Solution of System of Fractional Differential Equations Using Variational Iteration Method
DOI:
https://doi.org/10.26438/ijcse/v6i8.797802Keywords:
Variational iteration method, System of fractional differential equation, Caputo fractional derivativeAbstract
In this paper, an approximate method to solve system of fractional differential equations analytically, fractional heat-like two-dimensional equation and stiff system of differential equations of nonlinear kind is obtained using variational iteration method (VIM). The results demonstrate that our approach is straightforward, effective and very simple. The numerical findings for different cases of problems are presented graphically. The outcomes reveal that the VIM is convenient, stable and performs extremely good in terms of simplicity and efficiency
References
[1] S. G. Samko, A. A. Kilbas, and O. I. Marichev, "Fractional integrals and derivatives, translated from the 1987 Russian original" Gordon and Breach, Yverdon, 1993.
[2] K. B. Oldham and J. Spanier The Fractional Calculus, Academic Press, New York , 1974.
[3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[4] J.H. He, “ Nonlinear oscillation with fractional derivative and its applications”, International Conference on Vibrating Engineering 98, Dalian, China, Vol. 28, pp. 288-291.
[5] N. Ferreira, F. Duarte, M. Lima, M. Marcos and J.T. Machado, “Application of fractional calculus in the dynamical analysis and control of mechanical manipulators”, Fract. Calc. Appl. Anal., Vol. 11, no.1, pp. 91-11, 2008.
[6] Magin, Richard L. "Fractional calculus in bioengineering, part 1,Critical Reviews in Biomedical Engineering 32.1, 2004.
[7] L. Debnath, “Recent applications of fractional calculus to science and engineering”, Int. J. Math. Math. Sci., Vol. 54, pp. 3413-3442, 2003.
[8] Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic riccati differential equation of fractional order”, Chaos Soli. Fract, Vol. 36, no.1, pp. 167-174, 2008.
[9] M. Zurigat, S. Momani, Z. Odibat and A. Alawneh “The homotopy analysis method for handling systems of fractional differential equations”, Appl. Math. Model., Vol. 34, no.1, pp.24-35, 2010.
[10] H.Jafari and V.D. Gejji., “Solving a system of nonlinear fractional differential equations using adomain decomposition”. Appl. Math. Comput., Vol. 196, no., pp.644–651,2006.
[11] J. A. Patel., G. R Khatri, H. P. Bhathawala, "Adomian Decomposition Method for Multiphase Miscible Flow the Porous Media",International Journal of Scientific Research in Computer Science, Engineering and Information Technology (IJSRCSEIT), Volume 3, Issue 1, pp.1227-1232, 2018
[12] S. Momani and Z. Odibat, “ Analytical approach to linear fractional partial differential equations arising in fluid mechanics”, Physics Letters A, Vol. 355, no. 4-5, pp. 271-279, 2006.
[13] S. W Abbasbandy, “A new application of He`s variational iteration method for quadratic Riccati differential equation by using Adomian`s polynomials”, Journal of Computational and Applied Mathematics, Vol. 207, no.1, pp.59-63, 2007.
[14] M. Javidi and A.Golbabai, “Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos, Solitons & Fractals, Vol. 36, no.2, pp.309-313, 2008.
[15] M. Dehghan and F. Shakeri, “Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via he’s variational iteration technique”. Comm in Numer. Meth. in Eng., Vol. 26, pp.705-715, 2008.
[16] Li Yun-dong and Yang Yi-ren, “Vibration Analysis of Conveying Fluid Pipe via He’s Variational Iteration Method”, Applied Mathematical Modelling, Vol. 43, pp. 409-420 ,2016.
[17] Y.Chen, J. Zhangand H.Zhang, “Free vibration analysis of rotating tapered Timoshenko beams via variational iteration method”. Journal of Vibration and Control, Vol. 23, no. 2, pp. 220-234, 2017.
[18] H.Ghaneai and M.M. Hosseini, “Solving Differential-Algebraic Equations through Variational Iteration Method with an Auxiliary Parameter”, Applied Mathematical Modelling, Vol. 40, no. 5-6, pp. 3991-4001, 2016.
[19] I. Ali and N. A. Malik, “ Hilfer Fractional Advection-Diffusion Equations with Power-Law Initial Condition; A Numerical Study using Variational Iteration Method”, Computers & Mathematics with Applications, Vol. 68, no. 10, pp. 1161-1179, 2014.
[20] F.S. Aski, S.J. Nasirkhani, E. Mohammadian and A.Asgari, “ Application of Adomian decomposition method for micropolar flow in a porous channel”, Propulsion and Power Research, Vol. 3, no.1, pp.15-21,2014.
[21] S. H. Chang, “Convergence of Variational Iteration Method Applied to Two-Point Diffusion Problems”, Applied Mathematical Modelling, Vol. 40, no. 15-16, pp. 6805-6810, 2016.
[22] M. Di Paola, F.P. Pinnola and M. Zingales, “Fractional differential equations and related exact mechanical models”. Computers & Mathematics with Applications, Vol. 66, no. 5, pp. 608-620, 2013.
[23] S. S. Siddiqi and M. Iftikhar, “Variational Iteration Method for the Solution of Seventh Order Boundary Value Problems using He’s Polynomials”, Journal of the Association of Arab Universities for Basic and Applied Sciences, Vol.18, pp. 60-65, 2015 .
[24] D. K. Salkuyeh and T. Ali, “Interpolated Variational Iteration Method for Initial Value Problems”, Applied Mathematical Modelling, Vol. 40, no. 5-6, pp. 3979-3990, 2016.
[25] R. Gorenflo and F. Mainardi, “Fractional calculus: Int and differential equations of fractional order. in:” A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 1997.
[26] A. Golbabai and M. Javidi, “A variational iteration method for solving parabolic partial differential equations”, Int. J. Computers and Mathematics with Applications Vol. 54 , pp. 987–992, 2007.
[27] M. G. Sakar, F. Erdoganand and A.Yıldırım, “Variational iteration method for the time-fractional Fornberg-Whitham equation”, Computers & Mathematics with Applications, Vol. 63, no. 9, pp. 382–1388.
[28] T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, “Comparing numerical methods for ordinary differential equations”, SIAM J. on Numer. Anal., Vol. 9, no.4, pp. 603-637, 1972.
[29] D.D Ganji., M. Nourollahi, and E. Mohseni, “Application of he’s methods to nonlinear chemistry problem”, Comp. and Math. with Appl., Vol. 54, no.7-8, pp.1122-1132.
Downloads
Published
How to Cite
Issue
Section
License

This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors contributing to this journal agree to publish their articles under the Creative Commons Attribution 4.0 International License, allowing third parties to share their work (copy, distribute, transmit) and to adapt it, under the condition that the authors are given credit and that in the event of reuse or distribution, the terms of this license are made clear.
