Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Padé technique

In this paper, the variational iteration method (VIM) is reintroduced with Laplace transforms and the Pade 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 KdV and mKdV equations are given to show the effectiveness of the coupled VIM-Laplace-Pade and VIM-Pade techniques.

[1]  C. Dang,et al.  An aftertreatment technique for improving the accuracy of Adomian's decomposition method , 2002 .

[2]  M. A. Abdoua,et al.  Variational iteration method for solving Burger ’ s and coupled Burger ’ s equations , 2005 .

[3]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[4]  Magdy A. El-Tawil,et al.  Solving nonlinear partial differential equations using the modified variational iteration Padé technique , 2007 .

[5]  R. Hirota.  Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告) , 1976 .

[6]  Ji-Huan He.  SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS , 2006 .

[7]  G. Adomian.  A new approach to nonlinear partial differential equations , 1984 .

[8]  Hyman,et al.  Compactons: Solitons with finite wavelength. , 1993, Physical review letters.

[9]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[10]  G. A. Baker.  Essentials of Padé approximants , 1975 .

[11]  Abdul-Majid Wazwaz,et al.  A sine-cosine method for handlingnonlinear wave equations , 2004, Math. Comput. Model..

[12]  Abdul-Majid Wazwaz,et al.  Nonlinear dispersive special type of the Zakharov-Kuznetsov equation ZK(n, n) with compact and noncompact structures , 2005, Appl. Math. Comput..

[13]  Abdul-Majid Wazwaz,et al.  The tanh method for traveling wave solutions of nonlinear equations , 2004, Appl. Math. Comput..

[14]  Tamer A. Abassy,et al.  Toward a modified variational iteration method , 2007 .

[15]  G. Adomian,et al.  Solving frontier problems of physics : the decomposition methoc [i.e. method] , 1994 .

[16]  V. Marinca,et al.  An Approximate Solution for One-Dimensional Weakly Nonlinear Oscillations , 2002 .

[17]  W. Malfliet,et al.  The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations , 2004 .

[18]  Ji-Huan He,et al.  Construction of solitary solution and compacton-like solution by variational iteration method , 2006 .

[19]  Ji-Huan He,et al.  Variational iteration method for autonomous ordinary differential systems , 2000, Appl. Math. Comput..

[20]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[21]  W. Malfliet.  Solitary wave solutions of nonlinear wave equations , 1992 .

[22]  Tamer A. Abassy,et al.  The solution of Burgers’ and good Boussinesq equations using ADM–Padé technique , 2007 .

[23]  Hsu-Shih Shih,et al.  Computers and Mathematics with Applications , 2008 .

[24]  Magdy A. El-Tawil,et al.  The Solution of KdV and mKdV Equations Using Adomian Pade Approximation , 2004 .

[25]  C. Bai.  New explicit and exact travelling wave solutions for a system of dispersive long wave equations , 2004 .