A nonlinear second derivative method with a variable step-size based on continued fractions for singular initial value problems

We propose a nonlinear one-step second derivative method (NSDM) with a variable step-size implementation based on continued fractions for the numerical solution of singular initial value problems (IVPs). The singular IVPs typically originate from models in Mathematical Biology which are represented by reaction diffusion partial differential equations (PDEs). These PDEs are then discretized in space through finite difference methods to yield a system of ordinary differential equations with singular behavior associated with finite time blow-up. We are motivated by the fact that numerical methods based on polynomial approximation perform poorly in the proximity of a singular point when applied to singular IVPs. Therefore, a continuous method based on a finite continued fraction approximation is derived and used to obtain the NSDM. The NSDM is implemented in a self-starting mode and in a predictor-corrector mode, equipped with a variable step-size implementation. The superiority of the NSDM over standard methods in the literature is established numerically.