One-point pseudospectral collocation for the one-dimensional Bratu equation. (English) Zbl 1222.65070

The approximate solution of the well-known Bratu problem
\[ u_{xx}+\lambda \exp (u) =0, \quad u(\pm 1)=0, \]
is revisited. As the first main result, it is shown that over the entire lower branch, and most of the upper branch, the solution is well approximated by a parabola, \(u(x)\approx u_0 (1-x^2)\), where \(u_0\) is determined by collocation at a single point \(x=\xi\). Then, the choice of \(\xi\) is discussed. A high order approximation of the solution by a series of Chebyshev polynomials is investigated, as well. It is concluded that the solution is so well approximated by a parabola that it is not a good test for numerical methods. The second main result consists of some new contributions to the theory of the Bratu equation. It is shown that the solution can be written in terms of a single, parameter-free function defined by an initial value problem. For evaluating the analytical solution \(u(x,\lambda)\) throughout the entire parameter space, three overlapping perturbative approximations are derived. Comparisons with other methods are discussed as well.


65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34L30 Nonlinear ordinary differential operators
Full Text: DOI


[1] Aregbesola, Y. A.S., Numerical solution of Bratu problem using the method of weighted residual, Electron. J. South. Afr. Math. Sci., 3, 1-7 (2003)
[2] Barray, D. A.; Parlange, J. Y.; Li, L.; Prommer, H.; Cunningham, C. J.; Stagnitti, E., Analytical approximations for real values of the Lambert W-function, Math. Comput. Simulat., 53, 95-103 (2000)
[3] Boyd, J. P., An analytical and numerical study of the two-dimensional Bratu equation, J. Sci. Comput., 1, 183-206 (1985) · Zbl 0649.65057
[4] Boyd, J. P., Chebyshev and Legendre spectral methods in algebraic manipulation languages, J. Symb. Comput., 16, 377-399 (1993) · Zbl 0793.65084
[5] Boyd, J. P., Global approximations to the principal real-valued branch of the Lambert W-function, Appl. Math. Lett., 11, 27-31 (1998), errata: in Eq. (4), 11/36 should be \(\sqrt{2} 11 / 36\) · Zbl 0940.65018
[6] Boyd, J. P., Chebyshev and Fourier Spectral Methods (2001), Dover: Dover Mineola, New York · Zbl 0987.65122
[7] Boyd, J. P., Rational Chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial-growth special basis functions, Comput. Math. Appl., 41, 1293-1315 (2001) · Zbl 0987.65122
[8] 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., 143, 189-200 (2002) · Zbl 1025.65042
[9] Bratu, G., Sur les équations intégrales non linéaires, Bull. Soc. Math. France, 43, 113-142 (1914) · JFM 45.1306.01
[10] Chang, S.-H., A variational iteration method for solving troeschs problem, J. Comput. Appl. Math., 234, 3043-3047 (2010) · Zbl 1191.65101
[11] Corless, R. M.; Gonnet, G. H.; Hare, D. E.G.; Jeffrey, D. J.; Knuth, D. E., On the Lambert W function, Adv. Comput. Math., 5, 329-359 (1996) · Zbl 0863.65008
[12] E. Deeba, S. Khuri, S. Xie, An algorithm for solving boundary value problems, J. Comput. Phys. 159 (19) 125-138.; E. Deeba, S. Khuri, S. Xie, An algorithm for solving boundary value problems, J. Comput. Phys. 159 (19) 125-138. · Zbl 0959.65091
[13] Finlayson, B. A., The Method of Weighted Residuals and Variational Principles (1973), Academic: Academic New York, p. 412
[14] Finlayson, B. A., Orthogonal collocation in chemical-reaction engineering, Catal. Rev. Sci. Eng., 10, 69-138 (1974)
[15] Gelfand, I. M., Some problems in the theory of quasi-linear equations, Trans. Amer. Math. Soc. Ser., 2, 295-381 (1963) · Zbl 0127.04901
[16] Hassan, I.; Erturk, V., Applying differential transformation method to the one-dimensional planar Bratu problem, Int. J. Contemp. Math. Sci., 2, 1493-1504 (2007) · Zbl 1152.34008
[17] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, 20, 1141-1199 (2006) · Zbl 1102.34039
[18] Keller, H. B., Numerical Methods for Two-Point Boundary-Value Problems (1992), Dover: Dover New York · Zbl 0172.19503
[19] Khuri, S. A., A numerical algorithm for solving the Troesch’s problem, Int. J. Comput. Math., 80, 493-498 (2003) · Zbl 1022.65084
[20] Khuri, S. A., A new approach to Bratu’s problem, Appl. Math. Comput., 1477, 131-136 (2004) · Zbl 1032.65084
[21] Li, S.; Liao, S., An analytic approach to solve multiple solutions of a strongly nonlinear problem, Appl. Math. Comput., 169, 854-865 (2005) · Zbl 1151.35354
[22] Mounim, A.; de Dormale, B., From the fitting techniques to accurate schemes for the Liouville-Bratu-Gelfand problem, Numer. Meth. Part. Diff. Eq., 22, 761-775 (2006) · Zbl 1099.65098
[23] Scott, M. R., On the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms, (Aziz, A. K., Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations (1975), Academic Press: Academic Press New York), 89-146 · Zbl 0335.65032
[24] Scott, M. R.; Watts, H. A., A systematized collection of codes for solving two-point boundary-value problems on the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms, (Lapidus, L.; Schiesser, W. E., Numerical Methods for Differential Systems (1976), Academic Press: Academic Press New York)
[25] Scott, M. R.; Watts, H. A., Computational solution of linear two-point boundary value problems via orthonormalization, SIAM J. Numer. Anal., 14, 40-70 (1977) · Zbl 0357.65058
[26] Syam, M.; Hamdan, A., An efficient method for solving Bratu equations, Appl. Math. Comput., 176, 704-713 (2006) · Zbl 1093.65108
[27] Wazwaz, A. M., Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput., 166, 652-663 (2005) · Zbl 1073.65068
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