Numerical Analysis Of Semi-discretization And Full Discretization For Linear Delay Partial Differential Equations

Delay partial differential equations are widely used in real life, but it is very difficult to obtain the exact solution of the equations. Therefore, it is very necessary to study the numerical solution of delay partial differential equations. In this thesis, we study the properties of the numerical solutions of three kinds of linear delay partial differential equations. These three kinds of equations are parabolic delay differential equation, hyperbolic delay differential equation and a class of integro-differential equation with delay.The content of this thesis includes three main parts, and the structure of this thesis is as follows.The first part of this thesis is to study parabolic delay differential equation. Firstly,we give a general semi-discretization method for the parabolic delay differential equation by using the linear multi-step method. And we obtain a necessary and sufficient condition for the semi-discretization method is of order p. By using this condition, we obtain that the order of semi-discretization derived from central difference scheme is two, and the order of semi-discretization method obtained by five point formula is four. Secondly, a sufficient condition for the semi-discretization method to be asymptotically stable is established by using the Fourier method. The semi-discretization methods derived from central difference scheme and five point formula are asymptotically stable. Finally, we give a general full discretization method of the parabolic delay differential equation by using the linear multi-step method. Fourier method is used to obtain the sufficient condition for the asymptotic stability of full discretization method. And sufficient conditions are given on the asymptotic stability of Euler method and Crank-Nicolson method.The second part of this thesis is to study hyperbolic delay differential equation. Firstly, we give a general semi-discretization method for the hyperbolic delay differential equation by using the linear multi-step method. And we obtain a necessary and sufficient condition for the semi-discretization method is of order p. By using this condition, we obtain that the order of semi-discretization derived from forward difference scheme is one, and the order of semi-discretization method obtained by central difference scheme is two. Secondly, we obtain a sufficient condition for the asymptotic stability of the semidiscretization method by using the Fourier method. A sufficient condition for the asymptotic stability of the semi-discretization methods derived from forward difference scheme is obtained, and we obtain that the semi-discretization methods derived from forward difference scheme is unstable. Finally, we give a general full discretization method of the hyperbolic delay differential equation by using the linear multi-step method. Fourier method is used to obtain the sufficient condition for the asymptotic stability of full discretization method. A sufficient condition is given on the asymptotic stability of Euler method. And the Crank-Nicolson method is unstable.The third part of this thesis is to study a class of integro-differential equation with delay. Firstly, we give a semi-discretization method for this equation by using the linear multi-step method. And we obtain a necessary and sufficient condition for the semidiscretization method is of order p. By using this condition, we obtain that the order of semi-discretization derived from central difference is two, and the order of semidiscretization method obtained by five point formula is four. Secondly, we obtain a sufficient condition for the asymptotic stability of the semi-discretization method. The semi-discretization methods derived from forward difference scheme is asymptotically stable. Finally, we give a full discretization method of the delay integro-differential equation by using the trapezium rule. And a sufficient condition is given on the asymptotic stability of trapezium rule.