本文選題：隨機(jī)延遲微分方程 + 擴(kuò)散的分裂步θ方法�。� 參考：《哈爾濱工業(yè)大學(xué)》2017年碩士論文

【摘要】：隨機(jī)延遲微分方程作為一種重要的數(shù)學(xué)模型在物理學(xué),生物學(xué),金融學(xué),控制論以及醫(yī)學(xué)等諸多領(lǐng)域具有廣泛的應(yīng)用。這一類方程既考慮了滯后對(duì)系統(tǒng)的作用,同時(shí)考慮了外界環(huán)境對(duì)系統(tǒng)性質(zhì)所造成的影響。因此,隨機(jī)延遲微分方程更加準(zhǔn)確的模擬了自然生活。在實(shí)際應(yīng)用中,隨機(jī)延遲微分方程精確解的顯式表達(dá)式很難求出,或表達(dá)式很復(fù)雜,因此構(gòu)造適用的數(shù)值方法并研究數(shù)值方法的性質(zhì)具有重要意義。近年來(lái)許多學(xué)者研究了隨機(jī)常延遲微分方程及數(shù)值方法,對(duì)于隨機(jī)變延遲微分方程及數(shù)值方法的研究剛剛開始。本文探討了隨機(jī)變延遲微分方程分裂步θ方法的收斂性和穩(wěn)定性。本文分別研究了擴(kuò)散的分裂步θ方法和漂移的分裂步θ方法的收斂性和穩(wěn)定性。首先,當(dāng)隨機(jī)延遲微分方程的系數(shù)滿足全局Lipschitz條件和線性增長(zhǎng)條件時(shí),研究了擴(kuò)散的分裂步θ方法的均方收斂性,并得出該方法的均方收斂階為1/2;隨后本文在單調(diào)條件和線性增長(zhǎng)條件下探討了擴(kuò)散的分裂步θ方法的穩(wěn)定性,證明了該方法對(duì)于一定范圍內(nèi)的步長(zhǎng)是均方指數(shù)穩(wěn)定的。其次,當(dāng)隨機(jī)延遲微分方程的系數(shù)滿足全局Lipschitz條件和線性增長(zhǎng)條件時(shí),分析了漂移的分裂步θ方法的均方收斂性,并得出該方法的均方收斂階為1/2;同時(shí)在單調(diào)條件和線性增長(zhǎng)條件下,本文探討了漂移的分裂步θ方法的穩(wěn)定性,證明了當(dāng)θ∈(1/2,1]時(shí),該方法依任意步長(zhǎng)保持均方指數(shù)穩(wěn)定;當(dāng)θ∈[0,1/2]時(shí),存在h0,使得當(dāng)h∈(0,h0)時(shí),漂移的分裂步θ方法是均方指數(shù)穩(wěn)定的。
[Abstract]:As an important mathematical model, stochastic delay differential equations are widely used in physics, biology, finance, cybernetics and medicine. This kind of equation not only considers the effect of hysteresis on the system, but also considers the influence of the external environment on the properties of the system. Therefore, the stochastic delay differential equations more accurately simulate the natural life. In practical application, the explicit expression of exact solution of stochastic delay differential equation is very difficult to obtain, or the expression is very complicated, so it is very important to construct the suitable numerical method and study the properties of the numerical method. In recent years, many scholars have studied stochastic constant delay differential equations and numerical methods, and the research on stochastic variable delay differential equations and numerical methods has just begun. In this paper, the convergence and stability of the splitting step 胃 method for stochastic variable delay differential equations are discussed. In this paper, the convergence and stability of diffusion splitting 胃 method and drift splitting step 胃 method are studied respectively. Firstly, when the coefficients of the stochastic delay differential equation satisfy the global Lipschitz condition and the linear growth condition, the mean square convergence of the split step 胃 method of diffusion is studied. The mean square convergence order of the method is 1 / 2. Then the stability of the diffusion splitting step 胃 method is discussed under the monotone condition and the linear growth condition. It is proved that the method is exponentially stable for the step size in a certain range. Secondly, when the coefficients of the stochastic delay differential equation satisfy the global Lipschitz condition and the linear growth condition, the mean square convergence of the drift splitting step 胃 method is analyzed. At the same time, under monotone condition and linear growth condition, the stability of the drift splitting step 胃 method is discussed, and it is proved that the mean square exponent stability of the method is maintained according to arbitrary step size when 胃 鈭，

本文編號(hào)：1786084