Multistep Multiderivative Methods For Linear Functional Differential Equations With Piecewise Continuous Arguments

Compared with ordinary differential equations, delay differential equations can depict a variety of complicated phenomena in the real life much better. So it is widely used in many fields, such as ecology, economics, management science, chemistry and medicine.As a special kind of delay differential equations, the linear functional differential equations with piecewise continuous arguments are often applied in the fields of control, biomedicine,physics, economics and others. Hence, the researches on this kind of equations attract many researchers' interest. In fact, it is very difficult to obtain their theoretical solutions. As a result, the studies on the numerical solution for such equation are quickly developed. In addition, convergence, stability, boundedness, periodicity, oscillation and dissipativity of many properties also have been studied. But no scholar has tried to use multistep multiderivative methods to solve such equations.In this paper, the multistep multiderivative methods for solving ordinary differential equations are extended to solve the linear functional differential equations with piecewise continuous arguments. The generalized multistep multiderivative methods are obtained,and we have proved variety of the relevant properties. This paper has five chapters. In the first chapter, we review the current status of related research briefly for the equations and the methods. Finally, we explain the main research work of this paper. The second chapter gives the model problems which are considered in this paper, then we apply the extended multistep multiderivative methods to this problem and compute the corresponding format. Then we discuss the convergence of this scheme. In the third chapter, we further discuss the stability and boundedness of the methods. In the fourth chapter, some numerical experiments are provided to demonstrate our theoretical results, including convergence,asymptotic stability and boundedness. In the fifth chapter, on the basis of this paper, the summaries and prospects are made. Then we point out the work we may do further.