Abstract
In this paper, a new algorithm for the numerical solution of the Bratu-type and the Lane–Emden problems with boundary conditions is presented. The proposed algorithm is based on the variational iteration method (VIM), where all the boundary conditions are used before designing the recursive scheme for the approximate solutions of considered boundary value problems. Unlike the VIM, the proposed algorithm avoids solving a sequence of nonlinear algebraic or (transcendental) equations for the undetermined coefficients. Convergence and error analysis of the proposed method is also given. Illustrative examples, of two different models, are examined to demonstrate the accuracy, applicability, and generality of the proposed method.
Similar content being viewed by others
References
H. Caglar, N. Caglar, M. Özer, A. Valarıstos, A.N. Anagnostopoulos, B-spline method for solving Bratu’s problem. Int. J. Comput. Math. 87(8), 1885–1891 (2010)
A. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166(3), 652–663 (2005)
J. Rashidinia, K. Maleknejad, N. Taheri, Sinc-Galerkin method for numerical solution of the Bratu’s problems. Numer. Algorithms 62(1), 1–11 (2013)
S. Khuri, A new approach to Bratu’s problem. Appl. Math. Comput. 147(1), 131–136 (2004)
M.I. Syam, A. Hamdan, An efficient method for solving Bratu equations. Appl. Math. Comput. 176(2), 704–713 (2006)
A.-M. Wazwaz, A reliable study for extensions of the Bratu problem with boundary conditions. Math. Methods Appl. Sci. 35(7), 845–856 (2012)
R. Buckmire, Investigations of nonstandard. Mickens-type, finite-difference schemes for singular boundary value problems in cylindrical or spherical coordinates. Numer. Methods Partial Differ. Equ. 19(3), 380–398 (2003)
C. Hsiao, Haar wavelet approach to linear stiff systems. Math. Comput. Simul. 64(5), 561–567 (2004)
J. Jacobsen, K. Schmitt, The Liouville–Bratu–Gelfand problem for radial operators. J. Differ. Equ. 184(1), 283–298 (2002)
A.S. Mounim, B. De Dormale, From the fitting techniques to accurate schemes for the Liouville–Bratu–Gelfand problem. Numer. Methods Partial Differ. Equ. 22(4), 761–775 (2006)
A. Wazwaz, A new method for solving singular initial value problems in the second-order ordinary differential equations. Appl. Math. Comput. 128(1), 45–57 (2002)
A. Wazwaz, R. Rach, Comparison of the Adomian decomposition method and the variational iteration method for solving the Lane–Emden equations of the first and second kinds. Kybernetes 40(9/10), 1305–1318 (2011)
R. Rach, J.-S. Duan, A. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the adomian decomposition method. J. Math. Chem. 52(1), 255–267 (2014)
H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, vol. 971 (Courier Corporation, Mineola, 1962)
S. Chandrasekhar, An Introduction to the Study of Stellar Structure, vol. 2 (Courier Corporation, Mineola, 1957)
O. Richardson, The emission of electricity from hot bodies. Arch. Radiol. Electrother. 26(10), 325–326 (1922)
R. Singh, J. Kumar, An efficient numerical technique for the solution of nonlinear singular boundary value problems. Comput. Phys. Commun. 185(4), 1282–1289 (2014)
A. Wazwaz, The variational iteration method for solving the volterra integro-differential forms of the Lane–Emden equations of the first and the second kind. J. Math. Chem. 52(2), 613–626 (2014)
P.M. Lima, L. Morgado, Numerical modeling of oxygen diffusion in cells with Michaelis-Menten uptake kinetics. J. Math. Chem. 48(1), 145–158 (2010)
H. Caglar, N. Caglar, M. Özer, A. Valaristos, A.N. Miliou, A.N. Anagnostopoulos, Dynamics of the solution of Bratu’s equation. Nonlinear Anal. Theory Methods Appl. 71(12), e672–e678 (2009)
R. Buckmire, Application of a mickens finite-difference scheme to the cylindrical Bratu–Gelfand problem. Numer. Methods Partial Differ. Equ. 20(3), 327–337 (2004)
S. Abbasbandy, M. Hashemi, C.-S. Liu, The ie-group shooting method for solving the Bratu equation. Commun. Nonlinear Sci. Numer. Simul. 16(11), 4238–4249 (2011)
R. Jalilian, Non-polynomial spline method for solving Bratu’s problem. Comput. Phys. Commun. 181(11), 1868–1872 (2010)
S. Venkatesh, S. Ayyaswamy, S. Raja Balachandar, The Legendre wavelet method for solving initial value problems of Bratu-type. Comput. Math. Appl. 63(8), 1287–1295 (2012)
B. Batiha, Numerical solution of Bratu-type equations by the variational iteration method. Hacet. J. Math. Stat 39(1), 23–29 (2010)
S. Li, S.-J. Liao, An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl. Math. Comput. 169(2), 854–865 (2005)
J.-H. He, Variational iteration method-a kind of non-linear analytical technique: some examples. Int. J. Non-linear Mech. 34(4), 699–708 (1999)
M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics. Var. Method Mech. Solids 33(5), 156–162 (1978)
J.-H. He, Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20(10), 1141–1199 (2006)
J. He, Variational approach to the Bratu’s problem, in Journal of Physics: Conference Series, vol 96 (IOP Publishing, 2008), p. 012087
L. Elsgolts, Differential equations and the calculus of variations, mir, moscow, translated from the Russian by g (1977)
Y. Aksoy, M. Pakdemirli, New perturbation-iteration solutions for Bratu-type equations. Comput. Math. Appl. 59(8), 2802–2808 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Das, N., Singh, R., Wazwaz, AM. et al. An algorithm based on the variational iteration technique for the Bratu-type and the Lane–Emden problems. J Math Chem 54, 527–551 (2016). https://doi.org/10.1007/s10910-015-0575-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-015-0575-6