本文選題：一維空間分數(shù)階擴散方程 + 一維空間分數(shù)階對流擴散方程�。� 參考：《寧夏大學》2016年博士論文

【摘要】：近幾十年來,由于分數(shù)階導數(shù)可以用來刻畫反常擴散現(xiàn)象,而許多復雜動力系統(tǒng)都包含著反常擴散,故分數(shù)階動力學方程是描述復雜系統(tǒng)的有效方法,從而使得分數(shù)階偏微分方程在自然科學和社會科學得到廣泛應用,由于其解析解不容易求出來,所以分數(shù)階偏微分方程數(shù)值解的研究就顯得尤為重要。求解分數(shù)階微分方程數(shù)值解需要克服一些困難,其一,分數(shù)階算子具有非局部記憶性質(zhì),導致分數(shù)階微分方程的數(shù)值求解不穩(wěn)定；其二,分數(shù)階微分方程的數(shù)值求解通常需要求解全系數(shù)線性方程組,一般需要用O(N3)的計算時間和O(N2)的存儲空間,其中N是網(wǎng)格點數(shù)；其三,對于給出的數(shù)值格式進行理論分析比較困難。能否構(gòu)造既穩(wěn)定又節(jié)省計算時間和存儲空間的數(shù)值方法,而且還方便進行數(shù)值格式的理論分析?最大值原理做為整數(shù)階微分方程數(shù)值方法中的經(jīng)典理論分析工具為我們構(gòu)造這樣的格式提供了可能性,它不僅保證了格式的穩(wěn)定性,而且可以用最大值原理對差分格式做一些先驗估計,并且依次可以對差分格式的收斂性、穩(wěn)定性等做進一步的證明；在此基礎上,結(jié)合整數(shù)階微分方程的一些經(jīng)典數(shù)值方法,可以構(gòu)造一些既節(jié)省計算時間、存儲空間又方便理論分析的穩(wěn)定的數(shù)值方法用以求解分數(shù)階偏微分方程。本文主要基于Riemann-Loiuville分數(shù)階導數(shù)定義,以最大值原理為基礎,結(jié)合一階精度的Grunwald公式和移位Grunwald公式,構(gòu)造二階精度的算子近似分數(shù)階導數(shù),再利用隱式歐拉方法和Saul'ev算法,給出了求解一維空間分數(shù)階擴散方程及對流擴散方程的幾種有限差分格式,另外介紹了求解空間分數(shù)階擴散方程的滿足最大值原理的中心差分格式,且對格式的穩(wěn)定性、收斂性等進行了詳盡的理論分析。本論文的主要內(nèi)容及創(chuàng)新點如下：一,介紹了分數(shù)階微分方程數(shù)值解相關(guān)的研究背景、意義、研究問題的提出以及國內(nèi)外研究現(xiàn)狀,并給出了本文需要用到的理論預備知識,即分數(shù)階導數(shù)的幾種常用定義及等價性關(guān)系、性質(zhì)和最大值原理的基本知識。二,提出了滿足最大值原理的二階有限差分算子離散分數(shù)階導數(shù),結(jié)合隱式歐拉公式構(gòu)造了解一維單邊、雙邊空間分數(shù)階擴散方程及一維單邊、雙邊空間分數(shù)階對流擴散方程的二階有限差分格式,并且用最大值原理或者能量不等式法進行了穩(wěn)定性證明和收斂性分析。三,由上一章提出來的滿足最大值原理的二階算子結(jié)合Saul'ev算法,構(gòu)造了一種對稱半隱格式求解空間分數(shù)階擴散方程及空間分數(shù)階對流擴散方程。此格式形式上是隱格式,但是計算過程是顯式的,大大節(jié)省了計算量和存儲量。對此數(shù)值方法的穩(wěn)定性、誤差分析進行了詳盡的分析證明,并通過理論證明和數(shù)值算例均驗證了此半隱格式在l2范數(shù)意義下的誤差估計式為C(△t2h-2(1-α)+△t+h2),其中α是分數(shù)階導數(shù)的階且△t,h分別是時間和空間步長。四,給出一個非整數(shù)節(jié)點上的二階有限差分算子離散分數(shù)階導數(shù)項,并結(jié)合最大值原理,構(gòu)造了解單邊空間分數(shù)階擴散方程的一個定義在非整數(shù)節(jié)點的有限差分格式,并且用最大值原理進行了穩(wěn)定性、收斂性分析。
[Abstract]:In recent decades, due to the fractional derivative can be used to describe the anomalous diffusion phenomenon, and many complex dynamical systems contain anomalous diffusion, the fractional kinetic equations is an effective method to describe the complex system, which makes the fractional partial differential equations will be widely used in science, natural science and society, because of its analytical solution is not easy seek out, so the fractional partial differential equation of numerical solution is particularly important. The numerical solution of fractional differential equations need to overcome some difficulties, a fractional order operator with non local memory properties, resulting in the numerical solution of fractional differential equations is not stable; second, the numerical solution of fractional differential equations usually require calculate the total coefficient of linear equations, the general need to use O (N3) of the O (N2) computing time and storage space, where N is the grid points; thirdly, the numerical lattice is given in Type analysis is difficult. Whether the structure is stable and save the numerical method of computing time and storage space, but also facilitate the analysis of numerical theory? The maximum principle for the classical theory of numerical methods for integer order differential equation analysis tools provide the possibility for us to construct such a format, it not only guarantees the format the stability, but also the principle of