线性延迟偏微分方程的半离散及全离散格式数值分析
发布时间：2018-12-10 23:24
【摘要】：延迟偏微分方程在现实生活中应用比较广泛,而方程本身的理论解一般很难得到,所以对延迟偏微分方程数值解的研究就非常必要。本论文主要研究了三类线性延迟偏微分方程的数值解的性质,这三类方程分别是抛物型延迟微分方程、双曲型延迟微分方程和一类带延迟的积分微分方程。本论文的研究内容包括三个主要部分,其结构安排如下:第一部分研究抛物型延迟微分方程。首先用线性多步法给出抛物型延迟微分方程的半离散格式,得到半离散格式的方法阶是p的充分必要条件,运用这个条件举例得出由中心差分格式和五点格式构造的半离散方法的方法阶分别为二阶和四阶;其次用Fourier方法得出半离散方法渐近稳定的充分条件,得到由中心差分格式和五点格式构造的半离散方法都是渐近稳定的;最后用线性多步法给出方程的全离散格式,用Fourier方法得出全离散格式渐近稳定的一个充分条件,并分析了向前Euler方法和Crank-Nicolson方法的渐近稳定性。第二部分研究双曲型延迟微分方程。首先用线性多步法给出双曲型延迟微分方程的半离散格式,得到半离散格式的方法阶是p的充分必要条件,运用这个条件举例得出向前差分格式和中心差分格式构造的半离散方法的方法阶分别为一阶和二阶;其次用Fourier方法得出半离散方法渐近稳定的充分条件,得到由向前差分格式构造的半离散方法的渐近稳定的一个充分条件,并且由中心差分格式得出的半离散方法是不稳定的;最后用线性多步法给出方程的全离散格式,用Fourier方法给出全离散格式渐近稳定的一个充分条件,得出向前Euler方法渐近稳定的充分条件,并且Crank-Nicolson方法不是渐近稳定的。第三部分研究一类带延迟的积分微分方程。首先用线性多步法给出该方程的半离散格式,得到半离散格式的方法阶是p的充分必要条件,运用这个条件举例得出中心差分格式和五点格式构造的半离散方法的方法阶分别为二阶和四阶;其次给出半离散方法渐近稳定的充分条件,得到由向前差分格式构造的半离散方法渐近稳定的一个充分条件;最后分析一种具体格式的全离散方法——梯形方法的稳定性,给出梯形方法渐近稳定的一个充分条件。
[Abstract]:The delay partial differential equation is widely used in real life, but the theoretical solution of the equation itself is generally difficult to obtain, so it is very necessary to study the numerical solution of the delay partial differential equation. In this paper, we study the properties of numerical solutions of three kinds of linear delay partial differential equations, which are parabolic delay differential equations, hyperbolic delay differential equations and a class of integro-differential equations with delay. There are three main parts in this thesis. The structure of this thesis is as follows: the first part studies parabolic delay differential equations. First, the semi-discrete scheme of parabolic delay differential equation is given by linear multi-step method, and the necessary and sufficient condition for the method order of semi-discrete scheme to be p is obtained. By using this condition, the method order of the semi-discrete method constructed by the central difference scheme and the five-point scheme is obtained to be the second order and the fourth order, respectively. Secondly, the sufficient condition of asymptotic stability of semi-discrete method is obtained by Fourier method, and the semi-discrete method constructed by central difference scheme and five-point scheme is asymptotically stable. Finally, the full discrete scheme of the equation is given by using the linear multistep method. A sufficient condition for the asymptotic stability of the full discrete scheme is obtained by using the Fourier method, and the asymptotic stability of the forward Euler method and the Crank-Nicolson method is analyzed. In the second part, hyperbolic delay differential equations are studied. First, the semi-discrete scheme of hyperbolic delay differential equation is given by linear multistep method, and the necessary and sufficient condition for the method order of semi-discrete scheme to be p is obtained. By using this condition, the method order of the semi-discrete method constructed by the forward difference scheme and the central difference scheme is obtained, which is the first order and the second order, respectively. Secondly, a sufficient condition for asymptotic stability of semi-discrete method is obtained by Fourier method, a sufficient condition for asymptotic stability of semi-discrete method constructed by forward difference scheme is obtained, and the semi-discrete method obtained from central difference scheme is unstable. Finally, the full discrete scheme of the equation is given by the linear multistep method, a sufficient condition for the asymptotic stability of the full discrete scheme is given by using the Fourier method, and a sufficient condition for the asymptotic stability of the forward Euler method is obtained, and the Crank-Nicolson method is not asymptotically stable. In the third part, we study a class of integro-differential equations with delay. First, the semi-discrete scheme of the equation is given by linear multistep method, and the necessary and sufficient conditions for the order of the semi-discrete scheme to be p are obtained. By using this condition, the method order of the semi-discrete method constructed by the central difference scheme and the five-point scheme is obtained to be the second order and the fourth order, respectively. Secondly, a sufficient condition for asymptotic stability of semi-discrete method is given, and a sufficient condition for asymptotic stability of semi-discrete method constructed by forward difference scheme is obtained. Finally, the stability of the trapezoidal method is analyzed, and a sufficient condition for the asymptotic stability of the trapezoidal method is given.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82
