Approximate analytic solutions of fractional ZakharovKuznetsov equations by fractional complex transform
 Abstract
 Keywords
 References

Abstract
In this paper, fractional complex transform (FCT) with help of variational iteration method (VIM) is used to obtain numerical and analytical solutions for the fractional ZakharovKuznetsov equations. Fractional complex transform (FCT) is proposed to convert fractional ZakharovKuznetsov equations to its differential partner and then applied VIM to the new obtained equations. Several examples are given and the results are compared to exact solutions. The results reveal that the method is very effective and simple.

Keywords
Fractional complex transform; Variational iteration method; Fractional Zakharov–Kuznetsov equations.

References
[1] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[2] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002)367386.
[3] J.H. He, Nonlinear oscillation with fractional derivative and its applications. In: International Conference on Vibrating Engineering’98, Dalian,China, 1998, pp. 288291.
[4] J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15 (2) (1999) 8690.
[5] F. Mainardi, Fractional calculus: ’Some basic problems in continuum and statistical mechanics’, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, SpringerVerlag, New York, 1997, pp. 291348.
[6] R. Gorenflo, Afterthoughts on interpretation of fractional derivatives and integrals, in: P. Rusev, I. Dimovski, V. Kiryakova (Eds.), Transform Methods and Special Functions, Varna 96, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Sofia, 1998, pp. 589591.
[7] A. Luchko, R. Groneflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A0898, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998.
[8] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993.
[9] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[10] M. Caputo, Linear models of dissipation whose Q is almost frequency independent. Part II, J. Roy. Astral. Soc. 13 (1967) 529539.
[11] L. Debnath, D. Bhatta, Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics, Frac. Calc. Appl. Anal. 7 (2004) 153192.
[12] M.M. Meerschaert, C. Tadjeran, Finite difference approximations for twosided spacefractional partial differential equations, Appl. Numer. Math. 56 (2006) 8090.
[13] C. Tadjeran, M.M. Meerschaert, A secondorder accurate numerical method for the twodimensional fractional diffusion equation, J. Comput. Phys. 220 (2007) 813823.
[14] V.E. Lynch, B.A. Carreras, D. delCastilloNegrete, K.M. FerrieraMejias, H.R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys. 192 (2003) 406421.
[15] S. Momani, K. AlKhaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput. 162 (3) (2005) 13511365.
[16] S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul. 70 (2) (2005) 110118.
[17] S. Momani, Nonperturbative analytical solutions of the space and timefractional Burgers equations, Chaos Soliton. Fract. 28 (4) (2006) 930937.
[18] S. Momani, Z. Odibat, Analytical solution of a timefractional NavierStokes equation by Adomian decomposition method, Appl. Math. Comput. 177 (2006) 488494.
[19] Z. Odibat, S. Momani, Approximate solutions for boundary value problems of timefractional wave equation, Appl. Math. Comput. 181 (2006) 1351 1358.
[20] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlin. Sci. Numer. Simul. 7 (1) (2006) 2734.
[21] S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Soliton. Fract. 31 (5) (2007) 12481255.
[22] S. Momani, Z. Odibat, Numerical approach to differential equations of fractional order, J. Comput. Appl. Math. 207 (1) (2007) 96110.
[23] Z. Odibat, S. Momani, Numerical methods for solving nonlinear partial differential equations of fractional order, Appl. Math. model. 32 (1) (2008) 28 39.
[24] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Soliton. Fract. 36 (1) (2008) 167174.
[25] S. Momani, Z. Odibat, Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl. 54 (78) (2007) 910919.
[26] S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A 365 (56) (2007) 345350.
[27] M.S. Chowdhury, I. Hashim, S. Momani, The multistage homotopy perturbation method: a powerful scheme for handling the Lorenz system, Chaos Soliton. Fract. 40 (4) (2009) 19291937.
[28] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Comm. Nonlin. Sci. Numer. Simul. 14 (3) (2009) 674684.
[29] J. Cang, Y. Tan, H. Xu, S.J. Liao, Series solutions of nonlinear Riccati differential equations with fractional order, Chaos Soliton. Fract. 40 (1) (2009) 19.
[30] Bekir A. New exact travelling wave solutions of some complex nonlinear equations, Communications in Nonlinear Science and Numerical Simulation, 14(4) (2009) 10691077.
[31] Cassol M, Wortmann S, Rizza U. Analytic modeling of two dimensional transient atmospheric pollutant dispersion by double GITT and Laplace Transform techniques, Environmental Modelling & Software, 24(1) (2009) 144151.
[32] Sejdi E, Djurovi I, et al. Fractional Fourier transform as a signal processing tool: An overview of recent developments, Signal Processing, 91(6) (2011) 13511369.
[33] Gordoa PR, Pickering A, Zhu ZN. B?cklund transformations for a matrix second Painlev´ e equation, Physics Letters A, 374 (34) (2010) 34223424.
[34] Cotta RM, Mikhailov MD, Integral transform method, Applied Mathematical Modelling, 17 (3) (1993) 156161.
[35] Yang X, Local Fractional Integral Transforms, Progress in Nonlinear Science, 4(2011)1225.
[36] [40] Li ZB, He JH. Fractional Complex Transform for Fractional Differential Equations, Mathematical and Computational Applications, 15 (5) (2010) 970973.
[37] [41] Li ZB. An Extended Fractional Complex Transform, Journal of Nonlinear Science and Numerical Simulation, 11 (2010) 03350337.
[38] [42] He JH, Li ZB. Converting Fractional differential equations into partial differential equations, Thermal Science, DOI REFERENCE: 10.2298/TSCI110503068H
[39] [43] He JH. A Short Remark on Fractional Variational Iteration Method, Physics Letters A, DOI: 10.1016/j.physleta.2011.07.033
[40] K. Batiha, Approximate analytical solution for the ZakharovKuznetsov equations with fully nonlinear dispersion, J. Comput. Appl. Math. 216(1) (2009) 157–163.
[41] R. Yulita Molliq, M.S.M. Noorani, I. Hashim, R.R. Ahmad, Approximate solutions of fractional Zakharov–Kuznetsov equations by VIM, Journal of Computational and Applied Mathematics 233 (2009) 103–
108
[42] S. Munro, E.J. Parkes, The derivation of a modified ZakharovKuznetsov equation and the stability of its solutions, J. Plasma Phys. 62(3) (1999) 305–317.
[43] S. Munro, E.J. Parkes, Stability of solitarywave solutions to a modified Zakharov–Kuznetsov equation, J. Plasma Phys. 64(4) (2000) 411–426.
[44] J.H. He, Variational iteration methodsome recent results and new interpretations, J. Comput. Appl. Math. 207(1) (2007) 3–17.
[45] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. NematNassed (Ed.), Variational Method in The Mechanics of Solids, Pergamon Press, Oxfort
33(5) 1978, pp.156–162.
[46] J.H. He, A new approach to linear partial differential equations, Commun. Nonlinear Sci. Numer. Simulat. 2(4) (1997) 230–235.
[47] J.H. He, Approximate analytical solutions for seepage flow with fractional derivatives in porous media, Comput. Meth. Appl. Mech. Engng 167 (1–2) (1998) 57–68.
[48] M. Inc, Exact solutions with solitary patterns for the ZakharovKuznetsov equations with fully nonlinear dispersion, Chaos Solitons Fractals 33(5) (2007) 1783–1790

View 
Download  Article ID: 5759 