difference schemes with some priori estimates of maximum value, and can turn on the convergence of the difference scheme, the stability of further proof; on this basis, some classical numerical method combining integer order differential equations, which can be constructed to save computing time, storage space and convenient stable numerical method of theoretical analysis for solving fractional partial differential equations. In this paper the definition of fractional derivative based on Riemann-Loiuville, with the largest value Based on the principle, and shift Grunwald formula with Grunwald formula of first order accuracy, operator two order accuracy approximation of fractional derivative, and then using the implicit Euler method and Saul'ev algorithm, gives several finite difference for solving one-dimensional space fractional diffusion equations and convection diffusion equation format, also introduces the solution space fractional the diffusion equation satisfies the maximum principle of the central difference scheme and stability of the format, the convergence is analyzed theoretically in detail. The main contents of this paper and innovation are as follows: first, this paper introduces the research background, the related numerical solution of fractional differential equations, put forward the research problem and research at home and abroad the status quo, and the need to use the theory of knowledge, namely different definitions and equivalence relation of fractional order derivative, the nature and basic knowledge of the principle of maximum two, Put forward to meet the two order finite difference maximum principle of discrete fractional derivative operator, combined with implicit Euler formula about one-dimensional unilateral, bilateral space fractional diffusion equation and one dimensional unilateral, bilateral two order finite difference space fractional convection diffusion equation and using the format, analyzed to prove the stability and convergence the principle or method of the maximum energy inequality. Three, proposed by the previous chapter meet the two order operator maximum principle based on Saul'ev algorithm, constructed a symmetric semi implicit scheme for solving the space fractional diffusion equation and space fractional convection diffusion equation. This format is implicit, but the calculation process is explicit, greatly reduces the amount of calculation and storage stability. This numerical method, error analysis carried out a detailed analysis of the proof, and through theoretical proof and numerical examples have verified this half The error implicit in the sense of L2 norm estimation for C (delta t2h-2 (1- alpha) + t+h2), where alpha is the fractional derivative order and delta T, h are the time and space step. Four, two order finite difference gives a non integer node of discrete fractional operator derivative, and combined with the maximum principle, construct a definition of unilateral understanding of space fractional diffusion equation in the non integer node finite difference scheme, and the maximum principle for the stability and convergence analysis.

【學位授予單位】：寧夏大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O241.82
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本文編號：1743940