Abstract
The homotopy analysis method of Liao has proven useful in obtaining analytical solutions to various nonlinear differential equations. As discussed in the preceding chapter, in this method, one has great freedom to select auxiliary functions, operators, and parameters in order to ensure the convergence of the approximate solutions and to increase both the rate and region of convergence. In the present chapter, we discuss the selection of the initial approximation, auxiliary linear operator, auxiliary function, and convergence control parameter in the application of the homotopy analysis method, in a fairly general setting.
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References
R.A. Van Gorder and K. Vajravelu, Analytic and numerical solutions to the Lane-Emden equation, Physics Letters A, 372 (2008) 6060.
J.I. Ramos, Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method, Chaos, Solitons and Fractals, 38 (2008) 400.
J.H. Lane, On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, The American Journal of Science & Arts, 50 (1870) 57.
E. Momoniat and C. Harley, Approximate implicit solution of a Lane-Emden equation, New Astronomy, 11 (2006) 520.
S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Physics Letters A, 360 (2006) 109.
M. Sajid, T. Hayat and S. Asghar, Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt, Nonlinear Dynamics, 50 (2007) 27.
S.J. Liao, A new analytic algorithm of Lane-Emden type equations, Applied Mathematics and Computation, 142 (2003) 1.
G. Schenkel, Plastics Extrusion Technology and Theory, Iliffe, London, 1966.
J. Vleggaar, Laminar boundary-layer behavior on continuous accelerating surfaces, Chem. Eng. Sci., 32 (1977) 1517.
C.Y. Wang, Fluid film sprayed on a stretching surface, Chem. Eng. Comm., 107 (1991) 11.
L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys., 21 (1970) 645.
N. Afzal and I.S. Varshney, The cooling of low heat resistance stretching sheet moving through a fluid, Warme-und Stoffubertragung, 14 (1980) 289.
S.J. Liao and I. Pop, Explicit analytic solution for similarity boundary layer equations, Int. J. Heat Mass Transfer, 47 (2004) 75.
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© 2012 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
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Vajravelu, K., van Gorder, R.A. (2012). Methods for the Control of Convergence in Obtained Solutions. In: Nonlinear Flow Phenomena and Homotopy Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32102-3_3
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DOI: https://doi.org/10.1007/978-3-642-32102-3_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32101-6
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