Momani, Shaher; Odibat, Zaid; Alawneh, Ahmed Variational iteration method for solving the space- and time-fractional KdV Equation. (English) Zbl 1130.65132 Numer. Methods Partial Differ. Equations 24, No. 1, 262-271 (2008). Summary: This paper presents numerical solutions for the space- and time-fractional Korteweg-de Vries equation (KdV for short) using the variational iteration method. The space- and time-fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and accurate when applied to space- and time-fractional KdV equations. The method introduces a promising tool for solving many space-time fractional partial differential equations. Cited in 37 Documents MSC: 65R20 Numerical methods for integral equations 45K05 Integro-partial differential equations 35Q53 KdV equations (Korteweg-de Vries equations) 26A33 Fractional derivatives and integrals 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs Keywords:KdV equation; variational iteration method; Lagrange multiplier; fractional differential equations; Caputo fractional derivative; Korteweg-de Vries equation; numerical results PDFBibTeX XMLCite \textit{S. Momani} et al., Numer. Methods Partial Differ. Equations 24, No. 1, 262--271 (2008; Zbl 1130.65132) Full Text: DOI References: [1] Gardner, Commun Pure Appl Math pp 97– (1974) [2] Khater, Math Meth Appl Sci 21 pp 713– (1998) [3] Kaya, Phys Lett A 299 pp 201– (2002) [4] Kaya, Int J Comput Math 72 pp 531– (1999) [5] Rasulov, Appl Math Comp 102 pp 139– (1999) [6] Taha, J Comput Phys 55 pp 231– (1984) [7] Ciesielski, Signal Process 86 pp 2619– (2006) [8] Momani, Math Comput Simulat 70 pp 110– (2005) [9] He, Commun Nonlin Sci Numer Simulat 2 pp 235– (1997) [10] He, Int J Turbo Jet-Engines 14 pp 23– (1997) [11] He, Computer Meth Appl Mech Eng 167 pp 69– (1998) [12] He, Computer Meth Appl Mech Eng 167 pp 57– (1998) [13] He, Int J Nonlinear Mech 34 pp 699– (1999) [14] He, Appl Math Comput 114 pp 115– (2000) [15] He, Chaos Solitons Fractals 29 pp 108– (2006) [16] Momani, Chaos Solitons Fractals 27 pp 1119– (2006) [17] Moghimi, Chaos Solitons Fractals 33 pp 1756– (2007) [18] Odibat, Int J Nonlinear Sci Numer Simulat 1 pp 15– (2006) [19] Momani, Chaos Solitons Fractals 31 pp 1248– (2007) [20] Momani, Phys Lett A 355 pp 271– (2006) [21] and , The fractional calculus, Academic Press, New York, 1974. [22] and , An introduction to the fractional calculus and fractional differential equations, John Wiley, New York, 1993. · Zbl 0789.26002 [23] and , The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08 – 98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998. [24] Fractional differential equations, Academic Press, San Diego, CA, 1999. [25] Caputo, J Roy Astronom Soc 13 pp 529– (1967) · doi:10.1111/j.1365-246X.1967.tb02303.x [26] , and , General use of the Lagrange multiplier in non-linear mathematical physics, in editor, Variational method in the mechanics of solids, Pergamon Press, Oxford, 1978, pp. 156–162. [27] He, Int J Nonlin Sci Numer Simulat 2 pp 309– (2001) [28] He, Int J Nonlinear Sci Numer Simulat 4 pp 313– (2003) [29] He, Chaos Solitons Fractals 19 pp 847– (2004) [30] Wazwaz, J Computat Appl Math [31] Abassy, J Computat Appl Math 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.