×
Wazwaz, Abdul-Majid

The variational iteration method for rational solutions for KdV, \(K(2,2)\), Burgers, and cubic Boussinesq equations. (English) Zbl 1119.65102

J. Comput. Appl. Math. 207, No. 1, 18-23 (2007).
Summary: The reliable variational iteration method is used to determine rational solutions for the Korteweg-de Vries (KdV), the \(K(2,2)\), the Burgers, and the cubic Boussinesq equations. The study highlights the efficiency of the method and its dependence on the Lagrange multiplier. Rational solutions are obtained directly in a straightforward manner.

Cited in 53 Documents

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)

Keywords:

variational iteration method; KdV equation; \(K(2,2)\) equation; Burgers equation; cubic Boussinesq equation; rational solutions
PDF BibTeX XML Cite
\textit{A.-M. Wazwaz}, J. Comput. Appl. Math. 207, No. 1, 18--23 (2007; Zbl 1119.65102)
Full Text: DOI

References:

[1] Abdou, M. A.; Soliman, A. A., Variational iteration method for solving Burgers’ and coupled Burgers’ equation, J. Comput. Appl. Math., 181, 245-251 (2005) · Zbl 1072.65127
[2] Abulwafa, E. M.; Abdou, M. A.; Mahmoud, A. A., The solution of nonlinear coagulation problem with mass loss, Chaos Solitons and Fractals, 29, 313-330 (2006) · Zbl 1101.82018
[3] Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501-544 (1988) · Zbl 0671.34053
[4] Adomian, G., Solving Frontier Problems of Physics: the Decomposition Method (1994), Kluwer: Kluwer Boston · Zbl 0802.65122
[5] He, J. H., A new approach to nonlinear partial differential equations, Comm. Nonlinear Sci. Numer. Simul., 2, 4, 203-205 (1997)
[6] He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng., 167, 57-68 (1998) · Zbl 0942.76077
[7] He, J. H., A variational iteration approach to nonlinear problems and its applications, Mech. Appl., 20, 1, 30-31 (1998)
[8] He, J. H., Variational iteration method—a kind of nonlinear analytical technique: some examples, Int. J. Non-Linear Mech., 34, 708-799 (1999)
[9] He, J. H., Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114, 2/3, 115-123 (2000) · Zbl 1027.34009
[10] He, J. H., Homotopy perturbation method: a new nonlinear technique, Appl. Math. Comput., 135, 73-79 (2003) · Zbl 1030.34013
[11] He, J. H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput., 151, 287-292 (2004) · Zbl 1039.65052
[12] He, J. H., A generalized variational principle in micromorphic thermoelasticity, Mech. Res. Comm., 32, 1, 93-98 (2005) · Zbl 1091.74012
[13] He, J. H., Some asymptotic methods for strongly nonlinearly equations, Internat. J. Modern Math., 20, 10, 1141-1199 (2006) · Zbl 1102.34039
[14] Kaya, D., Explicit solutions of generalized Boussinesq equations, J. Appl. Math., 1, 1, 29-37 (2001) · Zbl 0976.35066
[15] Momani, S.; Abusaad, S., Application of he’s variational-iteration method to Helmholtz equation, Chaos Solitons and Fractals, 27, 5, 1119-1123 (2005) · Zbl 1086.65113
[16] Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A, 1, 53, 1-9 (2006)
[18] Rosenau, P.; Hyman, J. M., Compactons: solitons with finite wavelengths, Phys. Rev. Lett., 70, 5, 564-567 (1993) · Zbl 0952.35502
[19] Wazwaz, A. M., Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput., 81, 265-274 (1997) · Zbl 0882.65132
[20] Wazwaz, A. M., A First Course in Integral Equations (1997), World Scientific: World Scientific Singapore
[21] Wazwaz, A. M., Analytical approximations and Padé’ approximants for Volterra’s population model, Appl. Math. Comput., 100, 13-25 (1999) · Zbl 0953.92026
[22] Wazwaz, A. M., A new technique for calculating Adomian polynomials for nonlinear polynomials, Appl. Math. Comput., 111, 1, 33-51 (2000)
[23] Wazwaz, A. M., Partial Differential Equations: Methods and Applications (2002), Balkema Publishers: Balkema Publishers The Netherlands · Zbl 0997.35083
[24] Wazwaz, A. M., A new method for solving singular initial value problems in the second order differential equations, Appl. Math. Comput., 128, 47-57 (2002) · Zbl 1030.34004
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.