Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain

Abstract

We describe a four-step algorithm for solving ordinary differential equation nonlinear boundary-value problems on infinite or semi-infinite intervals. The first step is to compute high-order Taylor series expansions using an algebraic manipulation language such as Maple or Mathematica. These expansions will contain one or more unknown parameters z which will be determined by the boundary condition at infinity. The second step is to convert the Taylor expansions into diagonal Padé approximants. The boundary condition that u(x) decays to zero at infinity becomes the condition that the coefficient of the highest power of x in the numerator polynomial must be zero. The third step is to solve this equation for the free parameter z. The final step is to evaluate each of the multiple solutions of this equation for physical plausibility and convergence (as N increases). This algorithm can be implemented in as few as seven lines of Maple (sample program provided!). We illustrate the method with three examples: the Flierl–Petviashvili vortex of geophysical fluid dynamics, the quartic oscillator of quantum mechanics, and the Blasius function for the boundary layer above a semi-infinite plate in fluid mechanics. Methods for nonlinear problems are almost always iterative and need a first guess to initialize the iteration. The Padé algorithm is unusual in that it is a direct method that requires no a priori information about the solution. © 1997 American Institute of Physics.