【摘要】：延遲偏微分方程在現(xiàn)實(shí)生活中應(yīng)用比較廣泛,而方程本身的理論解一般很難得到,所以對(duì)延遲偏微分方程數(shù)值解的研究就非常必要。本論文主要研究了三類線性延遲偏微分方程的數(shù)值解的性質(zhì),這三類方程分別是拋物型延遲微分方程、雙曲型延遲微分方程和一類帶延遲的積分微分方程。本論文的研究?jī)?nèi)容包括三個(gè)主要部分,其結(jié)構(gòu)安排如下:第一部分研究拋物型延遲微分方程。首先用線性多步法給出拋物型延遲微分方程的半離散格式,得到半離散格式的方法階是p的充分必要條件,運(yùn)用這個(gè)條件舉例得出由中心差分格式和五點(diǎn)格式構(gòu)造的半離散方法的方法階分別為二階和四階;其次用Fourier方法得出半離散方法漸近穩(wěn)定的充分條件,得到由中心差分格式和五點(diǎn)格式構(gòu)造的半離散方法都是漸近穩(wěn)定的;最后用線性多步法給出方程的全離散格式,用Fourier方法得出全離散格式漸近穩(wěn)定的一個(gè)充分條件,并分析了向前Euler方法和Crank-Nicolson方法的漸近穩(wěn)定性。第二部分研究雙曲型延遲微分方程。首先用線性多步法給出雙曲型延遲微分方程的半離散格式,得到半離散格式的方法階是p的充分必要條件,運(yùn)用這個(gè)條件舉例得出向前差分格式和中心差分格式構(gòu)造的半離散方法的方法階分別為一階和二階;其次用Fourier方法得出半離散方法漸近穩(wěn)定的充分條件,得到由向前差分格式構(gòu)造的半離散方法的漸近穩(wěn)定的一個(gè)充分條件,并且由中心差分格式得出的半離散方法是不穩(wěn)定的;最后用線性多步法給出方程的全離散格式,用Fourier方法給出全離散格式漸近穩(wěn)定的一個(gè)充分條件,得出向前Euler方法漸近穩(wěn)定的充分條件,并且Crank-Nicolson方法不是漸近穩(wěn)定的。第三部分研究一類帶延遲的積分微分方程。首先用線性多步法給出該方程的半離散格式,得到半離散格式的方法階是p的充分必要條件,運(yùn)用這個(gè)條件舉例得出中心差分格式和五點(diǎn)格式構(gòu)造的半離散方法的方法階分別為二階和四階;其次給出半離散方法漸近穩(wěn)定的充分條件,得到由向前差分格式構(gòu)造的半離散方法漸近穩(wěn)定的一個(gè)充分條件;最后分析一種具體格式的全離散方法——梯形方法的穩(wěn)定性,給出梯形方法漸近穩(wěn)定的一個(gè)充分條件。
[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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241.82

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