Homotopy analysis method for solving foam drainage equation with space- and time-fractional derivatives

International Journal of Differential Equations

Volumn 2011, Issue , 2011, Pages

  Fadravi, Hadi Hosseini ^a   Nik, Hassan Saberi ^a   Buzhabadi, Reza ^a  

^a ISLAMIC AZAD UNIVERSITY   (Iran)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84861575887     PISSN: 16879643     EISSN: 16879651     Source Type: Journal    
DOI: 10.1155/2011/237045     Document Type: Article

References (29)

1. L. Blank, "Numerical treatment of differential equations of fractional order," Numerical Analysis Report 287, The University of Manchester, Manchester, UK, 1996.
2. M. Caputo, "Linear models of dissipation whose Q is almost frequency independent," Journal of the Royal Australian Historical Society, vol. 13, part 2, pp. 529-539, 1967.
3. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
4. S. Momani and N. T. Shawagfeh, "Decomposition method for solving fractional Riccati differential equations," Applied Mathematics and Computation, vol. 182, no. 2, pp. 1083-1092, 2006.
5. Z. Odibat and S.Momani, "Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order," Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 167-174, 2008.
6. Z. Odibat and S. Momani, "Application of variation iteration method to nonlinear differential equations of fractional order," International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 7, pp. 15-27, 2006.
7. A. S. Bataineh, A. K. Alomari, M. S. M. Noorani, I. Hashim, and R. Nazar, "Series solutions of systems of nonlinear fractional differential equations," Acta Applicandae Mathematicae, vol. 105, no. 2, pp. 189-198, 2009.
8. M. Dehghan, J. Manafian, and A. Saadatmandi, "Solving nonlinear fractional partial differential equations using the homotopy analysis method," Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 448-479, 2010.
9. S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, 1992.
10. S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003.
11. S. J. Liao, "On the homotopy analysis method for nonlinear problems," Applied Mathematics and Computation, vol. 147, no. 2, pp. 499-513, 2004.
12. S. J. Liao, "Comparison between the homotopy analysis method and homotopy perturbation method," Applied Mathematics and Computation, vol. 169, no. 2, pp. 1186-1194, 2005.
13. S. J. Liao, "Homotopy analysis method: a new analytical technique for nonlinear problems," Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 2, pp. 95-100, 1997.
14. T. Hayat, M. Khan, and M. Ayub, "On non-linear flows with slip boundary condition," Zeitschrift f ür Angewandte Mathematik und Physik, vol. 56, no. 6, pp. 1012-1029, 2005.
15. S. Abbasbandy, "Soliton solutions for the 5th-order KdV equation with the homotopy analysis method," Nonlinear Dynamics, vol. 51, no. 1-2, pp. 83-87, 2008.
16. H. Jafari, M. Saeidy, and M. A. Firoozjaee, "The homotopy analysis method for solving higher dimensional initial boundary value problems of variable coefficients," Numerical Methods for Partial Differential Equations, vol. 26, no. 5, pp. 1021-1032, 2010.
17. S. P. Zhu, "An exact and explicit solution for the valuation of American put options," Quantitative Finance, vol. 6, no. 3, pp. 229-242, 2006.
18. D.Weaire and S. Hutzler, The Physic of Foams, Oxford University Press, Oxford, UK, 2000.
19. D.Weaire, S.Hutzler, S. Cox,M. D. Alonso, and D. Drenckhan, "The fluid dynmaics of foams," Journal of Physics: Condensed Matter, vol. 15, pp. 65-72, 2003.
20. M. A. Helal and M. S. Mehanna, "The tanh method and Adomian decomposition method for solving the foam drainage equation," Applied Mathematics and Computation, vol. 190, no. 1, pp. 599-609, 2007.
21. S. Hilgenfeldt, S. A. Koehler, and H. A. Stone, "Dynamics of coarsening foams: accelerated and selflimiting drainage," Physical Review Letters, vol. 86, no. 20, pp. 4704-4707, 2001.
22. G. Verbist, D. Weaire, and A. M. Kraynik, "The foam drainage equation," Journal of Physics Condensed Matter, vol. 8, no. 21, pp. 3715-3731, 1996.
23. Z. Dahmani, M. M. Mesmoudi, and R. Bebbouchi, "The foam drainage equation with time- and space-fractional derivatives solved by the Adomian method," Electronic Journal of Qualitative Theory of Differential Equations, vol. 30, pp. 1-10, 2008.
24. Z. Dahmani and A. Anber, "The variational iteration method for solving the fractional foam drainage equation," International Journal of Nonlinear Science, vol. 10, no. 1, pp. 39-45, 2010.
25. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003.
26. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
27. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
28. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
29. J. H. He, "Homotopy perturbation technique," ComputerMethods in AppliedMechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999.