Hybrid numerical methods for exponential models of growth
Introduction
Classical growth models, such as the logistic, Gompertz and Richards equations, are widely and frequently used to describe various biological and physical processes [1], [2], [3], [4], [5]. The model equations of the fiber amplifiers, including Raman amplifiers and erbium-doped fiber amplifiers (EDFAs) [3], [6], can be described by an exponential model of growth, namely,Because the light amplification or attenuation resembles the exponential pattern with propagation distance, usually, the exponential change of variable is advantageous in amplifier simulations [3]. Although the single-step, multi-step and predictor–corrector methods for the fiber amplifier propagation equations were proposed [3], [6], they are limited into the range where all solutions are positive or negative.
The motivation of the letter is to find a series of novel numerical methods for the ordinary differential equations (ODEs) with the property of growth models. Numerical experiments show that the proposed methods for solving problem (1) obviously increase the computing speed and accuracy in comparison with the classical numerical methods.
Section snippets
Hybrid single-step method
The model equations of problems involving differential equations can always be put into a set of first-order ODEs [7], [8], [9], [10], [11]. The typical form can be described asHere y is a vector including elements y = {y1(t), y2(t), … , yn(t)}T. Our objective is to construct an approximate representation of the function y(t) over some interval of interest t ∈ [t0, tf] that satisfy the initial conditions, given by another vector .
Although some simple
Hybrid predictor–corrector methods
To decrease the local truncation error of the Euler’s method (i.e., the single-step method), the multistep methods were proposed [6]. By using Newton forward and backward interpolating polynomials to approximate the function f(t, y), one can develop a mechanism for obtaining a whole class of implicit and multistep methods. The Adams methods are the most elementary multistep methods [6], [14], [15], since they use the interpolation on the immediate previous approximation. The k-step
Conclusions
In this letter, we have proposed a series of novel numerical methods for ODEs. Based on these methods, hybrid Euler-type method (i.e., (12)) and hybrid predictor–corrector methods (i.e., (20), (21)) are constructed. Numerical experiments show that the hybrid numerical methods cannot only increase the accuracy and decrease the CPU time on the same conditions, but also enlarge the stability region. All ODEs, which the traditional numerical methods can solve, can be solved by the proposed hybrid
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