## Homotopy analysis method for solving foam drainage equation with space- and time-fractional derivatives.(English)Zbl 1234.35298

Summary: The analytical solution of the foam drainage equation with time- and space-fractional derivatives was derived by means of the homotopy analysis method (HAM). The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made; the comparisons show that the homotopy analysis method is very effective and convenient. By choosing different values of the parameters $$\alpha, \beta$$ in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained.

### MSC:

 35R11 Fractional partial differential equations 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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### References:

  L. Blank, “Numerical treatment of differential equations of fractional order,” Numerical Analysis Report 287, The University of Manchester, Manchester, UK, 1996. · Zbl 0870.65137  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.  K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011  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. · Zbl 1107.65121  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. · Zbl 1152.34311  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. · Zbl 1378.76084  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. · Zbl 1187.34007  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. · Zbl 1185.65187  S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, 1992.  S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003. · Zbl 1051.76001  S. J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499-513, 2004. · Zbl 1086.35005  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. · Zbl 1082.65534  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. · Zbl 0927.65069  T. Hayat, M. Khan, and M. Ayub, “On non-linear flows with slip boundary condition,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, no. 6, pp. 1012-1029, 2005. · Zbl 1097.76007  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. · Zbl 1170.76317  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. · Zbl 1197.65151  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. · Zbl 1136.91468  D. Weaire and S. Hutzler, The Physic of Foams, Oxford University Press, Oxford, UK, 2000.  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.  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. · Zbl 1120.76051  S. Hilgenfeldt, S. A. Koehler, and H. A. Stone, “Dynamics of coarsening foams: accelerated and self-limiting drainage,” Physical Review Letters, vol. 86, no. 20, pp. 4704-4707, 2001.  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.  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. · Zbl 1181.26013  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. · Zbl 1226.35082  B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003.  K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993. · Zbl 0789.26002  S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. · Zbl 0818.26003  I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008  J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017
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