Variational iteration method for the Burgers' flow with fractional derivatives-New Lagrange multipliers

Applied Mathematical Modelling

Volumn 37, Issue 9, 2013, Pages 6183-6190

  Wu, Guo Cheng ^a,b   Baleanu, Dumitru ^c,d,e  

^a NEIJIANG NORMAL UNIVERSITY   (China)

^b SICHUAN UNIVERSITY   (China)

^c CANKAYA UNIVERSITY   (Turkey)

^d INSTITUTE OF SPACE SCIENCES   (Romania)

^e KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

Author keywords

Caputo derivative; Fractional differential equation; Variational iteration method

Indexed keywords

APPROXIMATE SOLUTION; CAPUTO DERIVATIVES; FLOW THROUGH POROUS MEDIAS; FRACTIONAL DIFFERENTIAL EQUATIONS; FRACTIONAL MODEL; MEMORY EFFECTS; VARIATIONAL ITERATION METHOD;

DIFFERENTIAL EQUATIONS; LAGRANGE MULTIPLIERS;

ITERATIVE METHODS;

EID: 84875378030     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2012.12.018     Document Type: Article

References (32)

1. Diethelm K., Ford N.J., Freed A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 2002, 29:3-22.
2. Diethelm K., Ford N.J. Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 2004, 154:621-640.
3. Liu F.W., Anh V., Turner I. Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 2004, 166:209-219.
4. Tadjeran C., Meerschaert M.M., Scheffler H.P. A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 2006, 213:205-213.
5. Deng W. Short memory principle and a predictor-corrector approach for fractional differential equations. J. Comput. Appl. Math. 2007, 206:174-188.
6. Baleanu D., Diethelm K., Scalas E., Trujillo J. Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos 2012, World Scientific, Boston, Mass, USA.
7. Shawagfeh N.T. Analytical approximate solutions for nonlinear fractional differential equations. Appl. Math. Comput. 2002, 131:517-529.
8. Momani S., Odibat Z. Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 2007, 365:345-350.
9. Zhuang P., Liu F.W., Anh V., Turner I. New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 2008, 46:1079-1095.
10. Li C.P., Wang Y.H. Numerical algorithm based on Adomian decomposition for fractional differential equations. Comput. Math. Appl. 2009, 57:1672-1681.
11. Hristov J. Approximate solutions to fractional subdiffusion equations. Eur. Phys. J. Special Topics 2011, 193:229-243.
12. Duan J.S., Rach R., Buleanu D., Wazwaz A.M. A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Fract. Calc. 2012, 3:73-99.
13. He J.H. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 1998, 167:57-68.
14. He J.H. Variational iteration method - a kind of non-linear analytical technique: some examples. Int. J. Non-Linear Mech. 1999, 34:699-708.
15. Abbasbandy S. An approximation solution of a nonlinear equation with Riemann-Liouville's fractional derivatives by He's variational iteration method. J. Comput. Appl. Math. 2007, 207:53-58.
16. Momani S., Odibat Z. Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Phys. Lett. A 2006, 355:271-279.
17. Inc M. The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl. 2008, 345:476-484.
18. Yulita M.R., Noorani M.S.M., Hashim I. Variational iteration method for fractional heat- and wave-like equations. Nonlinear Anal. Real World Appl. 2009, 10:1854-1869.
19. Nawaz Y. Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations. Comput. Math. Appl. 2011, 61:2330-2341.
20. Sakar M.G., Erdogan F., Yildirim A. Variational iteration method for the time-fractional Fornberg-Whitham equation. Comput. Math. Appl. 2012, 63:1382-1388.
21. Podlubny I. Fractional Differential Equations 1999, Academic press, San Diego.
22. Debnath L., Bhatta D. Integral Transforms and their Applications 2006, Chapman & Hall/CRC.
23. Sheng H., Li Y., Chen Y.Q. Application of numerical inverse Laplace transform algorithms in fractional calculus. J. Franklin Inst. 2011, 348:315-330.
24. Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 1996, 7:1461-1477.
25. He J.H., Wu X.H. Variational iteration method: new development and applications. Comput. Math. Appl. 2007, 54:881-894.
26. Diethelm K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition using Differential Operators of Caputo type 2010, Springer.
27. Ruiz-Medina M.D., Angulo J.M., Anh V.V. Scaling limit solution of a fractional Burgers equation. Stochastic Process. Appl. 2001, 93:285-300.
28. Hayat T., Khan M., Asghar S. On the MHD flow of fractional generalized Burgers' fluid with modified Darcy's law. Acta. Mech. Sin. 2007, 23:257-261.
29. Xue C., Nie J., Tan W. An exact solution of start-up flow for the fractional generalized Burgers' fluid in a porous half-space. Nonlinear Anal.-Theory Methods Appl. 2008, 69:2086-2094.
30. Khan M., Ali S.H., Qi H. On accelerated flows of a viscoelastic fluid with the fractional Burgers' model. Nonlinear Anal.-Real World Appl. 2009, 10:2286-2296.
31. Shah S.H.A.M. Some helical flows of a Burgers' fluid with fractional derivative. Meccanica 2010, 45:143-151.
32. Chen Y., An H.L. Numerical solutions of coupled Burgers equations with time-and space-fractional derivatives. Appl. Math. Comput. 2008, 200:87-95.