Exponential fitting Runge-Kutta methods for the delayed recruitment/renewal equation

The so-called delayed recruitment/renewal equation provides the mathematical model in a diverse spectrum of practical applications and may become singularly perturbed when the time-lag is large relative to the reciprocal of the decay rate. In order to accurately capture its solution features numerically, we design a family of exponential fitting Runge-Kutta methods of collocation type to obtain the numerical approximation. The exponential fitting approximations are proved to have higher order of uniform accuracy. We demonstrate the efficiency of this family of exponential fitting Runge-Kutta methods for the delayed recruitment/renewal equation via application to some important problems.

[1]  David R. Appleton,et al.  Modelling Biological Populations in Space and Time , 1993 .

[2]  Gabil M. Amiraliyev,et al.  Uniform numerical method for singularly perturbed delay differential equations , 2007, Comput. Math. Appl..

[3]  H. De Meyer,et al.  Exponentially fitted Runge-Kutta methods , 2000 .

[4]  Vinod Kumar,et al.  An ϵ-uniform hybrid scheme for singularly perturbed delay differential equations , 2011, Appl. Math. Comput..

[5]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[6]  M. Mackey,et al.  Mixed Feedback: A Paradigm for Regular and Irregular Oscillations , 1987 .

[7]  Asymptotic expansion for the solution of singularly perturbed delay differential equations , 2003 .

[8]  Brian J. McCartin.  Exponential fitting of the delayed recruitment/renewal equation , 2001 .

[9]  R. D. Driver,et al.  Ordinary and Delay Differential Equations , 1977 .

[10]  Shui-Nee Chow,et al.  Singularly Perturbed Delay-Differential Equations , 1983 .

[11]  G Kember,et al.  A mathematical analysis of the Grodins model of respiratory control. , 1993, IMA journal of mathematics applied in medicine and biology.

[12]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[13]  M. Mackey,et al.  Dynamical Diseases , 1987, Annals of the New York Academy of Sciences.

[14]  JohnM . Miller,et al.  Robust Computational Techniques for Boundary Layers , 2000 .

[15]  Hongjiong Tian.  The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag , 2002 .

[16]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.