One-step Multistage Multiderivative Methods For Linear Functional Differential Equations With Piecewise Continuous Arguments

In this paper, we mainly discuss linear functional differential equations with piecewise continuous arguments. This kind of equations widely exist in all aspects of the real life, it can be used to describe engineering, economy and a lot of problems in biomedical fields.Therefore the study of this kind of equations have important practical significance. The theoretical solutions of delay differential equations are not easy to get, hence the research of the numerical methods are particularly important. In this article, we generalize the one-step multistage multiderivative methods of ordinary differential equations, obtain the extended one-step multistage multiderivative methods. Applied the methods to solving linear functional differential equations with piecewise continuous arguments, and studies the convergence and numerical stability.In the first chapter, we mainly reviews the research background and some results of one-step multistage multiderivative methods for the functional differential equations with piecewise continuous arguments, including the convergence, boundedness and stability etc. In the second chapter, we derive the extended one-step multistage multiderivative methods for solving the linear functional differential equations with piecewise continuous arguments.Then we discuss the convergence of the methods. In the third chapter, we mainly study the boundedness and asymptotic stability of the numerical methods. In the fourth chapter, a few of numerical experiments are carried out to support the theoretical analysis of the research of convergence and stability in the of the second chapter and the third chapter respectively. In the fifth chapter, we briefly summary of this article, points out the limitations and give the future work we can continue to do.