The Methods Using Cubic Spline For Two Classes Of Parabolic Equations With Time Delay

This paper deals with numerical methods and theoretical analysis for Delay Parabolic Equations. Delay Parabolic Equation is one of delay differential equations (DDEs). DDEs are widely used in Population dynamics,Infectious Diseases and En-vironmental Engineering. In many cases, it is difficult for Delay Parabolic Equations to give an accurate expression to an analytical solution. So In dealing with practical issues ,we often use numerical methods for Delay Parabolic Equations. With the growing emphasis on delay,Researching on the Delay Parabolic Equations has im-portant practical significance.There are many ways dealing with Delay Parabolic Equations, such as Finite Difference Methods, Finite Element Methods, etc.In this paper, according to Finite Difference Methods and the good approaching nature of the use of cubic spline inter-polation function, obtained the cubic spline interpolation method of Delay Parabolic Equations.In chapter one, it introduces the academic background, theory and practical sig-nificance about the Neutral Parabolic Equations and Delay Differential Equations, and research methods and results of numerical solution for the equations and describe the basics knowledge needed . it introduces the main content and structure of this paper at the end of the chapter.The second chapter studies the spline function method of the initial boundary value problem of a class of neutral parabolic. First, we discrete time derivative with difference methods, and discrete space derivative with Interpolation methods, then we have an implicit format. Next we prove the stability of the format. Finally, a numeri-cal simulation proves the validity of the method.Chapter III gives a class of Delay parabolic boundary value problems with cubic spline precise integration method. First of all discreteing space derivative with Inter-polation methods, thus get a delay ordinary differential equations. Next, using the precise integration method, we obtain the iterative formula and prove the stability of the formula. Finally, we give a numerical example.