Spectral Method For Solving Two Types Of Delay Differential Equations

In the real world many problems have a lag, therefor delay differential equations widely used in applied science, such as control theory, economics, fluid mechanics, atmospheric science, ecology, etc. In recent decades many numerical methods have been used to solve differential equations, and has obtained great success. For example Runge-Kutta method,0-method, one-leg methods, linear-multi step methods, etc. However, these traditional methods also have many shortcomings, such as low precision, slow convergence, to solve some special problems may have difficult. Here we use spectral method to solve delay differential equations is to achieve the spectral accuracy and exponential convergence, and compared to traditional methods be able to deal with some difficult problem. Can be used as a supplement to traditional methods.The basic idea of spectral methods is to use the global sufficiently smooth trial functions approach the whole true solution of the problem. Therefor as long as the problem sufficiently smooth, spectral method can be used on a few nodes do a very good approximation to the true solution, also to the delay item, delay differential item(including the higher order differential delay item). Therefore, as long as the numerical methods are properly designed, spectral method for solving smooth delay differential equations can be a powerful tool.In this paper we designed the corresponding spectral numerical meth-ods for the two types of delay differential equations model, linear variable coefficient variable delay differential equation model (3.1.1) and linear vari-able coefficient neutral variable delay differential equation model (4.1.1), and gave the convergence analysis and some representative numerical examples. On the other hand, these convergence conclusions and numerical examples also verify that using spectral method to solve the delay differential equations can be achieved spectral accuracy and exponential convergence. It also shows that spectral methods can achieve good results to some difficult problem for traditional methods.