【摘要】：在求解隨機(jī)延遲微分方程(SDDE)中,許多學(xué)者構(gòu)造了多種形式的線(xiàn)性多步法,并研究了它們的穩(wěn)定性和收斂性,但是在它們針對(duì)的SDDE中,漂移系數(shù)和擴(kuò)散系數(shù)的延遲項(xiàng)是相同的,然而在實(shí)際中,它們的延遲項(xiàng)是不相同的,且是任意正常數(shù).對(duì)此尚未研究.因此本文考慮了一種新的非線(xiàn)性SDDE,其中漂移系數(shù)和擴(kuò)散系數(shù)的延遲項(xiàng)是不同的,分別用τ1,τ2表示,τ1,τ2可取任意正常數(shù).本文將常微分方程的k步BDF法推廣到這類(lèi)非線(xiàn)性SDDE中,構(gòu)造了新的隨機(jī)k步BDF法,并研究了它的均方穩(wěn)定性,均方收斂性.再將隨機(jī)k步BDF法運(yùn)用到一維SDDE中,獲得了該數(shù)值算法的均方相容條件和均方收斂階.第一部分為緒論.主要介紹隨機(jī)延遲微分方程的相關(guān)背景和國(guó)內(nèi)外研究現(xiàn)狀,本文的創(chuàng)新之處和主要內(nèi)容,以及本文涉及的符號(hào)說(shuō)明.第二部分簡(jiǎn)要介紹了本文新構(gòu)造的隨機(jī)k步BDF法,并給出了它均方穩(wěn)定,均方相容,均方收斂的相關(guān)定義和結(jié)論.第三部分證明了隨機(jī)k步BDF法的均方穩(wěn)定和均方收斂定理,給出了穩(wěn)定性不等式.第四部分將隨機(jī)k步BDF法運(yùn)用到一維SDDE中,獲得了隨機(jī)k步BDF法的收斂階.第五部分構(gòu)造隨機(jī)3步BDF法,通過(guò)Matlab軟件,用數(shù)值試驗(yàn)驗(yàn)證它的均方穩(wěn)定性和均方收斂階.
[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.
【學(xué)位授予單位】：廣西師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O211.63

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相關(guān)期刊論文 前1條

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本文編號(hào)：2375495