A fractional variational iteration method for solving fractional nonlinear differential equations. (English) Zbl 1219.65085

Wu, Guo-Cheng

Comput. Math. Appl. 61, No. 8, 2186-2190 (2011).

Summary: Recently, fractional differential equations have been investigated by employing the famous variational iteration method. However, all the previous works avoid the fractional order term and only handle it as a restricted variation. A fractional variational iteration method was first proposed by the author and E.W.M. Lee [Phys. Lett. A 374, 2506–2509 (2010)] and gave a generalized Lagrange multiplier. In this paper, two fractional differential equations are approximately solved with the fractional variational iteration method.

MSC:

65L99	Numerical methods for ordinary differential equations
34A08	Fractional ordinary differential equations
34A45	Theoretical approximation of solutions to ordinary differential equations
45J05	Integro-ordinary differential equations

Keywords:

modified Riemann-Liouville derivative; fractional corrected functional; fractional variational iteration method

References:

[1]	He, J. H., Variational iteration method: a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech., 34, 699-708 (1999) · Zbl 1342.34005
[2]	Safari, M.; Ganji, D. D.; Moslemi, M., Application of He’s variational iteration method and Adomian’s decomposition method to the fractional KdV-Burgers-Kuramoto equation, Comput. Math. Appl., 58, 2091-2097 (2009) · Zbl 1189.65255
[3]	Odibat, Z.; Momani, S., The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. Appl., 58, 2199-2208 (2009) · Zbl 1189.65254
[4]	He, J. H.; Wu, G. C.; Austin, F., The variational iteration method which should be followed, Nonlinear Sci. Lett. A, 1, 1-30 (2010)
[5]	Das, S., Analytical solution of a fractional diffusion equation by variational iteration method, Comput. Math. Appl., 57, 483-487 (2009) · Zbl 1165.35398
[6]	Khan, N. A.; Ara, A.; Ali, S. A., Analytical study of Navier-Stokes equation with fractional orders using He’s homotopy perturbation and variational iteration methods, Int. J. Nonlinear Sci. Numer., 10, 1127-1134 (2009)
[7]	Wu, G. C.; Lee, E. W.M., Fractional variational iteration method and its application, Phys. Lett. A, 374, 2506-2509 (2010) · Zbl 1237.34007
[8]	Kolwankar, K. M.; Gangal, A. D., Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6, 505-513 (1996) · Zbl 1055.26504
[9]	Kolwankar, K. M.; Gangal, A. D., Local fractional Fokker-Planck equation, Phys. Rev. Lett., 80, 214-217 (1998) · Zbl 0945.82005
[10]	Chen, Y.; Yan, Y.; Zhang, K. W., On the local fractional derivative, J. Math. Anal. Appl., 362, 17-33 (2010) · Zbl 1196.26011
[11]	Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. Math. Appl., 51, 1367-1376 (2006) · Zbl 1137.65001
[12]	Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131, 517-529 (2002) · Zbl 1029.34003
[13]	Almeida, R.; Malinowska, A. B.; Torres, D. F.M., A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys., 51, 033503 (2010) · Zbl 1309.49003
[14]	Malinowska, A. B.; Ammi, M. R.S.; Torres, D. F.M., Composition functionals in fractional calculus of variations, Commun. Frac. Calc., 1, 32-40 (2010)