Introduction

It is well-known that nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) for boundary-value problems are much more difficult to solve than linear ODEs and PDEs, especially by means of analytic methods. Traditionally, perturbation (Van del Pol, 1926; Von Dyke, 1975; Nayfeh, 2000) and asymptotic techniques are widely applied to obtain analytic approximations of nonlinear problems in science, finance and engineering. Unfortunately, perturbation and asymptotic techniques are too strongly dependent upon small/large physical parameters in general, and thus are often valid only for weakly nonlinear problems. For example, the asymptotic/perturbation approximations of the optimal exercise boundary of American put option are valid only for a couple of days or weeks prior to expiry, as shown in Fig. 1.1. Another famous example is the viscous flow past a sphere in fluid mechanics: the perturbation formulas of the drag coefficient are valid only for rather small Reynolds number Re ≪ 1. Thus, it is necessary to develop some analytic approximation methods, which are independent of any small/large physical parameters at all and besides valid for strongly nonlinear problems.