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72 Mathematical Techniques of Fractional Order Systems

Gorder, R., 2012. Gaussian waves in the fitzhugh-nagumo equation demonstrate one role of the
auxiliary function h(x,t) in the homotopy analysis method. Commun. Nonlinear Sci. Num.
Simul. 17 (3), 1233 1240.
Graef, J.R., Qian, C., Yang, B., 2003. A three point boundary value problem for nonlinear forth
order differential equations. J. Math. Anal. Applicat. 287, 217 233.
Graef, J.R., Qian, C., Yang, B., 2004. Multiple positive solutions of a boundary value prolem for
ordinary differential equations. Electr. J. Qualit. Theory Different. Eq. 11, 1 13.
Hafez, T., Ganji, D.D., Babazadeh, H., 2007. The application of he’s variational iteration method
to nonlinear equations arising in heat transfer. Phys. Lett. A 363, 213 217.
Hassan, H.N., El-Tawil, M.A., 2011a. An efficient analytic approach for solving two-point non-
linear boundary value problems by homotopy analysis method. Math. Methods Appl. Sci. 34
(8), 977 989.
Hassan, H.N., El-Tawil, M.A., 2011b. A new technique of using homotopy analysis method for
solving high-order nonlinear differential equations. Math Methods Appl. Sci. 34 (6),
728 742.
Hassan, H.N., Semary, M.S., 2013. Analytic approximate solution for the bratus problem by
optimal homotopy analysis method. Commun. Numer. Anal. 2013, 1 14.
Hassan, H.N., Semary, M.S., 2014. The application of he’s homotopy perturbation method to
nonlinear equations arising in heat transfer equations by optimal homotopy analysis method.
Walailak J. Sci. Technol. 11 (8), 659 677.
Hayat, T., Sajid, M., 2007. On analytic solution for thin film flow of a fourth grade fluid down a
vertical cylinder. Phys. Lett. A 361 (4-5), 316 322.
He, J.H., 1999. Variational iteration method - a kind of non-linear analytical technique: some
examples. Int. J. Non-Linear Mech. 34, 699 708.
Jacobsen, J., Schmitt, K., 2002. The liouville-bratu-gelfand problem for radial operators. J. Diff.
Eq. 184, 283 298.
Jalilian, R., 2010. Non-polynomial spline method for solving bratu’s problem. Comput. Phys.
Commun. 181, 1868 1872.
Liao, S., 1992. The proposed homotopy analysis technique for the solution of nonlinear pro-
blems. Ph.D. thesis, Shanghai Jiao Tong University.
Liao, S., 2003. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman
& Hall/CRC Press, London/Boca Raton, FL.
Liao, S., 2005. A new branch of solutions of boundary-layer flows over an impermeable
stretched plate. Int. J. Heat Mass Tran. 48, 977 989.
Liao, S., 2009. Notes on the homotopy analysis method: some definitions and theorems.
Commun. Nonlinear Sci. Num. Simul. 14 (4), 983 997.
Liao, S., 2012. Homotopy Analysis Method in Nonlinear Differential Equations. Springer,
Heidelberg Dordrecht London New York.
Liu, Y., LiZ, B., 2009. The homotopy analysis method for approximating the solution of the
modified korteweg-de vries equation. Chaos Solitons Fractals 39 (1), 1 8.
Magyari, E., 2008. Exact analytical solution of a nonlinear reaction-diffusion model in porous
catalysts. Chem. Eng. J. 143 (1-2), 167 171.
Monje, C.A., Chen, Y., Vinagre, B., Xue, D., Fileu, V., 2010. Fractional Order Controls:
Fundamentals and Application. Springer Verlag, Berlin.
Motsa, S.S., Sibanda, P., Shateyi, S., 2010. A new spectral homotopy analysis method for solv-
ing a nonlinear second order bvp. Commun. Nonlinear Sci. Num. Simul. 15, 2293 2302.
Odibat, Z.M., 2010. Analytic study on linear systems of fractional differential equations.
Comput. Math. Applicat. 59 (3), 1171 1183.