Quasilinearization method: Nonperturbative approach to physical problems

Abstract

The general properties of the quasilinearization method (QLM), particularly its fast quadratic convergence, monotonicity, and numerical stability, are analyzed and illustrated on different physical problems. The method approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of linear ones and is not based on the existence of a small parameter. It is shown that QLM gives excellent results when applied to different nonlinear differential equations in physics, such as Blasius, Lane-Emden, and Thomas-Fermi equations, as well as in computation of ground and excited bound-state energies and wave functions in quantum mechanics (where it can be applied by casting the Schrödinger equation in the nonlinear Riccati form) for a variety of potentials most of which are not treatable with the help of perturbation theory. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and the first few iterations already yield extremely precise results. The QLM approximations, unlike the asymptotic series in perturbation theory and 1/N expansions, are not divergent at higher orders. The method sums many orders of perturbation theory as well as of the WKB expansion. It provides final and accurate answers for large and infinite values of the coupling constants and is able to handle even supersingular potentials for which each term of the perturbation series is infinite and the perturbation expansion does not exist.