Adams Method For Solving Stiff Delay Differential Equations

The delay integro-differential equations have been widely applied to the physics, biology, ecology, control science and other scientific domains. It is usually difficult to obtain the theory solution of these kinds of equations, so studying the numerical methods for such equations becomes extremely significant. In order to solve these equations, scholars have designed a lot of numerical methods, such as Runge-Kutta method, linear multi-step methods, Rosonbrock method and so on.The Adams method is a special form of the linear multi-step methods. Because the Adams method has good stability and is convenient for changing the stepsize and the order,it shows many advantages when used to solve stiff differential equations. However, we have to meet a serious problem that is how to choose the stepsize and the order. It is quite important to choose the suitable stepsize and order which will take direct influence to the precision and efficiency of the Adams method. Therefore, how to choose the length of stride and order becomes the focus of many researches.Firstly, we have studied how to use the extended Adams method to solve a sort of the Volterra discrete-distributed type delay-integro-differential equations. Adopting the matching order quadrature formula to disperse the distribution delay function, we have expanded the ordinary implicit Adams method. We use the Newton iterative method to execute the computation here.Secondly, we have constructed the variable step and variable order Adams method to solve stiff delay differential equations. We use the Nordsieck method that can easily change the stepsize and the order to implement the transform strategy. Then we introduce two small parameters to alter the Adams method. Taking advantages of the parameters, we can control the stability and the convergence of the method of the stiff equations.