Abstract
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 meth...
Highlights
We consider the initial value problems (IVPs) y = f (x, y), y(a) = y0, y, f ∈ Rm, x ∈ [a, b], a, b ∈ R. (1)Several physical processes in science and engineering can be modeled in the form (1) which could be non-stiff, stiff or singular in nature
We have proposed an accurate nonlinear one-step second derivative method with a variable step-size implementation based on continued fractions for the numerical solution of singular initial value problems
The singular initial value problems typically originate from models in Mathematical Biology which are represented by reaction diffusion partial differential equations discretized in space through finite difference or finite element methods to yield a system of ordinary differential equations with singular behavior associated with finite time blow-up
Summary
In order to determine the coefficients (b0, b1, b2, b3) in (5), we demand that the following conditions must be satisfied up(xn) = yn, u p(xn) = fn, u p (xn)(xn) = gn, u p (xn) = zn, and lead to a system of four equations and four unknowns, which is solved to obtain the coefficients (b0, b1, b2, b3) These coefficients are known and substituted into in (5) to yield the continuous form of the predictor up(x) which is evaluated at xn+1 to yield ynp+1 =.
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