本文選題：分?jǐn)?shù)階微分 + 再生核理論　； 參考：《哈爾濱工業(yè)大學(xué)》2017年碩士論文

【摘要】：分?jǐn)?shù)階微分算子因其非局部性,更適合用來描述實際生活中一些復(fù)雜的動態(tài)行為。近年來,分?jǐn)?shù)階微分方程的應(yīng)用已遍及眾多科學(xué)領(lǐng)域,但是方程的解析解較難得到。因此,探究方程的數(shù)值算法成為定性地研究此類方程的一個重要手段,而如何構(gòu)造高精度的數(shù)值算法需要進(jìn)一步地探究。基于樣條思想與再生核理論,本文系統(tǒng)地研究了兩類時間分?jǐn)?shù)階偏微分方程的數(shù)值算法,即變時間分?jǐn)?shù)階Mobile-Immobile對流擴(kuò)散方程和非線性時間分?jǐn)?shù)階薛定諤方程。主要結(jié)果如下:首先,本文討論了變時間分?jǐn)?shù)階Mobile-Immobile對流擴(kuò)散方程解的光滑性,由此可以用樣條函數(shù)來逼近方程的解�；谠偕撕瘮�(shù)和樣條多項式,本文構(gòu)造了此類方程的一個級數(shù)形式的近似解表達(dá)式。同時,構(gòu)建了一個簡便的數(shù)值算法來得到ε-近似解。為了說明算法的有效性,本文給出了算法的收斂性及穩(wěn)定性分析。其次,本文研究了一類非線性時間分?jǐn)?shù)階薛定諤方程的數(shù)值算法。此類方程的解通常具有弱奇性,本文引入分?jǐn)?shù)階積分算子作用原來微分方程的兩端,得到積分方程再進(jìn)行求解。與一般的分?jǐn)?shù)階微分方程不同,此類方程的解屬于復(fù)數(shù)域。本文將方程解的實虛部拆分,在相應(yīng)的再生核空間中分別導(dǎo)出了實虛部易于計算的解表達(dá)式。同時,討論了方程ε-近似解的存在性。數(shù)值結(jié)果表明,該算法具有很好的收斂性和較高的數(shù)值精度。
[Abstract]:Fractional differential operators are more suitable to describe some complex dynamic behaviors in real life because of their nonlocality. In recent years, fractional differential equations have been widely used in many scientific fields, but the analytical solutions of the equations are difficult to obtain. Therefore, exploring the numerical algorithm of equations becomes an important means to qualitatively study such equations, and how to construct high-precision numerical algorithms needs to be further explored. Based on spline theory and reproducing kernel theory, two kinds of numerical algorithms for fractional partial differential equations of time, namely, variable time fractional Mobile-Immobile convection-diffusion equation and nonlinear time fractional Schrodinger equation, are studied systematically in this paper. The main results are as follows: firstly, we discuss the smoothness of the solution of the Mobile-Immobile convection-diffusion equation with variable time fractional order, so that the solution of the equation can be approximated by spline function. Based on the reproducing kernel function and spline polynomial, an approximate solution expression of a series form of this kind of equation is constructed in this paper. At the same time, a simple numerical algorithm is constructed to obtain the 蔚-approximate solution. In order to illustrate the validity of the algorithm, the convergence and stability analysis of the algorithm are given in this paper. Secondly, the numerical algorithm for a class of nonlinear time fractional Schrodinger equations is studied. The solution of this kind of equation is usually weak singularity. In this paper, the fractional integral operator is introduced to act on the two ends of the original differential equation, and then the integral equation is obtained and solved. Unlike ordinary fractional differential equations, the solutions of such equations belong to the complex field. In this paper, the real imaginary part of the solution of the equation is split, and the expressions of the real imaginary part are derived in the corresponding reproducing kernel space. At the same time, the existence of 蔚-approximate solution of the equation is discussed. Numerical results show that the algorithm has good convergence and high numerical accuracy.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.82

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相關(guān)期刊論文 前1條

1 默會霞;余東艷;隋鑫;;利用Adomain分解法求時間分?jǐn)?shù)階薛定諤方程的近似解[J];純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué);2014年05期

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本文編號：2017069