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

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.