Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation

摘要

We introduce two methods for simultaneously approximating both branches of a two-branched function using Chebyshev polynomials. Both schemes remove the pernicious, convergence-wrecking effects of the square root singularity at the limit point where the two branches meet. The “Chebyshev–Shafer” method gives the approximants as the solution to a quadratic equation; the “mapping” algorithm makes a quadratic change of variable. For both, the only input information is the set of values of f(x) at a set of discrete points. There is little to choose between the two schemes in accuracy, but the single expansion/mapping method is more flexible in that it can accommodate different ranges on the two branches. The eigenrelation for the one-dimensional Bratu equation is an interesting example because the upper branch is also singular at another point besides the limit point; this, too, can be removed by subtracting the asymptotic solution, which is the Lambert W-function, from the upper branch only. When the domain of the variable is infinite, the quadratic change of variable can still be applied by substituting rational Chebyshev functions, which are basis functions for an unbounded interval, for Chebyshev polynomials. We illustrate this by approximating the real-valued root of the Brill quintic, u5−u−λ, λ∈[−∞,∞], which was first solved by Hermite using elliptic modular functions more than a century ago.