The Spectral Methods For Several Classes Of Delay Differential Equations

Delay differential equations have been used extensively in chemistry, biology, electri-cal system and machinery control systems modeling. In most cases, it is difficult to get the analytical solution of this kind of the problem directly. This restricts the development of theoretical studies, also cause some inconvenience in dealing with practical problems. In this case, the numerical solution can not only help reveal the theoretical nature, and can be directly applied to practical problems. Therefore, whether from the perspective of theoret-ical research or practical application, the numerical solution of delay differential equations are worth studying. When computing the numerical solution of the delay differential equa-tion, we need to store multiple layers of time. Althougt the high precision algorithms can reduce the storage, it also might lead to increase the computational costs. So, on the premise of no significant increase in the amount of the computation, it is necessary to find the high precision algorithms to solve delay differential equations.This thesis focuses on the construction of the spectral methods for several classes of delay differential equations and the corresponding analysis in theory.In Chapter2, based on the Legendre-Gauss-Radau interpolation and the thought of do-main decomposition, we present a multi-domain spectral collocation method for the neutral differential equation with piecewise delay. Through the convergence analysis, it shows that the precision of the algorithm is high under some proposed conditions. Several numerical examples further illustrate the obtained theoretical results and the computational effective-ness of the method. We also take the proposed algorithm compared with some existing ones in the computational efficiency.In Chapter3, we deals with the numerical approximation of a class of nonlinear de-lay convection-diffusion-reaction equations. In order to derive an efficient numerical scheme to solve the equations, we first convert the original equation into an equivalent re-action-diffusion problem with an exponential transformation. Then, we propose a ful-ly discrete scheme by combining the Crank-Nicolson method and the Legendre spectral Galerkin method. The analytical and numerical stability criteria are obtained in L2-norm. It is proven under the suitable conditions that the method is convergent of second-order in time and of exponential order in space. Finally, several numerical experiments are given to illustrate the computational efficiency and the theoretical results.In Chapter4, we investigate the convergence of the semi-discrete finite element solu-tion for the non-Fickian reaction-diffusion models with a constant delay. With the aid of the elliptic projection with memory, the finite element solution has the convergent order of s, if the exact solution satisfies some proposed conditions. There are also two numerical experi-ments for biological models in dimension one and two. The numerical results are identical to our theoretical analysis.In Chapter5, A Fourier spectral method is applied to the Swift-Hohenberg equation. Due to such problems has the property of energy stability, we construct a semi-implicit and energy stable scheme in time direction, which is second order and suitable for Fourier spectral discretization. After that, the energy stability is proved in both semi-discrete and fully discrete forms. Under some conditions, the convergence analysis is carried out as well. Moreover, two numerical examples show the energy stability and the order of convergence of the numerical method.In the last chapter, we give a briefly conclusion of the main work, and make a prospect of the research for the future.