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The Homotopy Analysis Method for Solving Discontinued Problems Arising in Nanotechnology
This paper applies the homotopy analysis method method to a nonlinear differential-difference equation arising in nanotechnology. Continuum hypothesis on nanoscales is invalid, and a differential-difference model is considered as an alternative approach to describing discontinued problems. Comparison of the approximate solution with the exact one reveals that the method is very effective.


[1] S. Abbasbandy, Soliton solutions for the 5th-order KdV equation with the homotopy analysis method, Nonlinear Dyn 51. (2008) 83-87.
[2] T. Hayat, T. Javed, M. Sajid, Analytic solution for rotating ¤ow and heat transfer analysis of a third-grade ¤uid, Acta Mech. 191 (2007) 219-29.
[3] J.H. He, Y.Q. Wan, L. Xu, Nano-effects, quantum-like properties in electrospun nano£bers, Chaos Solitons Fractals 33 (2007) 26-37.
[4] J.H. He, Y.Y. Liu, L. Xu, et al., Micro sphere with nanoporosity by electrospinning, Chaos Solitons Fractals 32 (2007) 1096-1100.
[5] J.H. He, S.D. Zhu, Differential-difference model for nanotechnology, J. Phys. Conf. Ser. 96 (2008) 012189.
[6] M.S. El Naschie, Deterministic quantum mechanics versus classical mechanical indeterminism, Int. J. Nonlinear Sci. 8 (1) (2007) 5-10.
[7] M.S. El Naschie, A review of applications and results of E-in£nity theory, Int. J. Nonlinear Sci. 8 (1) (2007) 11-20.
[8] M.S. El Naschie, Probability set particles, Int. J. Nonlinear Sci. 8 (1) 117-119.
[9] M.S. El Naschie, Nanotechnology for the developing world, Chaos Solitons Fractals 30 (2006) 769-773.
[10] S.D. Zhu, Exp-function method for the Hybrid-Lattice system, Int. J. Nonlinear Sci. 8 (3) (2007) 461-464.
[11] S.P. Zhu, An exact and explicit solution for the valuation of American put options, Quantitative Finance. 6 (2006) 229-242.
[12] S.D. Zhu, Exp-function method for the discrete mKdV lattice, Int. J. Nonlinear Sci. 8 (3) (2007) 465-469.
[13] S.D. Zhu, Discrete (2+1) dimensional Toda lattice equation via Expfunction method, Phys. Lett. A 372 (2008) 654-657.
[14] S.D. Zhu, Yu-ming Chu, Song-liang Qiu. The homotopy perturbation method for discontinued problems arising in nanotechnology, Computers and Mathematics with Applications (2009), doi:10.1016/j.camwa.2009.03.048.
[15] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[16] S.J. Liao, Beyond perturbation: introduction to the homotopy analysis method, CRC Press, Boca Raton: Chapman & Hall; 2003.
[17] S.J. Liao, On the homotopy anaylsis method for nonlinear problems, Appl Math Comput 2004;147:499513.
[18] S.J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Appl Math Comput,169 (2005) 11861194.
[19] S.J. Liao, A new branch of solutions of boundary-layer ¤ows over an impermeable stretched plate, Int J Heat Mass Transfer ,48 (2005) 25292539.
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[21] Y. Liu, J.H. He, Bubble electrospinning for mass production of nano£bers, Int. J. Nonlinear Sci. 8 (2007) 393-396.
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