Abstract
In this paper, modifications of the quasilinearization method with higher-order convergence for solving nonlinear differential equations are constructed. A general technique for systematically obtaining iteration schemes of order m ( > 2) for finding solutions of highly nonlinear differential equations is developed. The proposed iterative schemes have convergence rates of cubic, quartic and quintic orders. These schemes were further applied to bifurcation problems and to obtain critical parameter values for the existence and uniqueness of solutions. The accuracy and validity of the new schemes is tested by finding accurate solutions of the one-dimensional Bratu and Frank-Kamenetzkii equations.
Similar content being viewed by others
References
Chun, C.: Iterative methods: improving Newton’s method by the decomposition method. Comput. Math. Appl. 50, 1559–1568 (2005)
Fang, L., He, G.: Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 228, 296–303 (2009)
Frontini, M., Sormani, E.: Some variant of Newtons method with third-order convergence. Appl. Math. Comput. 140, 419–426 (2003)
Frontini, M., Sormani, E.: Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput. 149, 771–782 (2004)
Kou, J.: The improvements of modified Newton’s method. Appl. Math. Comput. 189, 602–609 (2007)
Peng, W., Danfu, H.: A family of iterative methods with higher-order convergence. Appl. Math. Comput. 182(1), 474–477 (2006)
Adomian, G.: A review of the decomposition method and some recent results for nonlinear equation. Math. Comput. Model. 13(7), 17–43 (1990)
Adomian, G., Rach, R.: Noise terms in decomposition series solution. Comput. Math. Appl 24(11), 61–64 (1992)
Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994)
Bellman, R.E., Kalaba, R.E.: Quasilinearization and Nonlinear Boundary-Value Problems. Elsevier, New York (1965)
Krivec, R., Mandelzweig, V.B.: Numerical investigation of quasilinearization method in quantum mechanics. Comput. Phys. Commun. 138, 69–79 (2001)
Mandelzweig, V.B.: Quasilinearization method and its verification on exactly solvable models in quantum mechanics. J. Math. Phys. 40(12), 6266–6291 (1999)
Mandelzweig, V.B., Tabakin, F.: Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput. Phys. Commun. 141, 268–281 (2001)
Mandelzweig, V.B.: Quasilinearization method: nonperturbative approach to physical problems. Phys. At. Nucl. 68(7), 1227–1258 (2005)
Amore, P., Fernández, F.M.: The Virial Theorem for Nonlinear Problems. arXiv: 0904.3858v2 [math-ph] (2009)
Boyd, J.P.: Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation. Appl. Math. Comput. 142, 189–200 (2003)
Boyd, J.P.: One-point pseudospectral collocation for the one-dimensional Bratu equation. Appl. Math. Comput. 217, 5553–5565 (2011)
Chan, T.F.C., Keller, H.B.: Arc-length amd multi-grid techniques for nonlinear elliptic eigenvalue problems. SIAM J. Sci. Stat. Comput. 3, 173–194 (1982)
Hassan, I.H., Ertürk, V.S.: Applying differential transformation method to the one-dimensional planar Bratu problem. Int. J. Contemp. Math. Sci. 2, 1493–1504 (2007)
Makinde, O.D., Osalusi, E.: Exothermic explosions in symmetric geometries—an exploitation of perturbation technique. Rom. J. Phys. 50, 621–625 (2005)
Vázquez-Espí, C., Linán, A.: The effect of square corners on the ignition of solids. SIAM J. Appl. Math. 53(6), 1567–1590 (1993)
Wazwaz, A.M.: Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166, 652–663 (2005)
Harley, C., Momoniat, E.: Efficient boundary value problem solution for a Lane-Emden equation. Math. Comput. Appl. 15(4), 613–620 (2010)
Frank-Kamenetzkii, D.A.: Diffusion and Heat Transfer in Chemical Kinetics. Plenum Press, New York (1969)
Kubíček, M., Hlavačék, V.: Numerical Solution of Nonlinear Boundary Value Problems with Applications. Prentice-Hall, Englewood Cliffs (1983)
Wazwaz, A.: A new algorithm for solving singular initial value problems in the second-order differential equations. Appl. Math. Comput. 128, 45–57 (2002)
Wazwaz, A.M.: Adomian decomposition method for a reliable treatment of the Emden–Fowler equation. Appl. Math. Comput. 161, 543–560 (2005)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)
Li, S., Liao, S.J.: An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl. Math. Comput. 169, 854–865 (2005)
Abbaoui, K., Cherruault, Y., Seng, V.: Practical formulae for the calculus of multivariable Adomian polynomials. Math. Comput. Model. 22(1), 89–93 (1995)
Sanderson, B.G.: Order and resolution for computation ocean dynamics. J. Phys. Oceanogr. 28, 1271–1286 (1998)
Winther, N.G., Morel, Y.G., Evensen, G.: Efficiency of higher order numerical schemes for momentum advection. J. Mar. Syst. 67, 31–46 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Motsa, S.S., Sibanda, P. Some modifications of the quasilinearization method with higher-order convergence for solving nonlinear BVPs. Numer Algor 63, 399–417 (2013). https://doi.org/10.1007/s11075-012-9629-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-012-9629-z