Abstract
In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.
Highlights
Traces of ordinary differential equations can be found in various fields of mathematics, natural, or social sciences
The lack of analytical solutions for these complex and nonlinear equations has led to the introduction and development of numerical solutions [16,17,18,19,20,21]
By vanishing of the phase-lag and its derivatives up to five, we have developed a new eight-step singularly P-stable multiderivative method of algebraic order ten
Summary
The most natural mathematical form of many general laws of nature (in physics, chemistry, biology, and astronomy) is found in the language of differential equations. Various engineering fields including analytical mechanics and electrical engineering, geology, physics, chemistry (in the analysis of nuclear chain reactions), biology (in the modeling of infectious diseases and genetic changes), ecology (in population modeling), and economy (in the modeling of dividend and stock price changes) are some of the scientific branches in which the ordinary differential equations play an essential role [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. Due to significant advances in the processing capabilities and speed of the processors and computers in the late 20th century, numerical solutions became more prevalent and continue to
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