本文選題：分數(shù)階積分微分方程 + 再生核理論��； 參考：《哈爾濱工業(yè)大學》2015年碩士論文

【摘要】：分數(shù)階積分微分的出現(xiàn)發(fā)生在基礎(chǔ)物理學中,它的出現(xiàn)所帶來的新問題使數(shù)學家和物理學家對分數(shù)階微積分理論產(chǎn)生了極大的興趣。因為分數(shù)階微積分方程能夠更準確的描述實際現(xiàn)象的動力學行為,因此在工程學和物理學等其他領(lǐng)域都有著很廣泛的應(yīng)用,例如可以應(yīng)用非線性分數(shù)階微積分來模擬地震的非線性振動,并且在控制學中我們也可以發(fā)現(xiàn)分數(shù)階微分(FDES)的身影。但是通常情況下這類方程很難得到解析解,所以求解其數(shù)值解就變得非常重要且具有實際應(yīng)用價值。近幾十年來,學者們已經(jīng)提出了一些數(shù)值方法用于求解分數(shù)階積分微分方程,如Adomain分解法、有限差分法、多項式配置法、小波法等。但理論體系仍需進一步的完善。因此,本文對兩類分數(shù)階Volterra型積分微分方程的數(shù)值解法進行了探討,即非線性分數(shù)階Volterra型積分微分方程和分數(shù)階Volterra型人口增長模型。作為預備知識本文第2章節(jié),介紹了分數(shù)階導數(shù)基本定義,再生核理論的基本知識,為下面兩部分再生核的應(yīng)用作鋪墊。本文的第3章首先對模型進行了解釋說明,其次應(yīng)用再生核理論求解非線性分數(shù)階Volterra型積分微分方程并得到近似解。最后,由具體的數(shù)值算例驗證該算法的優(yōu)越性。本文的第4章主要研究的是分數(shù)階Volterra型人口增長模型的數(shù)值解法問題。本文基于再生核理論,對再生核方法進行改進求解此類型的方程,并建立了一套完整的理論體系。最后,數(shù)值實驗的結(jié)果表明本文所提出的算法十分有效且易于操作。
[Abstract]:The appearance of fractional integro-differential in basic physics brings about new problems which make mathematicians and physicists have great interest in fractional calculus theory. Because fractional calculus equations can more accurately describe the dynamics of actual phenomena, they are widely used in engineering, physics and other fields. For example, nonlinear fractional calculus can be used to simulate the nonlinear vibration of earthquakes, and the fractional differential FDESs can also be found in control. However, it is very difficult to obtain the analytical solution for this kind of equation, so it is very important to solve its numerical solution and has practical application value. In recent decades, some numerical methods have been proposed to solve fractional integrodifferential equations, such as Adomain decomposition method, finite difference method, polynomial collocation method, wavelet method and so on. However, the theoretical system still needs to be further improved. Therefore, in this paper, the numerical solutions of two kinds of fractional Volterra type integro-differential equations are discussed, that is, nonlinear fractional Volterra integral differential equations and fractional Volterra type population growth models. As the preparatory knowledge, this paper introduces the basic definition of fractional derivative and the basic knowledge of the theory of reproducing kernel, which paves the way for the application of the following two parts of reproducing kernels. In chapter 3, the model is explained firstly, and then the nonlinear fractional Volterra integro-differential equation is solved by using the reproducing kernel theory and the approximate solution is obtained. Finally, the superiority of the algorithm is verified by a numerical example. In chapter 4, the numerical solution of fractional Volterra population growth model is studied. Based on the reproducing kernel theory, the reproducing kernel method is improved to solve the equations of this type, and a complete theoretical system is established. Finally, the numerical results show that the proposed algorithm is very effective and easy to operate.
【學位授予單位】：哈爾濱工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O175.6

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本文編號：2029609