# 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)
Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Padé technique. (English) Zbl 1141.65382
Summary: The variational iteration method (VIM) is reintroduced with Laplace transforms and the Padé technique treatment to obtain closed form solutions of nonlinear equations. Some examples, including the coupled Burger’s equation, compacton $k(n,n)$ equation, Zakharov-Kuznetsov Zk$(n,n)$ equation, and Korteweg de Vries (KdV) and modified KdV equations are given to show the effectiveness of the coupled VIM-Laplace-Padé and VIM-Padé techniques.

##### MSC:
 65M70 Spectral, collocation and related methods (IVP of PDE) 35Q53 KdV-like (Korteweg-de Vries) equations 35A22 Transform methods (PDE) 44A10 Laplace transform 41A21 Padé approximation
Full Text:
##### References:
 [1] Ablowitz, M. J.; Clarkson, P. A.: Solitons: nonlinear evolution equations and inverse scattering. (1991) · Zbl 0762.35001 [2] Miura, M. R.: Bäcklund transformation. (1978) [3] Hirota, R.: Direct methods in soliton theory. Solitons (1980) [4] Gu, C. H.; Hu, H. S.; Zhou, Z. X.: Darboux transformation in solitons theory and geometry applications. (1999) [5] Olver, P. J.: Application of Lie group to differential equation. (1986) · Zbl 0588.22001 [6] Ablowitz, M. J.; Segur, H.: Solitons and the inverse scattering transform. (1981) · Zbl 0472.35002 [7] Malfliet, W.: The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations. Journal of computational and applied mathematics, 164-165 (2004) · Zbl 1038.65102 [8] Malfliet, W.: Solitary wave solutions of nonlinear wave equations. American journal of physics 60, No. 7, 650-654 (1992) · Zbl 1219.35246 [9] Wazwaz, A. M.: The tanh method for traveling wave solutions of nonlinear equationds. Applied mathematics and computation 154, No. 3, 713-723 (2004) · Zbl 1054.65106 [10] Chenglin, B.: New explicit and exact travelling wave solutions for a system of dispersive long wave equations. Reports on mathematical physics 53, No. 2, 291-299 (2004) · Zbl 1069.35068 [11] Wazwaz, A. M.: A sine cosine method for handling nonlinear wave equations. Mathematical and computer modelling 40, No. 5--6, 499-508 (2004) · Zbl 1112.35352 [12] Adomian, G.: Solving frontier problem of physics: the decomposition method. (1994) · Zbl 0802.65122 [13] He, J. H.: Some asymptotic methods for strongly nonlinear equations. International journal of modern physics B 20, No. 10, 1141-1199 (2006) · Zbl 1102.34039 [14] He, J. H.: A new approach to nonlinear partial differential equations. Communications in nonlinear science and numerical simulation 2, No. 4, 230-235 (1997) [15] He, J. H.: Variational iteration method -- A kind of non-linear analytical technique: some examples. International journal of non-linear mechanics 34, No. 4, 699-708 (1999) · Zbl 05137891 [16] He, J. H.: Variational iteration method for autonomous ordinary differential systems. Applied mathematics and computation 114, No. 2--3, 115-123 (2000) · Zbl 1027.34009 [17] Marinca, V.: An approximate solution for one-dimensional weakly nonlinear oscillations. International journal of nonlinear sciences and numerical simulation 3, 107-120 (2002) · Zbl 1079.34028 [18] He, J. H.; Wu, X. H.: Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, solitons and fractals 29, No. 1, 108-113 (2006) · Zbl 1147.35338 [19] T. Abassy, M.A. El-Tawil, H. El-Zoheiry, Toward a modified variational iteration method (MVIM), Journal of Computational and Applied Mathematics (in press), doi: 10.1016/j.cam.2006.07.019 · Zbl 1119.65096 [20] Abassy, T. A.; El-Tawil, M.; Kamel, H.: The solution of KdV and mkdv equations using Adomian Padé approximation. International journal of nonlinear sciences and numerical simulation 5, No. 4, 327-339 (2004) [21] Abassy, T. A.; El-Tawil, M.; Kamel, H.: The solution of Burgers and good Boussinesq equations using ADM--Padé technique. Chaos, solitons and fractals 32, 1008-1026 (2007) · Zbl 1130.35111 [22] T. Abassy, Magdy A. El-Tawil, Hanafy El-Zoheiry, Solving nonlinear P.D.E. using the modified variational iteration--Padé technique (MVIM--Padé), Journal of Computational and Applied Mathematics (in press), doi: 10.1016/j.cam.2006.07.024 [23] Baker, G. A.: Essentials of Padé approxmants. (1975) [24] Jiao, Y. C.; Yamamoto, Y.; Dang, C.; Hao, Y.: An after treatment technique for improving the accuracy of Adomian’s decomposition method. Computers and mathematics with applications 43, 783-798 (2002) · Zbl 1005.34006 [25] Rosenau, P.; Hyman, J. M.: Compactons: solitons with finite wavelength. Physical review letters 70, 564-567 (1993) · Zbl 0952.35502 [26] Abdou, M. A.; Soliman, A. A.: Variational iteration method for solving burger’s and coupled burger’s equations. Journal of computational and applied mathematics 181, No. 2, 245-251 (2005) · Zbl 1072.65127 [27] Wazwaz, A. -M.: Nonlinear dispersive special type of the Zakharov--Kuznetsov equation ZK (n,n) with compact and noncompact structures. Applied mathematics and computation 161, 577-590 (2005) · Zbl 1061.65105 [28] P.G. Drazin, R.S. Jonson, Soliton: An Introduction, Combridge, New York, 1993