×

zbMATH — the first resource for mathematics

Extended simplest equation method for nonlinear differential equations. (English) Zbl 1168.34003
The authors consider the equation
\[ P(y,y',y'',\dots)=0, \tag{1} \] where \(y=y(z)\) is an unknown function, \(P\) is a polynomial in the variable \(y\) and its derivatives and look for exact solutions \(y=y(z)\) of the form
\[ y(z)=\sum_{k=0}^NA_k\left( \frac{\psi '}{\psi} \right)^k, \tag{2} \] \(A_k= \text{const}\), \(A_N\neq 0\), where the function \(\psi=\psi(z)\) is the general solution of the linear ordinary differential equation
\[ \psi ''' +\alpha\psi '' +\beta \psi ' +\gamma \psi=0, \tag{3} \] \(\alpha, \beta, \gamma =\text{const}\). They propose the algorithm for searching the parameters \(N,A_k,\) \(k=1,\dots,N\), \(\alpha,\beta,\gamma\). This approach for the exact solution of the equation (1) the authors call the extended simplest equation method. They apply this method to the Sharma-Tasso-Olver and the Burgers-Huxley equations. New exact solutions of these equations are obtained.

MSC:
34A05 Explicit solutions, first integrals of ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M., Method for solving the Korteweg-de Vries equation, Phys. rev. lett., 19, 1095-1097, (1967) · Zbl 1061.35520
[2] Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H., Method for solving the sine – gordon equation, Phys. rev. lett., 30, 1262-1264, (1973)
[3] Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equation and inverse scattering, (1991), Cambridge University Press New York · Zbl 0762.35001
[4] Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. rev. lett., 27, 1192-1194, (1971) · Zbl 1168.35423
[5] Kudryashov, N.A., Analytical theory of nonlinear differential equations, (2004), Institute for Computer Investigations Moskow-Igevsk, (in Russian)
[6] Weiss, J.; Tabor, M.; Carnevalle, G., The painleve property for partial differential equations, J. math. phys., 24, 522-526, (1983) · Zbl 0514.35083
[7] Kudryashov, N.A., Special polynomials associated with some hierarchies, Phys. lett. A, 372, 945-1956, (2008) · Zbl 1220.34111
[8] He, J.H.; Wu, X.H., Exp-function method for nonlinear wave equations, Chaos solitons fract., 30, 700-708, (2006) · Zbl 1141.35448
[9] He, J.H.; Abdou, M.A., New periodic solutions for nonlinear evolution equations using exp-function method, Chaos solitons fract., 34, 1421-1429, (2007) · Zbl 1152.35441
[10] Ebaid, A., Exact solitary wave solutions for some nonlinear evolution equations via exp-function method, Phys. lett. A, 365, 213-219, (2007) · Zbl 1203.35213
[11] El-Wakil, S.A.; Madkour, M.A.; Abdou, M.A., Application of exp-function method for nonlinear evolution equations with variable coefficients, Phys. lett. A, 369, 62-69, (2007) · Zbl 1209.81097
[12] Abdou, M.A.; Soliman, A.A.; El-Basyony, S.T., New application of exp-function method for improved Boussinesq equation, Phys. lett. A, 369, 469-475, (2007) · Zbl 1209.81091
[13] El-Wakil, S.A.; Abdou, M.A.; Hendi, A., New periodic wave solutions via exp-function method, Phys. lett. A, 372, 830-840, (2008) · Zbl 1217.37070
[14] Zhang, S., Application of exp-function method to Riccati equation and new exact solutions with three arbitrary functions of broer – kaup – kupershmidt equations, Phys. lett. A, 372, 1873-1880, (2008) · Zbl 1220.37071
[15] Kudryashov, N.A., On types of nonlinear nonintegrable equations with exact solutions, Phys. lett. A, 155, 269-275, (1991)
[16] Kudryashov, N.A., Exact solutions of the generalized kuramoto – sivashinsky equation, Phys. lett. A, 147, 287-291, (1990)
[17] Parkes, E.J.; Duffy, B.R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. phys. commun., 98, 288-300, (1996) · Zbl 0948.76595
[18] El-Wakil, S.A.; El-Labany, S.K.; Zahran, M.A.; Sabry, R., Modified extended tanh-function method and its applications to nonlinear equations, Appl. math. comput., 161, 403-412, (2005) · Zbl 1062.35082
[19] Huber, A., Solitary solutions of some nonlinear evolution equations, Appl. math. comput., 166, 464-474, (2005) · Zbl 1080.35111
[20] Yusufogˇlu, E.; Bekir, A., A travelling wave solution to the Ostrovsky equation, Appl. math. comput., 186, 256-260, (2007) · Zbl 1110.76010
[21] Liu, S.; Fu, Z.; Liu, S.; Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. lett. A, 289, 69-74, (2001) · Zbl 0972.35062
[22] Fu, Z.; Liu, S.; Liu, S.; Zhao, Q., New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. lett. A, 290, 72-76, (2001) · Zbl 0977.35094
[23] Kudryashov, N.A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos solitons fract., 24, 1217-1231, (2005) · Zbl 1069.35018
[24] Kudryashov, N.A., Exact solitary waves of the Fisher equation, Phys. lett. A, 342, 99-106, (2005) · Zbl 1222.35054
[25] Bekir, A.; Boz, A., Exact solutions for nonlinear evolution equations using exp-function method, Phys. lett. A, 372, 1619-1625, (2008) · Zbl 1217.35151
[26] Lan, H.; Wang, K., Exact solutions for some nonlinear equations, Phys. lett. A, 137, 369-372, (1989)
[27] Ince, E.L., Ordinary differential equations, (1926), Longmans, Green & Co. London · Zbl 0063.02971
[28] Kamke, E., Differentialgleichungen, Lösungsmethoden und Lösungen I: gewöhnliche differentialgleichungen, (1943), Geest & Portig Leipzig · Zbl 0028.22702
[29] Kudryashov, N.A.; Demina, M.V., Polygons of differential equations for finding exact solutions, Chaos solitons fract., 33, 1480-1496, (2007) · Zbl 1133.35084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.