Spectral Deferred Correction Method For Several Classes Of Neutral Functional Differential Equations

Systems of delay differential equations now occupies a place of importance in all areas of science and particularly in the engineering, control and biological sciences. It is of great significance to study their numerical algorithms. Recently, interest in the numerical analysis of functional-integro-differential equations(FIDEs) and neutral delay differential equations(NDDEs) is rising rapidly. Spectral deferred correction(SDC) method, as a kind of algorithm used to solve the numerical solution of ordinary differential equations originally, has high precision in theory. This paper mainly introduces SDC method to compute the numerical solutions of these two types of delay systems. Then convergence and stability of the algorithms are investigated.In this paper, introduction expounds the application background and significance of delay differential equations. The research status of FIDEs, NDDEs and SDC algorithm are also described. Chapter 2 presents a simple introduction of SDC's principle, the basis of numerical analysis involved and theories that will be quoted for the analysis of numerical stability later. Chapter 3 gives the SDC method for a kind of nonlinear FIDEs. Convergence and numerical stability are studied, and we obtain the global stability criterion that is a little bit weaker than the condition gotten by other scholars for Pouzet-Runge-Kutta method. Because our quadrature method utilized to compute the integral term contained in the equation differs from what predecessors used. And the computation complexity is cut down by changing the explicit Euler method to be available to some stiff problems. Numerical experiments are made to verify the theortical results and the method's effectiveness later. In Chapter 4, combining with a new “variable linear ?-method”, SDC method is applied to a class of NDDEs. We also get a series of criteria about the numerical stability based on GS-, GAS(l)- and weakly GAS(l)-stability. Then the effectiveness of the algorithm is demonstrated by the experimental results, and we find that SDC method has some advantages over Runge-Kutta method in some cases. In the end, summing-up and expectations are presented.