Analysis of Exponential Runge–Kutta Methods for Differential Equations with Time Delay

Numerous mathematical models simulating the phenomenon in science and engineering use delay differential equations. In this paper, we focus on the semilinear delay differential equations, which include a wide range of mathematical models with time lags, such as reaction-diffusion equation with delay...

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Bibliographic Details
Published in:Mathematical problems in engineering 2022-10, Vol.2022, p.1-11
Main Author: Zhan, Rui
Format: Article
Language:English
Subjects:
Bacteria
Bioreactors
Convergence
Differential equations
Mathematical analysis
Mathematical models
Methods
Neural networks
Numerical analysis
Ordinary differential equations
Phages
Reaction-diffusion equations
Runge-Kutta method
Stability
Time lag
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Summary:Numerous mathematical models simulating the phenomenon in science and engineering use delay differential equations. In this paper, we focus on the semilinear delay differential equations, which include a wide range of mathematical models with time lags, such as reaction-diffusion equation with delay, model of bacteriophage predation on bacteria in a chemostat, and so on. This paper is concerned with the stability and convergence properties of exponential Runge–Kutta methods for semilinear delay differential equations. GDN-stability and D-convergence of exponential Runge–Kutta methods are investigated. These two concepts are generalizations of the classical AN-stability and B-convergence for ordinary differential equations to delay differential equations. Sufficient conditions for GDN-stability are given by a newly introduced concept of strong exponential algebraic stability. Further, with the aid of diagonal stability, we show that exponential Runge–Kutta methods are D-convergent. The D-convergent orders are also examined. Numerical experiments are presented to illustrate the theoretical results.
ISSN:1024-123X
1563-5147
DOI:10.1155/2022/2693940