Rational chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial-growth special basis functions

AbstractIn the method of matched asymptotic expansions, one is often forced to compute solutions which grow as a polynomial in y as |y| → ∞. Similarly, the integral or repeated integral of a bounded function f(y) is generally unbounded also. The kth integral of a function f(y) solves . We describe a two-part algorithm for solving linear differential equations on y ϵ [−∞, ∞] where u(y) grows as a polynomial as |y| → ∞. First, perform an explicit, analytic transformation to a new unknown v so that v is bounded. Second, expand v as a rational Chebyshev series and apply a pseudospectral or Galerkin discretization. (For our examples, it is convenient to perform a preliminary step of splitting the problem into uncoupled equations for the parts of u which are symmetric and antisymmetric with respect to y = 0, but although this is very helpful when applicable, it is not necessary.) For the integral and interated integrals and for constant coefficient differential equations in general, the Galerkin matrices are banded with very low bandwidth. We derive an improvement on the “last coefficient error estimate” of the author's book which applies to series with a subgeometric rate of convergence, as is normally true of rational Chebyshev expansions