Delay Parabolic Differential Equation Of The Stability Of The Numerical Method

This paper mainly studies the stability in the numerical solution and the con-vergence in the analytical solution of the parabolic delay diferential equation withpiecewise continuous arguments and the heat conduction equations with constantdelay.In the first chapter, we give the background and origin of the problem, and pointout the practical significance. We review the application of the delay diferentialequation and previous the stability and the convergence theory development andresearch process in the analytical solution and the numerical solution for the delaydiferential equations. Stability and convergence is one of the important aspectsof the qualitative theory research of the delay diferential equations, and the delaydiferential equation can depict many models of life. Therefore, the research ofstability and convergence for the delay diferential equation is very necessary.The second chapter, according to the parabolic delay diferential equation withpiecewise continuous arguments, we first use the separation variable method to thisequation, which turn into two equations of first order, one is ordinary equation, theother is delay diferential equation with piecewise continuous arguments. Then weuse the Runge-Kutta method discussion the conditions of the stability in the numeri-cal solution for the delay diferential equation after the separation. In this condition,we explain the necessary condition of the convergence in the analytical solution ofthe parabolic delay diferential equation with piecewise continuous arguments byusing the related properties of Fourier series.In the third chapter, we also apply the method of separation of variable to theheat conduction equations with constant delay, and discussion the stability of thedelay diferential equation with constant delay of first order by using the Runge-Kutta method. We discrete the heat conduction equations with constant delay,which translate into the parabolic delay diferential equation with piecewise contin-uous arguments. On this basis, we obtain the condition of the convergence in theanalytical solution of this equation by using the Fourier series about the nature.