One-point pseudospectral collocation for the one-dimensional Bratu equation John P. Boyd Department of Atmospheric, Oceanic and Space Science and Laboratory for Scientific Computation, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109, United States article info Keywords: Chebyshev polynomial Pseudospectral Collocation Orthogonal collocation Bratu equation abstract The one-dimensional planar Bratu problem is u xx + k exp(u) = 0 subject to u(±1) = 0. Because there is an analytical solution, this problem has been widely used to test numerical and perturbative schemes. We show that over the entire lower branch, and most of the upper branch, the solution is well approximated by a parabola, u(x)  u 0 (1  x 2 ) where u 0 is determined by collocation at a single point x = n. The collocation equation can be solved explicitly in terms of the Lambert W-function as u(0) W(k(1  n 2 )/2)/(1  n 2 ) where both real-valued branches of the W-function yield good approximations to the two branches of the Bratu function. We carefully ana- lyze the consequences of the choice of n. We also analyze the rate of convergence of a series of even Chebyshev polynomials which extends the one-point approximation to arbitrary accuracy. The Bratu function is so smooth that it is actually poor for compar- ing methods because even a bad, inefficient algorithm is successful. It is, however, a solution so smooth that a numerical scheme (the collocation or pseudospectral method) yields an explicit, analytical approximation. We also fill some gaps in theory of the Bratu equation. We prove that the general solution can be written in terms of a single, parameter-free b(x) without knowledge of the explicit solution. The analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly. We give three overlapping perturbative approximations to the eigen- relation, allowing the analytical solution to be easily evaluated throughout the entire parameter space. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction The one-dimensional Bratu problem has a long history. Bratu’s own article appeared in 1914 [9]; generalizations are sometimes called the ‘‘Liouville–Gelfand’’ or ‘‘Liouville–Gelfand–Bratu’’ problem in honor of Gelfand [15] and the nineteenth century work of the great French mathematician Liouville. In recent years, it has been a popular testbed for numerical and perturbation methods [1,17,16,27,21,26,20,12,22,12]. In this note, we apply very low order spectral methods. Our purpose is to illustrate three themes. First, Chebyshev collocation is not only a numerical method; sometimes it can be applied to gen- erate accurate, explicit, analytic approximations [4]. Second, the Bratu function is so well approximated by a parabola that it is a poor test for numerical methods because any scheme is accurate for such a smooth function. Third, we fill in some the- oretical gaps that still exist for this heavily-studied problem. 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.12.029 E-mail addresses: jpboyd@engin.umich.edu, jpboyd@umich.edu URLs: http://www.engin.umich.edu:/~jpboyd/ Applied Mathematics and Computation 217 (2011) 5553–5565 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc