zbl 0998.35046 · doi:10.1016/S0375-9601(02)01114-3
[16]Feng, Z.S., Wang, X.H.: The first integral method to the two-dimensional Burgers–Korteweg–de Vries equation. Phys. Lett. A 308, 173–178 (2003) · Zbl 1008.35062 · doi:10.1016/S0375-9601(03)00016-1
[17]Li, H., Guo, Y.: New exact solutions to the Fitzhugh–Nagumo equation. Appl. Math. Comput. 180, 524–528 (2006) · Zbl 1102.35315 · doi:10.1016/j.amc.2005.12.035
[18]Ding, T.R., Li, C.Z.: Ordinary Differential Equations. Peking University Press, Peking (1996)
[19]Bourbaki, N.: Commutative Algebra. Addison–Wesley, Paris (1972)
[20]Raslan, K.R.: Collocation method using cubic B-spline for the generalized equal width equation. Int. J. Simul. Process Model. 2, 37–44 (2006)