Research On Compact Difference Scheme For Solving A Class Of Delay Parabolic Equation

Time-delay is common and inevitable in nature,which is one of the main factors that affect the system performance and stability.The delay differential equation has been widely used in many fields,such as science,engineering and so on.In the past,people always assume that the considered system obeys a law that the future state of system is determined only by the current state and described through the correspond-ing model in the research the celestial mechanics,physics,dynamical systems,etc.However,with the more in-depth analysis of many natural phenomena,people find that the state of system not only depends on the current development state,but also relies on the past development in the real world.In most cases,the system will have a relatively large negative impact if ignore time-delay term to reduce the difficulty of the problem.Because of time-delay term,it is difficult to obtain the theoretical anal-ysis and the analytic expression of the exact solution.So,the exact solution of delay differential equation is usually replaced by the numerical solution when we solve prac-tical problems.This study makes up the shortcoming of the theory,and also has the important practical significance.This paper expounds how to construct the compact difference scheme for delay parabolic equations,and also introduces the corresponding theory analysis of the numerical schemes.In the first chapter,the research background and research significance of delay differential equations are introduced.Then we sketch the research progress of the domestic and foreign experts and scholars on the numerical method related to the delay differential equations.In addition,we also show the main research content and significance in this paper.The second chapter mainly discusses the initial-boundary value problems for the nonlinear one-dimensional delay parabolic equations,constructs a compact difference scheme by using the difference discrete method.We prove the existence and uniqueness of solutions under the difference scheme,the unconditional stability and the conver-gence order is O(?2+h4)in L? norm by the energy analysis method.Finally,we show that the scheme is by an example.The third chapter mainly constructs the compact difference scheme for the two-dimensional initial-boundary value of delay parabolic equations,and we use alternating direction techniques in order to improve the computational efficiency.We solve the equation under the compact difference scheme,and then analyze the priori estimates of the solution and the stability.Finally,we illustrate that the scheme is feasible through a numerical example.