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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.

MSC:

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
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