Abstract
In all numerical methods, it is necessary to ascertain the validity of any particular scheme. And this is possible to determine, by verifying the nature of the stability of that scheme. So the general stability function definition is given, from where an investigation is carried out on a class of rational integrator of order 15, to establish the region of absolute stability of the scheme, by constructing the Jordan curve. In the process of expanding the rational function, binomial theorem as well as the idea of combination process were introduced to ease the computation by using Maple-18 package. The simplification of the general rational integrator formula, is constructed from two processes namely through complex function, and then through polar analysis, The Jordan curve is constructed with the help of MATLAB package. Furthermore, it was discovered that the region of instability is on the positive side of the complex plane, while the region of absolute stability is outside the Jordan curve. Finally, it is further established that the encroachment point, τ, lie within the interval ± 140.6. And the encroachment point is visible from the corresponding values of ø and R at the extremes. The stability curve revealed that the integrator is not only A-stable, but also L- stable.
Highlights
Numerical methods for evaluating systems of Ordinary Differential Equations (ODEs) have been attracting much attention because they proffer the solutions of problems arising from the mathematical formulation of physical situations such as those in chemical kinetics, population, economic, political and social models
The rational interpolating method is used to find the numerical establishment of the stability function of a general rational integrator, reported in Aashikpelokhai [1], with particular attention to the case k=8 and show the region of Absolute stability of the integrator
1) It shows the region of absolute stability 2) The exterior of the curve represent the region of absolute stability (RAS) of the integrator
Summary
Numerical methods for evaluating systems of Ordinary Differential Equations (ODEs) have been attracting much attention because they proffer the solutions of problems arising from the mathematical formulation of physical situations such as those in chemical kinetics, population, economic, political and social models. American Journal of Mathematical and Computer Modelling 2020; 5(4): 102-108 favorably well with other existing methods without really establishing the stability region of the method All of these works were basically for the improvement of numerical solutions to initial value problems, which sets the foundation for the study. The rational interpolating method is used to find the numerical establishment of the stability function of a general rational integrator, reported in Aashikpelokhai [1], with particular attention to the case k=8 and show the region of Absolute stability of the integrator. As Aashikpelokhai [1] puts it, the smallness of the individual error is called accuracy but the ability to keep the effect of this error under control is called stability He further opined that the region of absolute stability (RAS) of rational interpolation methods always lie entirely on the left-half of the complex plane. The roots of the complex polynomial are determined to enable the plotting of the Jordan curve, which will show both the Region of Absolute stability (RAS) and the Region of instability (RIS)
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