Conformable variational iteration method

Abstract

In this study, we introduce the conformable variational iteration method based on new defined fractional derivative called conformable fractional derivative. This new method is applied two fractional order ordinary differential equations. To see how the solutions of this method, linear homogeneous and non-linear non-homogeneous fractional ordinary differential equations are selected. Obtained results are compared the exact solutions and their graphics are plotted to demonstrate efficiency and accuracy of the method.

Keywords

• Conformable fractional derivative,Factional derivative,Fractional differential equations,Variational iteration method

Kaynakça

1. I. Podlubny, Fractional differential equations, Academic Press, 1999.
2. W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989) 134-144.
3. H. Beyer, S. Kempfle, Definition of Physically Consistent Damping Laws with Fractional Derivatives, J. Appl. Math. Mech. 75 (1995) 623-635.
4. F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons & Fractals. 7 (1996) 1461-1477.
5. J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng. 167 (1998) 57-68.
6. J.H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Int. J. Non. Linear. Mech. 34 (1999) 699-708.
7. J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2000) 115-123.
8. J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B. 20 (2006) 1141-1199.

9. J. He, Variational Equations Iteration Method for Delay Differential, Commun. Nonlinear Sci. Numer. Simul. 1997 (1997) 1997-1998.
10. J.-H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Eng. 167 (1998) 69-73. doi:10.1016/S0045-7825(98)00109-1.
11. A.-M. Wazwaz, Solving Systems of Fourth-Order Emden–Fowler Type Equations by the Variational Iteration Method, Chem. Eng. Commun. (2016) (Just accepted).
12. Z. M. Odibat, S. Momani, Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order, Int. J. Nonlinear Sci. Numer. Simul. 7 (2006).
13. S.A. El-Wakil, M.A. Abdou, New applications of variational iteration method using Adomian polynomials, Nonlinear Dyn. 52 (2007) 41-49.
14. D.D. Ganji, G.A. Afrouzi, R.A. Talarposhti, Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations, Phys. Lett. A. 368 (2007) 450-457.
15. A.-M. Wazwaz, The variational iteration method for rational solutions for KdV, (2,2), Burgers, and cubic Boussinesq equations, J. Comput. Appl. Math. 207 (2007) 18-23.
16. S. a. Khuri, a. Sayfy, Variational iteration method: Green’s functions and fixed point iterations perspective, Appl. Math. Lett. 32 (2014) 28-34.
17. B. Ibiş, M. Bayram, Approximate solution of time-fractional advection-dispersion equation via fractional variational iteration method., Sci. World J. 2014 (2014). doi:10.1155/2014/769713.
18. B. İbiş, M. Bayram, Analytical approximate solution of time-fractional Fornberg–Whitham equation by the fractional variational iteration method, Alexandria Eng. J. 53 (2014) 911-915.
19. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65-70.
20. T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015) 57-66.
21. Y. Cenesiz, A. Kurt, The solutions of time and space conformable fractional heat equations with conformable Fourier transform, Acta Univ. Sapientiae, Math. 7 (2015) 130-140. doi:10.1515/ausm-2015-0009.
22. A. Kurt, Y. Çenesiz, O. Tasbozan, On the Solution of Burgers’ Equation with the New Fractional Derivative, Open Phys. 13 (2015) 355-360. doi:10.1515/phys-2015-0045.
23. O.S. Iyiola, G.O. Ojo, On the analytical solution of Fornberg – Whitham equation with the new fractional derivative, Pramana – J. Phys. 85 (2015) 567-575.
24. O. Acan, O. Firat, Y. Keskin, G. Oturanc, Solution of Conformable Fractional Partial Differential Equations by Reduced Differential Transform Method, Selcuk J. Appl. Math. (2016) (In press).
25. O. Acan, O. Firat, A. Kurnaz, Y. Keskin, Applications for New Technique Conformable Fractional Reduced Differential Transform Method, J. Comput. Theor. Nanosci. (2016) (Accepted).
26. E. Ünal, A. Gödogan, Solution of Conformable Fractional Ordinary Differential Equations via Differential Transform Method, arXiv Prepr. 1602.05605 (2016) 1-14.
27. A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math. 13 (2015).