The Analysis Of Convergence And Stablity Of Two-step Method For Heat Conduction Equation With Delay

In real life, scholars find that many problems in reality not only relate with their current status, but also closely relate with the state of the past. So when we use the mathematical model to describe this kind of problem, we need to add a term which is used to describe the state of the past in the mathematical model. This term is usually referred as the delay term. This thesis mainly deals with two-step methods for three kinds of heat conduction equationHeat conduction equation with delay is an important part of the delay partial differential equations, which can be applied in many fields, for example, electrodynamics, ecology, automatic control systems, nonlinear dynamical systems and financial field. It makes a great contribution to the progress of society. Generally speaking, the theoretical solution of heat conduction equation with delay is difficult to obtain. So we usually use numerical method to solve such equation.There are many numerical methods for solving the heat conduction equation. For example, the boundary element method, the finite difference method, the finite element method. In this thesis, the finite difference method is used to solve the heat conduction equation with delay. The main work can be divided into two parts.In the first part, firstly, we deduce asymptotic stability conditions of the theoretical solution. Under the conditions such that the theoretical solution of heat conduction equation with delay is asymptotically stable, we use two-step method to solve the equation and use the spectral radius to analyze the asymptotic stability of the numerical algorithms. Finally, the numerical examples are given to demonstrate the conclusions.In the second part, under the conditions such that the theoretical solution of heat conduction equations with delay is asymptotically stable, we use leap-frog method to solve the equation and will analyze the convergence of the method. The numerical examples are given to demonstrate the conclusions.