Abstract
Efficient fourth order iterative methods are presented to approximate the solution of the one dimensional Bratu problem. These iterative methods improve the efficiency index of the Chebyshev and Newton methods. A semilocal convergence analysis assuming the same conditions for Newton’s method is performed. Finally, a particular case of the 1D Bratu problem is solved using directly these efficient methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Altman, G.: Iterative methods of higher order. Bull. Acad. Pollon. Sci. (Série des Sci. Math. Astr. Phys. IX, 6268 (1961)
Amat, S., Hernández, M.A., Romero, N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)
Ascher, U.M., et al.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia (1995)
Bratu, G.: Sur les equation integrals non-lineaires. Bull. Math. Soc. France 42, 113–142 (1914)
Boyd, J.P.: Chebychev polynomial expansions for simultaneous approximation of two branches of a function with application of the one-dimensional Bratu equation. Appl. Math. Comput. 142, 189–200 (2003)
Boyd, J.P.: One-point pseudo spectral collocation for the one-dimensional Bratu equation. Appl. Math. Comput. 217, 5553–5565 (2011)
Caglar, H., Caglarb, N., Özerc, M., Valaristosd, A., Anagnostopoulos, A.N.: B-spline method for solving Bratu’s problem. Int. J. Comput. Math. 87, 1885–1891 (2010)
Ezquerro, J.A., Hernández, M.A.: An optimization of Chebyshev’s method. J. Complexity 25, 343–361 (2009)
Grau, M., Noguera, M.: A variant of Cauchy’s method with accelerated fifth-order convergence. Appl. Math. Lett. 17(5), 509–517 (2004)
Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41, 433–445 (2001)
Hernández, M.A., Romero, N.: On the efficiency index of one-point iterative processes. Numer. Algorithms 46, 35–44 (2007)
Mohsen, A.: A simple solution of the Bratu problem. Comput. Math. Appl. 67, 26–33 (2014)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)
Kumar, S., Sloan, I.H.: A new collocation-type method for Hammerstein integral equations. Math. Comput. 48(178), 585–593 (1987)
Hueso, J.L., Martnez, E., Torregrosa, J.R.: Third and fourth order iterative methods free from second derivative for nonlinear systems. Appl. Math. Comput. 211, 190–197 (2009)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Englewood Cliffs (1964)
Wan, Y.Q., Guo, Q., Pan, N.: Thermo-electro-hydro dynamic model for electro spinning process. Int. J. Nonlinear Sci. Numer. Simul. 5, 5–8 (2004)
Werner, W.: Newton-like methods for the computation of fixed points. Comput. Math. Appl. 10(1), 77–86 (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research has been supported by the Ministerio de Economía y Competitividad MTM2014-52016-C2-1-P.
Rights and permissions
About this article
Cite this article
Romero, N. Solving the one dimensional Bratu problem with efficient fourth order iterative methods. SeMA 71, 1–14 (2015). https://doi.org/10.1007/s40324-015-0041-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40324-015-0041-1