The meaning of equations which describe complex physical phenomena are hard to fathom unless good numerical solutions are available. The (Schroedinger) wave equation for quantum phenomena or the wave equations for geophysical applications are examples. A good numerical algorithm for solving a wave equation has been developed recently in conjunction with Professor I. Koltracht in the Mathematics department at UConn. It is based on the solution of the corresponding Lippman-Schwinger integral equation, and has spectral accuracy properties achieved by expanding the solution into Chebyshev polynomials on a mesh of partitions. Applications to the scattering of an electron from a hydrogen atom, to the collision of two atoms at ultra-cold temperatures, and to a resonance in a Morse potential will be presented.