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Approximate solutions of fractional Zakharov-Kuznetsov equations by VIM. (English) Zbl 1173.65066
Summary: This paper presents the approximate analytical solution of a fractional Zakharov-Kuznetsov equation with the help of the powerful variational iteration method (VIM). The fractional derivatives are described in the Caputo sense. Several examples are given and the results are compared to exact solutions. The results show that the variational iteration method is very effective, convenient and simple to use.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
45K05 Integro-partial differential equations
26A33 Fractional derivatives and integrals
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Ray, S.S., Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method, Commun. nonlinear sci. numer. simul., 14, 4, 1295-1306, (2009) · Zbl 1221.65284
[2] Abdulaziz, O.; Hashim, I.; Ismail, E.S., Approximate analytical solution to fractional modified KdV equations, Math. comput. modelling, 49, 136-145, (2009) · Zbl 1165.35441
[3] Hosseinnia, S.H.; Ranjbar, A.; Momani, S., Using an enhanced homotopy perturbation method in fractional equations via deforming the linear part, Comput. math. appl., 56, 3138-3149, (2008) · Zbl 1165.65375
[4] Abdulaziz, O.; Hashim, I.; Momani, S., Solving systems of fractional differential equations by homotopy-perturbation method, Phys. lett. A, 372, 451-459, (2008) · Zbl 1217.81080
[5] Abdulaziz, O.; Hashim, I.; Momani, S., Application of homotopy-perturbation method to fractional ivps, J. comput. appl. math., 216, 574-584, (2008) · Zbl 1142.65104
[6] Song, L.; Zhang, H., Application of homotopy analysis method to fractional kdv – burgers – kuramoto equation, Phys. lett. A, 11, 88-94, (2007) · Zbl 1209.65115
[7] Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional ivps, Commun. nonlinear sci. numer. simul., 14, 674-684, (2008) · Zbl 1221.65277
[8] Abdulaziz, O.; Hashim, I.; Saif, A., Series solutions of time-fractional PDEs by homotopy analysis method, Differ. equ. nonlinear mech., (2008) · Zbl 1172.35305
[9] Abdulaziz, O.; Hashim, I.; Chowdhury, M.S.H.; Zulkifle, A.K., Assessment of decomposition method for linear and nonlinear fractional differential equations, Far east J. appl. math., 28, 95-112, (2007) · Zbl 1134.26300
[10] Bataineh, A.S.; Alomari, A.K.; Noorani, M.S.M.; Hashim, I.; Nazar, R., Series solutions of systems of nonlinear fractional differential equations, Acta appl. math., 105, 189-198, (2009) · Zbl 1187.34007
[11] He, J.H., A new approach to linear partial differential equations, Commun. nonlinear sci. numer. simul., 2, 4, 230-235, (1997)
[12] He, J.H., Approximate analytical solutions for seepage flow with fractional derivatives in porous media, Comput. methods appl. mech. engrg., 167, 1-2, 57-68, (1998) · Zbl 0942.76077
[13] He, J.H., Variational iteration method-a kind of non-linear analytical technique: some examples, Internat. J. non-linear mech., 34, 4, 699-708, (1999) · Zbl 1342.34005
[14] He, J.H., Variational iteration method-some recent results and new interpretations, J. comput. appl. math., 207, 1, 3-17, (2007) · Zbl 1119.65049
[15] He, J.H., The variational iteration method for eighth-order initial-boundary value problems, Phys. scripta, 76, 680-682, (2007) · Zbl 1134.34307
[16] He, J.H., Variational iteration method for delay differential equations, Commun. nonlinear sci. numer. simulat, 2, 4, 235-236, (1997)
[17] He, J.H.; Wazwaz, A.M.; Xua, L., The variational iteration method: reliable, efficient, and promising, Comput. math. appl., 54, 7-8, 879-880, (2007) · Zbl 1138.65301
[18] Draganescu, G.E., Application of variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives, J. math. phys., 47, 3, 082902, (2006) · Zbl 1112.74009
[19] Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. nonlinear sci. numer. simulat., 7, 27-34, (2006)
[20] Inc, M., The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method, J. math. anal. appl., 345, 1, 476-484, (2008) · Zbl 1146.35304
[21] Song, L.; Wang, Q.; Zhang, H., Rational approximation solution of the fractional sharma – tasso – olever equation, J. comput. appl. math., 224, 1, 210-218, (2009) · Zbl 1157.65074
[22] Yulita Molliq, R.; Noorani, M.S.M.; Hashim, I., Variational iteration method for fractional heat- and wave-like equations, Nonlinear anal. RWA, 10, 3, 1854-1869, (2009) · Zbl 1172.35302
[23] Das, S., Analytical solution of a fractional diffusion equation by variational iteration method, Comput. math. appl., 53, 3, 483-487, (2009) · Zbl 1165.35398
[24] Batiha, K., Approximate analytical solution for the zakharov – kuznetsov equations with fully nonlinear dispersion, J. comput. appl. math., 216, 1, 157-163, (2009) · Zbl 1138.65092
[25] Munro, S.; Parkes, E.J., The derivation of a modified zakharov – kuznetsov equation and the stability of its solutions, J. plasma phys., 62, 3, 305-317, (1999)
[26] Munro, S.; Parkes, E.J., Stability of solitary-wave solutions to a modified zakharov – kuznetsov equation, J. plasma phys., 64, 4, 411-426, (2000)
[27] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[28] Gorenflo, R., Fractional calculus: some numerical methods, ()
[29] Caputo, M., Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. astron. soc., 13, 529-539, (1967)
[30] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in nonlinear mathematical physics, (), 156-162
[31] Inc, M., Exact solutions with solitary patterns for the zakharov – kuznetsov equations with fully nonlinear dispersion, Chaos solitons fractals, 33, 5, 1783-1790, (2007) · Zbl 1129.35450
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