The Numerical Method For Delay Parabolic Partial Differential Equations

Delay is inevitable in nature, in the feld of physical chemistry, engineeringtechnology and ecological system and so on, we often use a diferential equationwith time delay to establish mathematical model. Delay diferential equationwith respect to the diferential equations without delay, it can more accuratelydescribe the trend of the development and the change rule of objective things,because it describes a development system which is not only dependent on thecurrent state, but also relies on the state of the past. In general, only few delaydiferential equations are able to obtain accurate analytical solution, in the prac-tical application, we usually use numerical solution instead of the exact solutionof the problem. The study of the numerical algorithm of delay diferential equa-tions, not only has important theoretical signifcance, but also has very importantapplication value. So the study of numerical methods for delay diferential equa-tions is very necessary, it can make up for the inadequacy of theoretical research.Current research on numerical methods for delay diferential equations is rela-tively few, mainly is the standard fnite diference method, fnite element method,etc.This thesis mainly studied the numerical methods for delay parabolic partialdiferential equations, and carried on theoretical analysis and numerical experi-ments. The thesis included the following several parts:The frst part, briefy introduced the delay partial diferential equations andrelated background knowledge, theory and research signifcance, summarized thedomestic and foreign research status of numerical method. at the same time, themain structure and main work of this article were introduced.The second part researched a compact diference scheme for one-dimensionalnonlinear delay parabolic partial diferential equation with Neumann boundary conditions. Then the discrete energy method was used to prove the optimal ordererror estimate diference format. Finally, some numerical examples demonstratedthe efect of the high accuracy diference scheme.In the third part, studied the compact diference scheme for the neutraldelay parabolic partial diferential equation with initial-boundary value. Firstly,we constructed an unconditionally stable diference scheme with high accuracy.Then the stability of the format was proved. Finally, this method was verifedby a numerical example.The fourth part, it gave a compact diference scheme for the two-dimensionalnonlinear delay parabolic partial diferential equation, and proved the conver-gence of the format using discrete energy method. Finally, we used the alternat-ing direction implicit format to solve it. At last, a numerical example proved theefectiveness of the proposed format.