Pseudospectral Methods For Delay Differential Equations

Delay differential dynamical systems have appeared in areas of applications as diverseas neural networks, optics, biology, and automatic control theory, etc. The feature of delaydifferential systems is that the derivative of the current state depends on not only the currentstate but also the past state. In the recent 20 years, significant progress on basic theory ofsolution, stability theory, periodic solution theory and bifurcation theory of delay differentialsystems has been achieved.In this paper, we propose two long-time convergent numerical integration processesfor delay differential equations. We first construct an integration processes based on modi-fied Laguerre-Radau polynomials and establish its global convergence in certain weightedSobolev space. Then we propose another integration process based on the modifiedLaguerre-Radau functions and prove its global spectral accuracy in the space L2(0,∞). Wealso developed a technique for refinement of modified Laguerre-Radau interpolations. Nu-merical results demonstrate the spectral accuracy of the proposed method and coincide wellwith analysis.