Numerical solution of problems in unbounded regions: Coordinate transforms

We investigate the utility of mappings to solve numerically problems in infinite regions. it is demonstrated by six examples that mappings are very useful if the solution being sought behaves in a simple way at infinity; otherwise, they are not particularly helpful. Solutions that vanish rapidly or approach a constant at infinity are readily treated by mapping, but solutions that oscillate out to infinity are not so amenable to these techniques. The examples investigated in detail include a one-dimensional diffusion equation, the anharmonic oscillator eigenvalue problem, the Orr~Sommerfeld eigenvalue problem for the Blasius boundary layer flow, the Falkner-Skan equation, the one-dimensional wave equation, and Burgers' equation. For these examples, it is found that an algebraic mapping of the infinite region into a finite one is best.