解析逼近方法若干問題研究
發(fā)布時間：2019-03-12 15:55
【摘要】：現(xiàn)實(shí)世界中描述的許多現(xiàn)象都可歸結(jié)為非線性微分方程,求解非線性微分方程已經(jīng)成為研究者們面臨的關(guān)鍵問題.工程師、物理學(xué)家和應(yīng)用數(shù)學(xué)家們處理的許多物理問題顯示出某些基本特征,而這些基本特征使得相應(yīng)的問題無法求得精確的解析解.科學(xué)技術(shù)的發(fā)展和符號計算軟件的出現(xiàn)促進(jìn)了非線性微分方程解析逼近方法的發(fā)展.同倫分析方法(HAM)和Adomian分解法(ADM)是兩種比較常用的解析逼近方法.本文給出這兩種方法的一些重要理論改進(jìn)以及這些改進(jìn)的非平凡應(yīng)用.更具體地說,我們完成以下四部分內(nèi)容.1.提出了一種求解非線性初值問題的混合解析方法,此方法基于同倫分析方法和Laplace變換方法.首先,將一個初值問題轉(zhuǎn)化成初值點(diǎn)在零處的新問題,應(yīng)用標(biāo)準(zhǔn)同倫分析方法將新的非線性微分方程轉(zhuǎn)換為線性微分方程系統(tǒng).然后,應(yīng)用Laplace變換和Laplace逆變換求解所得到的線性初值問題.這些線性初值問題的解析逼近解能夠形成給定問題的一個收斂級數(shù)解.通過一些非平凡例子證明關(guān)于求解高階變形方程,混合解析逼近方法比標(biāo)準(zhǔn)同倫分析方法更有利.因此,新方法可以應(yīng)用于求解更復(fù)雜的非線性現(xiàn)實(shí)問題.2.同倫分析方法的突出特點(diǎn)之一是引入收斂控制參數(shù),收斂控制參數(shù)提供了一個簡便的方式來調(diào)節(jié)和控制所得到的級數(shù)解的收斂區(qū)域和速度.然而,從嚴(yán)格的數(shù)學(xué)觀點(diǎn)來看,收斂控制參數(shù)如何實(shí)現(xiàn)這個目標(biāo)?我們得到高階線性微分方程完整的理論結(jié)果.換言之,我們給出由同倫分析方法得到的級數(shù)解在某一區(qū)間上收斂的嚴(yán)格證明,該區(qū)間依賴于收斂控制參數(shù),并且得到該區(qū)間上逼近解的絕對誤差上界.此外,我們也給出一個確定收斂控制參數(shù)有效區(qū)域的方法,在所得區(qū)域上可以確保級數(shù)解收斂.3.基于同倫分析方法求解高階線性參數(shù)邊值問題.通過建立所得級數(shù)解的顯式表達(dá)式和大參數(shù)與收斂控制參數(shù)之間的關(guān)系,我們對所求問題的解結(jié)構(gòu)有更深刻的理解.對于大的參數(shù)值,通過適當(dāng)?shù)剡x擇收斂控制參數(shù)的值可以得到比較準(zhǔn)確的逼近解.與其他解析方法相比,該方法對于求解含大參數(shù)的高階線性邊值問題更有效.4.作為經(jīng)典偏微分方程的推廣,分?jǐn)?shù)階偏微分方程越來越多的被應(yīng)用于科學(xué)的不同領(lǐng)域.與經(jīng)典偏微分方程相比,分?jǐn)?shù)階偏微分方程可以更好的模擬現(xiàn)實(shí)問題.我們提出一個求解非線性分?jǐn)?shù)階偏微分方程的新方法.新方法的關(guān)鍵點(diǎn)是在傳統(tǒng)Adomian分解法中引入兩個參數(shù),稱為兩參數(shù)ADM.已證明兩參數(shù)ADM逼近解比傳統(tǒng)Adomian分解法逼近解更準(zhǔn)確.為了說明新方法的適用性和有效性,求解兩個非線性分?jǐn)?shù)階偏微分方程.
