论文信息 - On the Newton iteration in the application of collocation methods to implicit delay equations

On the Newton iteration in the application of collocation methods to implicit delay equations

We consider stiffly accurate collocation methods based on Radau nodes for the integration of initial value problems for implicit delay differential equations of the form My'(t) = f(t, y(t), y(α1(t, y(t))),...,y(αp(t, y(t)))), where M is a constant matrix and αi(t, y(t)) (i=1,....,p) denote the deviating arguments, which are assumed to satisfy the inequalities αi (t, y(t))≤t for all i. In a recent paper [Computing 67 (2001) 1-12] we have described how collocation methods based on Radau nodes can be applied to solve problems of this type.The aim of this paper is that of explaining the difficulties arising when solving the Runge-Kutta equations using stepsizes larger than delays and to design techniques able to efficiently overcome them.

Nicola Guglielmi | N. Guglielmi

[1] Heiko J. Luhmann,et al. Epileptiform activity in a neocortical network: a mathematical model , 2001, Biological Cybernetics.

[2] E. Hairer,et al. Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[3] Christopher T. H. Baker,et al. Issues in the numerical solution of evolutionary delay differential equations , 1995, Adv. Comput. Math..

[4] Alessandra Costanzo,et al. State-of-the-art harmonic-balance simulation of forced nonlinear microwave circuits by the piecewise technique , 1992 .

[5] C. Baker,et al. Differential algebraic equations with after-effect , 2002 .

[6] Wayne H. Enright,et al. A delay differential equation solver based on a continuous Runge–Kutta method with defect control , 1997, Numerical Algorithms.

[7] E. Hairer,et al. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[8] Ernst Hairer,et al. Implementing Radau IIA Methods for Stiff Delay Differential Equations , 2001, Computing.

[9] E. Hairer,et al. Stiff differential equations solved by Radau methods , 1999 .

[10] R. N. Castleton,et al. A First Order Method for Differential Equations of Neutral Type , 1973 .

[11] A. Bellen,et al. Numerical methods for delay differential equations , 2003 .

[12] Richard Bellman,et al. Differential-Difference Equations , 1967 .