{"title":"Quasilinearization\u2013Barycentric Approach for Numerical Investigation of the Boundary Value Fin Problem","authors":"Alireza Rezaei, Fatemeh Baharifard, Kourosh Parand","volume":50,"journal":"International Journal of Computer and Information Engineering","pagesStart":194,"pagesEnd":202,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/7635","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.<\/p>\r\n","references":" S. Abbasbandy, E. Shivanian, Exact analytical solution of a nonlinear\r\nequation arising in heat transfer, Phys. Lett. A. 347 (2010) 567-574.\r\n M. H. Chang, A numerical analysis to the non-linear fin problem, Int.\r\nJ. Heat. Mass. Tran. 48 (2005) 1819-1824.\r\n R. Cortell, A numerical analysis to the non-linear fin problem, J.\r\nZhejiang. Univ-Sc. A. 9 (2008) 648-653.\r\n S. D. Conte, C. de Boor, Elementary numerical analysis, McGrow Hill,\r\n1980.\r\n A. Ralston, P. Rabinowitz, A first course in numerical analysis, McGraw\r\nHill, 1988.\r\n R. E. Bellman, R. E. Kalaba, Quasilinearization and nonlinear Boundaryvalue\r\nproblems, Elsevier, New York, 1965.\r\n R. Kalaba, On nonlinear differential equations,the maximum operation\r\nand monotone convergence, J. Math. Mech. 8 (1968) 519-574.\r\n V. B. Mandelzweig, R. Krivec, Fast convergent quasilinearization approach\r\nto quantum problems, Aip. Conf. Proc. 768 (2005) 413-419.\r\n V. B. Mandelzweig, F. Tabakin, Quasilinearization approach to nonlinear\r\nproblems in physics with application to nonlinear odes, Comput. Phys.\r\nCommun. 141 (2001) 268-281.\r\n J. I. Ramos, Piecewise quasilinearization techniques for singular\r\nboundary-value problems, Comput. Phys. Commun. 158 (2003) 12-25.\r\n V. B. Mandelzweig, Quasilinearization method: Nonperturbative approach\r\nto physical problems, Phys. Atom. Nucl+. 68 (2005) 1227-1258.\r\n V. B. Mandelzweig, Comparison of quasilinear and wkb approximations,\r\nAnn. Phys-New York. 321 (2006) 2810-2829.\r\n P. J. Davis, Interpolation and approximation, Dover, New York, 1975.\r\n G. M. Phillips, Interpolation and approximation by polynomials,\r\nSpringer, New York, 2003.\r\n P. Henrici, Essentials of Numerical Analysis, Wiley, New York, 1982.\r\n H. Rutishauser, Vorlesungen \u252c\u00bfuber numerische Mathematik, Birkh\u252c\u00bfauser,\r\nBoston, 1990.\r\n H. Salzer, Lagrangian interpolation at the chebyshev points xn,\u256c\u00a2 \u2261 cos(\u256c\u00a2\u00a4\u00c7\/n), \u256c\u00a2 = 0(1)n; some unnoted advantages, Comput. J. 15\r\n(1972) 156-159.\r\n W. Werner, Polynomial interpolation: Lagrange versus newton, Math.\r\nComput. 43 (1984) 205-217.\r\n L. Winrich, Note on a comparison of evaluation schemes for the\r\ninterpolating polynomial, Comput. J. 12 (1969) 154-155.\r\n J. Berrut, L. Trefethen, Barycentric lagrange interpolation, SIAM. Rev.\r\n46 (2004) 501-517.\r\n N. J. Higham, The numerical stability of barycentric lagrange interpolation,\r\nIma. J. Numer. Anal. 24 (2004) 547-556.\r\n J. A. Taylor, F. S. Hover, Economical simulation in particle filtering\r\nusing interpolation, IEEE. Int. Conf. Info. Aut. (ICIA) (2009) 1326-\r\n1330.\r\n P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic\r\nPress, New York, 1975.\r\n S. Liaw, R. Yeh, Fins with temperature dependent surface heat flux\u00d4\u00c7\u00f6i:\r\nSingle heat transfer mode, Int. J. Heat. Mass. Tran. 37 (1994) 1509-\r\n1515.\r\n S. Liaw, R. Yeh, Fins with temperature dependent surface heat flux\u00d4\u00c7\u00f6ii:\r\nMulti-boiling heat transfer, Int. J. Heat. Mass. Tran. 37 (1994) 1517-\r\n1524.\r\n V. Lakshmikantham, A. S. Vatsala, Generalized Quasilinearization for\r\nNonlinear Problems (Mathematics and Its Applications), Kluwer Academic,\r\nDordrecht, 1998.\r\n M. N. Koleva, L. G. Vulkov, Two-grid quasilinearization approach to\r\nodes with applications to model problems in physics and mechanics,\r\nComput. Phys. Commun. 181 (2010) 663-670.\r\n J. L. Lagrange, Lec\u252c\u00a9ons 'el'ementaires sur les math'ematiques, donn'ees `a\r\nlEcole Normale en1795, Oeuvres VII, GauthierVillars, Paris, 1877.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 50, 2011"}