Research On Alternating Direction Implicit Scheme Of Partial Differential Equation With Time Delay

Delay is an inevitable phenomenon in our daily life, in the field of Natural eco-logical environment, Dynamical systems, Economics and so on, we often use a differ-ential equation with time delay to establish mathematical model. Comparing with the non-delay differential equation, it can more accurately describe the trend of the development and the change rule of objective things. Delay differential equation is applied in a wide range of fields, but in general, only few delay differential equations are able to obtain accurate analytical solution. In the practical application, we usually use the numerical solution instead of the exact solution. So the study of the numeri-cal methods for the delay differential equation has very important application value. Current research on numerical methods for the delay differential equation is relatively few, mainly is the standard finite difference method, the finite element method.The alternating direction method is introduced in this thesis belongs to the stan-dard finite difference method. Through the study of a class of delay partial differential equations, we have carried out a comprehensive theoretical study on this method. And the corresponding numerical examples are used to illustrate the practical application value.In the first chapter, the preface, introduces the background knowledge of the delay partial differential equations, the basic knowledge of the difference method, and the research status on numerical methods by domestic and foreign scholars, as well as a brief description of the structure of this paper and the main work.In the second chapter, an alternating direction implicit scheme is established for the initial boundary value problem of a two-dimensional delay parabolic partial differential equation, and solve the scheme. And then analyze the priori estimates of the solution and the stability. Finally, a numerical example is used to verify the accuracy of the proposed scheme.In the third chapter, an alternating direction implicit scheme is established for the initial boundary value problem of a three-dimensional delay partial differential equation with constant coefficients, and solve the scheme. The Fourier stability anal- ysis method is then used to analyze the stability of the proposed scheme. Finally, a numerical example is used to verify the accuracy of the proposed scheme.