---

Journal of Applied Nonlinear Dynamics 7(2) (2018) 189--196 | DOI:10.5890/JAND.2018.06.007

A. R. Gómez Plata„; E. Capelas de Oliveira

Department of Mathematics, Cajica, Universidad Militar Nueva Granada, 250247, Colombia Imecc, Campinas-SP, University of Campinas, 13083-859, Brazil

Download Full Text PDF

 

Abstract

The variational iteration method (VIM) is a analysis tool efficient for approximate non-linear fractional differential equations. Recently differents investigators are used this method in your works and we study the Lagrange multipliers of the variational iteration method for the time fractional Burgers equation and apply those in differents particular cases. In this conference we present approximations of the solutions for a particular case of the time fractional Burgers equation (BF), with the use of the variational iteration method, the Caputo derivate for 0 <α≤ 1, after make an comparation with the Adomian descomposition method (ADM).

References

1. [1]	Diethelm, K. (2010), The Analysis of Fractional Differential Equations: An Application- Oriented Exposition using Differential Operator of Caputo Type, Springer, Braunschweig.

2. [2]	Mainardi, F., Luchko, Y., and Pagnini, G. (2001), The fundamental solution of the space-time fractional diffusion equation, Frac. Calc. Appl. Anal., 4, 153-192.

3. [3]	Costa, F.S., Marao, J.A.P.F., Soares, J.C.A., and Capelas de Oliveira, E. (2015), Similarity solution to fractional nonlinear space-time, diffusion-wave equation, J. Math. Phys., 56, 033507.

4. [4]	Capelas de Oliveira, E., Mainardi, F., and Vaz Jr, J. (2014), Fractional models of anomalous relaxation based on the Kilbas and Saigo function, Meccanica (Milano. Print), 49, 2049-2060.

5. [5]	Figuereido Camargo, R., Capelas de Oliveira, E., and Vaz, Jr. J. (2009), On anamalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator, J. Math. Phys., 50, 123-518.

6. [6]	Diethelm, K. and Ford, N.J. (2004), Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154, 621-640.

7. [7]	Liu, F.W., Anh, V., and Turner, I. (2004), Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219.

8. [8]	Wazwaz, A.M. (2007), The variational iteration method for solving linear and nonlinear systems of PDEs, Comput. Math. Appl., 54, 895-902.

9. [9]	Ramos. J.I. (2008), On the variational iteration method and the other iterative techniques for nonlinear differential equations, Appl. Math. Comput., 199, 39-69.

10. [10]	Mainardi, F. (1996), Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7, 1461-1477.

11. [11]	Wazwaz, A.M., (2007), The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations, Comput. Math. Appl., 54, 926-932.

12. [12]	Wazwaz, A.M. (2007), The Variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations, Comput. Math. Appl., 54, 933-939.

13. [13]	Ozer, H. (2007), Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics, Int. J. Nonlinear Sci. Numer. Simul., 8, 513-518.

14. [14]	Sheng, H. and Chen, Y.Q. (2011), Application of numerical inverse Laplace transform algorithms in fractional calculus, J. Franklin Ins., 348, 315-330.

15. [15]	Ruiz-Medina, M.D., Angulo, J.M., and Anh, V.V. (2001), Scaling limit solution of a fractional Burgers’ equation, Stochastic Process. Appl., 93, 285-300.

16. [16]	Hayat, T., Khan, M., and Asghar, S. (2007), On the MHD flow of fractional generalized Burgers’ fluid with modified Darcy’s law, Acta. Mech. Sin., 23, 257-261.

17. [17]	Shah, S.H.A.M. (2010), Some helical flows of a Burgers’ fluid with fractional derivative, Meccanica, 45, 143-151.

18. [18]	Chen, Y. and H. L. (2008), An Numerical solutions of coupled Burgers’ equations with time and space-fractional derivatives, Appl. Math. Comput., 200, 87-95.

19. [19]	Jun, Y.J., Tenreiro Machado, J.A., and Hristov, J. (2015), Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow, Nonlinear Dyn., DOI 10.1007/s11071-015-2085-2.

20. [20]	Odibat, Z. and Momami, S. (2009), The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Comp. Math. Appl., 58, 2199-2208.

21. [21]	Inokuti, M., Sekine, H., and Mura, T. (1978), General use of the Lagrange multiplier in non-linear mathematical physics, in: S Nemat-Nasser (Ed), Variational Method in the Mechanics of Solid, Pergamon Press, Oxford, 156-162.

22. [22]	Wu, G.C. and Baleanu, D. (2013), Variational iteration method for Burgers’ flow with fractional derivatives-New Lagrange multipliers, App. Math. Mod., 37, 6183-6190.

23. [23]	Adomian, G. (1998), A review of the Decomposition method in applied mathematics, J. Math. Anal. App., 135, 501-544.

24. [24]	Adomian, G. (1994), Solving Frontier Problems of Physics: Decomposition Method, Springer: Athens, Georgia.

25. [25]	Wazwaz, A.M. (2000), A New algorithm for calculating Adomian polynomials for nonlinear operators, App. Math. Comp., 3, 33-51.

26. [26]	Wazwaz, A.M. and El-Sayed, S.M. (2001), A new Modification of the Adomian decomposition method for linear and nonlinear operators, Appl. Math. Comput., 122, 393-405.

27. [27]	Momani, S. (2006), Non-perturbative analytical solutions of the space-and time-fractional Burgers equations, Chaos, Solitons and Fractals, 28, 930-937.

28. [28]	Gómez Plata, A.R. (2016), Non-linear fractional differential equations, PhD. Thesis in applied Mathematics, Imecc-Unicamp, Campinas.