Numerical Methods For Two Classes Of Delay Differential Equations

In order to describe the complicated real world truly, delay differential equations came into being in the middle of the last century. As is well to known, it is very difficult to obtain the analytical solutions of delay differential equations in practical, even the simplest linear delay differential equations. Afterwards, numerical simulation breaks a feasible way for people. In 1930, Bergman proposed the concept and expression of the reproducing kernel firstly. Up to now, the theory of reproducing kernel has been developed very well, and highlights the great superiority in solving some boundary value problems for its good properties. In particular, the research of applying the reproducing kernel theory to study the fractional order differential equations is just started, so it has a very broad prospect.In the second part of this paper, the reproducing kernel Hilbert space method is presented for solving a class of liner impulsive delay differential equations. In this model, the “impulse” and “delay” exist simultaneously as two important factors, and the properties of solution become very complicated and get out of control, so it become more difficult to solve this kind of model. Therefore, this paper presents a new reproducing kernel method to solve this kind of liner impulsive delay differential equations. We define new inner products and norms, and obtain new reproducing kernel expressions. The new method avoids the complicated process of the discretization of the definition interval. We only need to select the appropriate reproducing kernel space to absorb the initial value condition, and then solve the equation piece by piece. The operation is simple and saves the calculation time. Finally, the numerical examples show the correctness and feasibility of the method.In recent years,the research of fractional order theory has received the widespread attention. Especially, the serious defect that the local property of the integer order derivative has a historical dependence on the system function is avoided perfectly as its global correlation. So we can get more consistent results with the actual ones using a small amount of parameters. In the third part of this paper, we solve a class of fractional order delay differential equations using the reproducing kernel Hilbert space method and cubic B-spline method respectively. At present, the theories of reproducing kernel and cublic B-spline are seldom used in the study of fractional order. Through the solving of this class of fractional order delay differential equations in this paper, we can see that the programs of the two methods are simpler, and the accuracy of the error obtained is higher.