[Abstract]:The delay partial differential equation is widely used in real life, but the theoretical solution of the equation itself is generally difficult to obtain, so it is very necessary to study the numerical solution of the delay partial differential equation. In this paper, we study the properties of numerical solutions of three kinds of linear delay partial differential equations, which are parabolic delay differential equations, hyperbolic delay differential equations and a class of integro-differential equations with delay. There are three main parts in this thesis. The structure of this thesis is as follows: the first part studies parabolic delay differential equations. First, the semi-discrete scheme of parabolic delay differential equation is given by linear multi-step method, and the necessary and sufficient condition for the method order of semi-discrete scheme to be p is obtained. By using this condition, the method order of the semi-discrete method constructed by the central difference scheme and the five-point scheme is obtained to be the second order and the fourth order, respectively. Secondly, the sufficient condition of asymptotic stability of semi-discrete method is obtained by Fourier method, and the semi-discrete method constructed by central difference scheme and five-point scheme is asymptotically stable. Finally, the full discrete scheme of the equation is given by using the linear multistep method. A sufficient condition for the asymptotic stability of the full discrete scheme is obtained by using the Fourier method, and the asymptotic stability of the forward Euler method and the Crank-Nicolson method is analyzed. In the second part, hyperbolic delay differential equations are studied. First, the semi-discrete scheme of hyperbolic delay differential equation is given by linear multistep method, and the necessary and sufficient condition for the method order of semi-discrete scheme to be p is obtained. By using this condition, the method order of the semi-discrete method constructed by the forward difference scheme and the central difference scheme is obtained, which is the first order and the second order, respectively. Secondly, a sufficient condition for asymptotic stability of semi-discrete method is obtained by Fourier method, a sufficient condition for asymptotic stability of semi-discrete method constructed by forward difference scheme is obtained, and the semi-discrete method obtained from central difference scheme is unstable. Finally, the full discrete scheme of the equation is given by the linear multistep method, a sufficient condition for the asymptotic stability of the full discrete scheme is given by using the Fourier method, and a sufficient condition for the asymptotic stability of the forward Euler method is obtained, and the Crank-Nicolson method is not asymptotically stable. In the third part, we study a class of integro-differential equations with delay. First, the semi-discrete scheme of the equation is given by linear multistep method, and the necessary and sufficient conditions for the order of the semi-discrete scheme to be p are obtained. By using this condition, the method order of the semi-discrete method constructed by the central difference scheme and the five-point scheme is obtained to be the second order and the fourth order, respectively. Secondly, a sufficient condition for asymptotic stability of semi-discrete method is given, and a sufficient condition for asymptotic stability of semi-discrete method constructed by forward difference scheme is obtained. Finally, the stability of the trapezoidal method is analyzed, and a sufficient condition for the asymptotic stability of the trapezoidal method is given.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82
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