[Abstract]:Many phenomena described in the real world can be reduced to nonlinear differential equations. Solving nonlinear differential equations has become a key problem faced by researchers. Many of the physical problems dealt with by engineers physicists and applied mathematicians show some basic characteristics which make it impossible to obtain exact analytical solutions for the corresponding problems. The development of science and technology and the emergence of symbolic computing software promote the development of analytical approximation methods for nonlinear differential equations. Homotopy analysis method (HAM) and Adomian decomposition method (ADM) are two more commonly used analytical approximation methods. In this paper, some important theoretical improvements of these two methods and their nontrivial applications are given. More specifically, we have completed the following four parts. A hybrid analytical method for solving nonlinear initial value problems is proposed. This method is based on homotopy analysis and Laplace transform. Firstly, the initial value problem is transformed into a new problem where the initial point is at zero. The standard homotopy analysis method is used to transform the new nonlinear differential equation into a linear differential equation system. Then, Laplace transform and Laplace inverse transform are used to solve the linear initial value problem. The analytic approximation solutions of these linear initial value problems can form a convergence series solution of a given problem. It is proved by some non-trivial examples that the mixed analytical approximation method is more advantageous than the standard homotopy method in solving higher-order deformation equations. Therefore, the new method can be applied to solve more complex nonlinear reality problems. One of the outstanding characteristics of homotopy analysis is the introduction of convergence control parameters, which provide a simple way to adjust and control the convergence region and speed of the obtained series solutions. However, from a strict mathematical point of view, how can convergence control parameters achieve this goal? We obtain the complete theoretical results of higher order linear differential equations. In other words, we give a strict proof that the series solution obtained by the homotopy analysis method converges on a certain interval, which depends on the convergence control parameters, and obtains the upper bound of the absolute error of the approximation solution on the interval. In addition, we also give a method to determine the effective region of convergence control parameters, on which the convergence of the series solution can be ensured. Based on the homotopy analysis method, the higher order linear parameter boundary value problem is solved. By establishing the explicit expression of the obtained series solution and the relationship between the large parameter and the convergence control parameter, we have a deeper understanding of the solution structure of the problem. For large parameter values, a more accurate approximate solution can be obtained by properly selecting the values of convergence control parameters. Compared with other analytical methods, this method is more effective for solving higher order linear boundary value problems with large parameters. 4. As a generalization of classical partial differential equations, fractional partial differential equations are more and more applied in different fields of science. Compared with classical partial differential equations, fractional partial differential equations can simulate practical problems better. We propose a new method for solving nonlinear fractional partial differential equations. The key point of the new method is to introduce two parameters into the traditional Adomian decomposition method, which is called two-parameter ADM.. It has been proved that the two-parameter ADM approximation solution is more accurate than the traditional Adomian decomposition method. In order to illustrate the applicability and effectiveness of the new method, two nonlinear fractional partial differential equations are solved.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175.29
本文編號：2438922
