A Numerical Method For A Class Of Nonlinear Delay Differential Algebraic Systems

Delay differential-algebraic equations(DDAEs),which has both delays and algebraic constraints.Because of of the complexity of DDAEs,it becomes quite difficult to obtain the numerical methods for DDAEs and they have become one of the most chief and principal methods to solve DDAEs.Moreover,effective and reliable computational methods and the stability of numerical methods are the key problems we have to face in the research of the numerical solution.In this paper,we discuss two kinds of nonlinear differential algebraic systems with time delay: single time delay,double delay differential algebra system.considering the stability and asymptotic stability for those two class of nonlinear DDAEs by means of linearization process in the neighborhood of stability solutions,using implicit function theory,vector inner product,the consistent initial vectors etc to get sufficient main results.Sufficient conditions for the stability and asymptotic stability of the analytic solutions of these two kinds of delay differential algebraic systems in the local neighborhood of steady-state solution are verified.Numerical methods to verify sufficient conditions of stability and asymptotic stability in the neighborhood of stability solutions for nonlinear DDAEs are implicit Euler methods and 2-step BDF methods.The key and difficult points of this article are the linearization process of nonlinear DDAEs,To discuss the stability and asymptotic stability of the original system by linearization in the local neighborhood stability solutions.In the numerical part,comparing the linearization methods of this paper with the similar problems in the current literature for the same system.It can be found that in the local neighborhood of stability solutions,linearization method can be used to obtain the same characteristics as the original system,and the calculation quantity is reduced,and the discrimination process is relatively simple.