非线性随机延迟微分系统的随机k步BDF法的稳定性与收敛性研究
发布时间：2018-12-13 00:13
【摘要】：在求解随机延迟微分方程(SDDE)中,许多学者构造了多种形式的线性多步法,并研究了它们的稳定性和收敛性,但是在它们针对的SDDE中,漂移系数和扩散系数的延迟项是相同的,然而在实际中,它们的延迟项是不相同的,且是任意正常数.对此尚未研究.因此本文考虑了一种新的非线性SDDE,其中漂移系数和扩散系数的延迟项是不同的,分别用τ1,τ2表示,τ1,τ2可取任意正常数.本文将常微分方程的k步BDF法推广到这类非线性SDDE中,构造了新的随机k步BDF法,并研究了它的均方稳定性,均方收敛性.再将随机k步BDF法运用到一维SDDE中,获得了该数值算法的均方相容条件和均方收敛阶.第一部分为绪论.主要介绍随机延迟微分方程的相关背景和国内外研究现状,本文的创新之处和主要内容,以及本文涉及的符号说明.第二部分简要介绍了本文新构造的随机k步BDF法,并给出了它均方稳定,均方相容,均方收敛的相关定义和结论.第三部分证明了随机k步BDF法的均方稳定和均方收敛定理,给出了稳定性不等式.第四部分将随机k步BDF法运用到一维SDDE中,获得了随机k步BDF法的收敛阶.第五部分构造随机3步BDF法,通过Matlab软件,用数值试验验证它的均方稳定性和均方收敛阶.
[Abstract]:In solving stochastic delay differential equations (SDDE), many scholars have constructed many kinds of linear multistep methods, and studied their stability and convergence, but in the SDDE for which they are aimed, The delay term of drift coefficient and diffusion coefficient is the same, however, in practice, their delay term is different and is an arbitrary normal number. This has not been studied. In this paper, we consider a new nonlinear SDDE, in which the delay terms of drift coefficient and diffusion coefficient are different. 蟿 1, 蟿 2, 蟿 1 and 蟿 2 can be used as arbitrary normal numbers, respectively. In this paper, the k-step BDF method for ordinary differential equations is extended to this kind of nonlinear SDDE. A new stochastic k-step BDF method is constructed, and its mean square stability and mean square convergence are studied. Then the stochastic k-step BDF method is applied to one-dimensional SDDE, and the mean square compatibility condition and mean square convergence order of the numerical algorithm are obtained. The first part is the introduction. This paper mainly introduces the background of stochastic delay differential equation and the current research situation at home and abroad, the innovations and main contents of this paper, and the symbolic explanation involved in this paper. In the second part, the new stochastic k-step BDF method is briefly introduced, and the definitions and conclusions of its mean square stability, mean square compatibility and mean square convergence are given. In the third part, the mean square stability and mean square convergence theorems of stochastic k-step BDF method are proved, and the stability inequality is given. In the fourth part, the stochastic k-step BDF method is applied to one-dimensional SDDE, and the convergence order of the stochastic k-step BDF method is obtained. In the fifth part, the stochastic three-step BDF method is constructed, and its mean square stability and mean square convergence order are verified by Matlab software.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63
本文编号：2375495
[Abstract]:In solving stochastic delay differential equations (SDDE), many scholars have constructed many kinds of linear multistep methods, and studied their stability and convergence, but in the SDDE for which they are aimed, The delay term of drift coefficient and diffusion coefficient is the same, however, in practice, their delay term is different and is an arbitrary normal number. This has not been studied. In this paper, we consider a new nonlinear SDDE, in which the delay terms of drift coefficient and diffusion coefficient are different. 蟿 1, 蟿 2, 蟿 1 and 蟿 2 can be used as arbitrary normal numbers, respectively. In this paper, the k-step BDF method for ordinary differential equations is extended to this kind of nonlinear SDDE. A new stochastic k-step BDF method is constructed, and its mean square stability and mean square convergence are studied. Then the stochastic k-step BDF method is applied to one-dimensional SDDE, and the mean square compatibility condition and mean square convergence order of the numerical algorithm are obtained. The first part is the introduction. This paper mainly introduces the background of stochastic delay differential equation and the current research situation at home and abroad, the innovations and main contents of this paper, and the symbolic explanation involved in this paper. In the second part, the new stochastic k-step BDF method is briefly introduced, and the definitions and conclusions of its mean square stability, mean square compatibility and mean square convergence are given. In the third part, the mean square stability and mean square convergence theorems of stochastic k-step BDF method are proved, and the stability inequality is given. In the fourth part, the stochastic k-step BDF method is applied to one-dimensional SDDE, and the convergence order of the stochastic k-step BDF method is obtained. In the fifth part, the stochastic three-step BDF method is constructed, and its mean square stability and mean square convergence order are verified by Matlab software.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63
【参考文献】
相关期刊论文 前1条
1 王文强;黄山;李寿佛;;非线性随机延迟微分方程Euler-Maruyama方法的均方稳定性[J];计算数学;2007年02期
，本文编号：2375495
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