Multistep Multiderivative Methods For Linear Neutral Equations

Delay differential equations can describe the objective world more accurately than ordinary differential equations, and it’s widely applied in many fields, such as ecology,economics, management science, chemistry, medicine and so on, and its importance is obvious. The linear neutral equations often applied in electrical engineering are one of important part of delay differential equations.Actually,we know it’s difficult to gain theoretical solutions of these equations,so it’s necessary to simulate and emulate theoretical solutions by numerical methods.The methods of multistep multiderivative which are often used to solve ordinary differential equations are used to solve linear neutral equations in this paper,and a series of research are as follows.The first chapter presents the brief summary of basic multistep multiderivative methods, the research background about delay differential equations, the research summary of numerical methods of delay differential equations and the main work of this paper. In chapter two, the problem in being discussed model and the extended multistep multiderivative methods are put forward, then the basic multistep multiderivative methods to solve the ordinary differential equations are briefly retrospected. In chapter three, the convergence character of multistep multiderivative methods for linear neutral equations is discussed. We gain a sufficient condition for the convergence of numerical solution of the equation. To fully verify the result,the collation maps between theoretical solutions and numerical solutions,table about convergence order and error maps taking different steps are given by some numerical experiments. In the fourth chapter, asymptotic stability and boundedness of numerical solution of multistep multiderivative methods for linear neutral equations are discussed, and some numerical examples are given to prove the correctness of the conclusion of asymptotic stability.