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Quasilinearization–Barycentric Approach for Numerical Investigation of the Boundary Value Fin Problem

Authors: Alireza Rezaei, Fatemeh Baharifard, Kourosh Parand

Abstract:

In this paper we improve the quasilinearization method by barycentric Lagrange interpolation because of its numerical stability and computation speed to achieve a stable semi analytical solution. Then we applied the improved method for solving the Fin problem which is a nonlinear equation that occurs in the heat transferring. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The modified QLM is iterative but not perturbative and gives stable semi analytical solutions to nonlinear problems without depending on the existence of a smallness parameter. Comparison with some numerical solutions shows that the present solution is applicable.

Keywords: Quasilinearization method, Barycentric lagrange interpolation, nonlinear ODE, fin problem, heat transfer.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1334161

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References:


[1] S. Abbasbandy, E. Shivanian, Exact analytical solution of a nonlinear equation arising in heat transfer, Phys. Lett. A. 347 (2010) 567-574.
[2] M. H. Chang, A numerical analysis to the non-linear fin problem, Int. J. Heat. Mass. Tran. 48 (2005) 1819-1824.
[3] R. Cortell, A numerical analysis to the non-linear fin problem, J. Zhejiang. Univ-Sc. A. 9 (2008) 648-653.
[4] S. D. Conte, C. de Boor, Elementary numerical analysis, McGrow Hill, 1980.
[5] A. Ralston, P. Rabinowitz, A first course in numerical analysis, McGraw Hill, 1988.
[6] R. E. Bellman, R. E. Kalaba, Quasilinearization and nonlinear Boundaryvalue problems, Elsevier, New York, 1965.
[7] R. Kalaba, On nonlinear differential equations,the maximum operation and monotone convergence, J. Math. Mech. 8 (1968) 519-574.
[8] V. B. Mandelzweig, R. Krivec, Fast convergent quasilinearization approach to quantum problems, Aip. Conf. Proc. 768 (2005) 413-419.
[9] V. B. Mandelzweig, F. Tabakin, Quasilinearization approach to nonlinear problems in physics with application to nonlinear odes, Comput. Phys. Commun. 141 (2001) 268-281.
[10] J. I. Ramos, Piecewise quasilinearization techniques for singular boundary-value problems, Comput. Phys. Commun. 158 (2003) 12-25.
[11] V. B. Mandelzweig, Quasilinearization method: Nonperturbative approach to physical problems, Phys. Atom. Nucl+. 68 (2005) 1227-1258.
[12] V. B. Mandelzweig, Comparison of quasilinear and wkb approximations, Ann. Phys-New York. 321 (2006) 2810-2829.
[13] P. J. Davis, Interpolation and approximation, Dover, New York, 1975.
[14] G. M. Phillips, Interpolation and approximation by polynomials, Springer, New York, 2003.
[15] P. Henrici, Essentials of Numerical Analysis, Wiley, New York, 1982.
[16] H. Rutishauser, Vorlesungen ¨uber numerische Mathematik, Birkh¨auser, Boston, 1990.
[17] H. Salzer, Lagrangian interpolation at the chebyshev points xn,╬¢ ≡ cos(╬¢¤Ç/n), ╬¢ = 0(1)n; some unnoted advantages, Comput. J. 15 (1972) 156-159.
[18] W. Werner, Polynomial interpolation: Lagrange versus newton, Math. Comput. 43 (1984) 205-217.
[19] L. Winrich, Note on a comparison of evaluation schemes for the interpolating polynomial, Comput. J. 12 (1969) 154-155.
[20] J. Berrut, L. Trefethen, Barycentric lagrange interpolation, SIAM. Rev. 46 (2004) 501-517.
[21] N. J. Higham, The numerical stability of barycentric lagrange interpolation, Ima. J. Numer. Anal. 24 (2004) 547-556.
[22] J. A. Taylor, F. S. Hover, Economical simulation in particle filtering using interpolation, IEEE. Int. Conf. Info. Aut. (ICIA) (2009) 1326- 1330.
[23] P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975.
[24] S. Liaw, R. Yeh, Fins with temperature dependent surface heat fluxÔÇöi: Single heat transfer mode, Int. J. Heat. Mass. Tran. 37 (1994) 1509- 1515.
[25] S. Liaw, R. Yeh, Fins with temperature dependent surface heat fluxÔÇöii: Multi-boiling heat transfer, Int. J. Heat. Mass. Tran. 37 (1994) 1517- 1524.
[26] V. Lakshmikantham, A. S. Vatsala, Generalized Quasilinearization for Nonlinear Problems (Mathematics and Its Applications), Kluwer Academic, Dordrecht, 1998.
[27] M. N. Koleva, L. G. Vulkov, Two-grid quasilinearization approach to odes with applications to model problems in physics and mechanics, Comput. Phys. Commun. 181 (2010) 663-670.
[28] J. L. Lagrange, Lec┬©ons 'el'ementaires sur les math'ematiques, donn'ees `a lEcole Normale en1795, Oeuvres VII, GauthierVillars, Paris, 1877.