# Approximate analytic solutions of fractional Zakharov-Kuznetsov equations by fractional complex transform

## DOI:

https://doi.org/10.14419/ijamr.v5i1.5759## Published:

2016-01-23## Keywords:

Fractional complex transform, Variational iteration method, Fractional Zakharovâ€“Kuznetsov equations.## 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 Zakharov-Kuznetsov equations. Fractional complex transform (FCT) is proposed to convert fractional Zakharov-Kuznetsov 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.

## 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)367-386.

[3] J.H. He, Nonlinear oscillation with fractional derivative and its applications. In: International Conference on Vibrating Engineeringâ€™98, Dalian,China, 1998, pp. 288-291.

[4] J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15 (2) (1999) 86-90.

[5] F. Mainardi, Fractional calculus: â€™Some basic problems in continuum and statistical mechanicsâ€™, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York, 1997, pp. 291-348.

[6] R. Gorenflo, Afterthoughts on interpretation of fractional derivatives and integrals, in: P. Rusev, I. Di-movski, V. Kiryakova (Eds.), Transform Methods and Special Functions, Varna 96, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Sofia, 1998, pp. 589-591.

[7] A. Luchko, R. Groneflo, 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.

[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) 529-539.

[11] L. Debnath, D. Bhatta, Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics, Frac. Calc. Appl. Anal. 7 (2004) 153-192.

[12] M.M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006) 80-90.

[13] C. Tadjeran, M.M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys. 220 (2007) 813-823.

[14] V.E. Lynch, B.A. Carreras, D. del-Castillo-Negrete, K.M. FerrieraMejias, H.R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys. 192 (2003) 406-421.

[15] S. Momani, K. Al-Khaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput. 162 (3) (2005) 1351-1365.

[16] S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul. 70 (2) (2005) 110-118.

[17] S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos Soliton. Fract. 28 (4) (2006) 930-937.

[18] S. Momani, Z. Odibat, Analytical solution of a time-fractional NavierStokes equation by Adomian decomposition method, Appl. Math. Comput. 177 (2006) 488-494.

[19] Z. Odibat, S. Momani, Approximate solutions for boundary value problems of time-fractional 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) 27-34.

[21] S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Soliton. Fract. 31 (5) (2007) 1248-1255.

[22] S. Momani, Z. Odibat, Numerical approach to differential equations of fractional order, J. Comput. Appl. Math. 207 (1) (2007) 96-110.

[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) 167-174.

[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 (7-8) (2007) 910-919.

[26] S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A 365 (5-6) (2007) 345-350.

[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) 1929-1937.

[28] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Comm. Nonlin. Sci. Numer. Simul. 14 (3) (2009) 674-684.

[29] J. Cang, Y. Tan, H. Xu, S.J. Liao, Series solutions of non-linear Riccati differential equations with fractional order, Chaos Soliton. Fract. 40 (1) (2009) 1-9.

[30] Bekir A. New exact travelling wave solutions of some complex nonlinear equations, Communications in Nonlinear Science and Numerical Simulation, 14(4) (2009) 1069-1077.

[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) 144-151.

[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) 1351-1369.

[33] Gordoa PR, Pickering A, Zhu ZN. B?cklund transformations for a matrix second PainlevÂ´ e equation, Physics Letters A, 374 (34) (2010) 3422-3424.

[34] Cotta RM, Mikhailov MD, Integral transform method, Applied Mathematical Modelling, 17 (3) (1993) 156-161.

[35] Yang X, Local Fractional Integral Transforms, Progress in Nonlinear Science, 4(2011)1-225.

[36] [40] Li ZB, He JH. Fractional Complex Transform for Fractional Differential Equations, Mathematical and Computational Applications, 15 (5) (2010) 970-973.

[37] [41] Li ZB. An Extended Fractional Complex Transform, Journal of Nonlinear Science and Numerical Simulation, 11 (2010) 0335-0337.

[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 Zakharov-Kuznetsov equations with fully nonlinear dispersion, J. Comput. Appl. Math. 216(1) (2009) 157â€“163.

[41] R. Yulita Molliq, M.S.M. Noorani, I. Hashim, R.R. Ahmad, Approximate solutions of fractional Zakharovâ€“Kuznetsov equations by VIM, Journal of Computational and Applied Mathematics 233 (2009) 103â€“

108

[42] S. Munro, E.J. Parkes, The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions, J. Plasma Phys. 62(3) (1999) 305â€“317.

[43] S. Munro, E.J. Parkes, Stability of solitary-wave solutions to a modified Zakharovâ€“Kuznetsov equation, J. Plasma Phys. 64(4) (2000) 411â€“426.

[44] J.H. He, Variational iteration method-some recent results and new interpretations, J. Comput. Appl. Math. 207(1) (2007) 3â€“17.

[45] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nassed (Ed.), Variational Method in The Mechanics of Solids, Pergamon Press, Oxfort

33(5) 1978, pp.156â€“162.

[46] J.H. He, A new approach to linear partial differential equations, Commun. Nonlinear Sci. Numer. Simulat. 2(4) (1997) 230â€“235.

[47] J.H. He, Approximate analytical solutions for seepage flow with fractional derivatives in porous media, Comput. Meth. Appl. Mech. Engng 167 (1â€“2) (1998) 57â€“68.

[48] M. Inc, Exact solutions with solitary patterns for the Zakharov-Kuznetsov equations with fully nonlinear dispersion, Chaos Solitons Fractals 33(5) (2007) 1783â€“1790

## License

Authors who publish with this journal agree to the following terms:

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [1]Â Â Â Â Â Â Â Â Â Â Â **Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under aÂ Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [2]Â Â Â Â Â Â Â Â Â Â Â **Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [3]Â Â Â Â Â Â Â Â Â Â Â **Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (SeeÂ The Effect of Open Access).