[Abstract]:Many phenomena described in the real world can be reduced to nonlinear differential equations. Solving nonlinear differential equations has become a key problem faced by researchers. Many of the physical problems dealt with by engineers physicists and applied mathematicians show some basic characteristics which make it impossible to obtain exact analytical solutions for the corresponding problems. The development of science and technology and the emergence of symbolic computing software promote the development of analytical approximation methods for nonlinear differential equations. Homotopy analysis method (HAM) and Adomian decomposition method (ADM) are two more commonly used analytical approximation methods. In this paper, some important theoretical improvements of these two methods and their nontrivial applications are given. More specifically, we have completed the following four parts. A hybrid analytical method for solving nonlinear initial value problems is proposed. This method is based on homotopy analysis and Laplace transform. Firstly, the initial value problem is transformed into a new problem where the initial point is at zero. The standard homotopy analysis method is used to transform the new nonlinear differential equation into a linear differential equation system. Then, Laplace transform and Laplace inverse transform are used to solve the linear initial value problem. The analytic approximation solutions of these linear initial value problems can form a convergence series solution of a given problem. It is proved by some non-trivial examples that the mixed analytical approximation method is more advantageous than the standard homotopy method in solving higher-order deformation equations. Therefore, the new method can be applied to solve more complex nonlinear reality problems. One of the outstanding characteristics of homotopy analysis is the introduction of convergence control parameters, which provide a simple way to adjust and control the convergence region and speed of the obtained series solutions. However, from a strict mathematical point of view, how can convergence control parameters achieve this goal? We obtain the complete theoretical results of higher order linear differential equations. In other words, we give a strict proof that the series solution obtained by the homotopy analysis method converges on a certain interval, which depends on the convergence control parameters, and obtains the upper bound of the absolute error of the approximation solution on the interval. In addition, we also give a method to determine the effective region of convergence control parameters, on which the convergence of the series solution can be ensured. Based on the homotopy analysis method, the higher order linear parameter boundary value problem is solved. By establishing the explicit expression of the obtained series solution and the relationship between the large parameter and the convergence control parameter, we have a deeper understanding of the solution structure of the problem. For large parameter values, a more accurate approximate solution can be obtained by properly selecting the values of convergence control parameters. Compared with other analytical methods, this method is more effective for solving higher order linear boundary value problems with large parameters. 4. As a generalization of classical partial differential equations, fractional partial differential equations are more and more applied in different fields of science. Compared with classical partial differential equations, fractional partial differential equations can simulate practical problems better. We propose a new method for solving nonlinear fractional partial differential equations. The key point of the new method is to introduce two parameters into the traditional Adomian decomposition method, which is called two-parameter ADM.. It has been proved that the two-parameter ADM approximation solution is more accurate than the traditional Adomian decomposition method. In order to illustrate the applicability and effectiveness of the new method, two nonlinear fractional partial differential equations are solved.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175.29
【相似文獻(xiàn)】
相關(guān)期刊論文 前10條
1 戴炳鑫，，陸建芳;一種雙園弧折線的逼近方法[J];寶雞文理學(xué)院學(xué)報(自然科學(xué)版);1996年02期
2 張宏偉;一種指數(shù)型逼近方法[J];吉林大學(xué)自然科學(xué)學(xué)報;1989年01期
3 劉曉俊;袁國強(qiáng);;兩階段模糊運(yùn)輸期望值模型及其逼近方法[J];模糊系統(tǒng)與數(shù)學(xué);2011年01期
4 陸建芳;一種三圓弧逼近方法[J];浙江工業(yè)大學(xué)學(xué)報;1997年03期
5 段火元;Reissner-Mindlin板模型的關(guān)于板厚一致收斂逼近方法[J];應(yīng)用力學(xué)學(xué)報;1999年02期
6 王華榮;俞建寧;劉一京;;黃河流速場數(shù)值模擬的逼近方法[J];蘭州鐵道學(xué)院學(xué)報;1991年02期
7 李貴杰;呂震宙;趙新攀;;基于加權(quán)線性響應(yīng)面的結(jié)構(gòu)可靠性估計的鞍點(diǎn)逼近方法[J];機(jī)械強(qiáng)度;2010年06期
8 王連堂;反演聲波阻尼系數(shù)的一個逼近方法[J];計算數(shù)學(xué);2000年03期
9 朱堯辰;用逼近方法建立代數(shù)無關(guān)性[J];數(shù)學(xué)學(xué)報;1998年02期
10 楊宏奇,李岳生;近似已知函數(shù)微商的穩(wěn)定逼近方法[J];自然科學(xué)進(jìn)展;2000年12期
相關(guān)會議論文 前1條
1 袁光偉;沈智鈞;周毓麟;;中子輸運(yùn)球諧逼近方法[A];中國工程物理研究院科技年報（1998）[C];1998年
相關(guān)博士學(xué)位論文 前1條
1 馬俊馳;解析逼近方法若干問題研究[D];大連理工大學(xué);2017年
相關(guān)碩士學(xué)位論文 前1條
1 虞婷;一類具擴(kuò)散的PEPA模型分析[D];揚(yáng)州大學(xué);2011年
本文編號：2438922
本文鏈接：http://sikaile.net/shoufeilunwen/jckxbs/2438922.html
最近更新
教材專著
