分数阶微分方程理论及其在生化反应中的应用

发布时间：2018-03-18 11:04

本文选题：次扩散　切入点：CTRWs　出处：《中国科学院大学(中国科学院武汉物理与数学研究所)》2017年博士论文　论文类型：学位论文

【摘要】：经典的扩散理论广泛应用自然科学的诸多领域,例如物理、生物、化学等,取得了很大的成就。然而,各种分形、多孔介质等复杂系统上的扩散表明经典的扩散理论不再适用。实验表明,此类扩散往往呈现反常扩散的特征。研究人员发现分数阶微分方程是刻画反常扩散的有效工具。由于分数阶微分算子的非局部性,其数值方法和分析研究(例如解的适定性、正则性等问题)仍处在初步阶段。本文主要研究次扩散条件下的双分子反应、分形中的次扩散方程的数值方法和初步探索分数阶微分方程解的适定性问题,具体如下:第一章,介绍了次扩散和分数阶微积分相关的研究背景和研究内容。第二章,简要介绍了分数阶微积分的基本定义和性质以及相关的特殊函数如Mittag-Leffler 函数和 Fox H 函数。第三章,研究次扩散条件下的双分子可逆反应A+B(?)C。利用CTRWs给出了反应次扩散过程的微观描述,基于CTRWs模型,建立了反应次扩散过程的分数阶微分方程宏观描述。最后,分析了粒子的统计性质和稳态解曲线,得到一类与时间成正比的分数阶矩。第四章,研究了一类分形介质中的分数阶次扩散方程的第一初边值问题。通过分数阶奇异Sturm-Liouville问题给出了其级数形式解,此外,我们建立了分形介质中的次扩散方程隐式数值格式,证明了其稳定性、收敛性。最后,我们定义了关于谱维数ds和次扩散指标dω的鲁棒性指标,通过数值分析,发现谱维数ds对最大绝对误差影响相比较而言更大。第五章第一部分,我们研究如下抽象非线性微分方程0CDtαu(t)= A(t,u)u(t)+ f(t,u(t),Bu(t)),0α1,0tT.的局部和非局部Cauchy问题。运用不动点理论,分别给出了方程经典解和适度解存在的条件,并证明了解对初值的连续依赖性。第五章第二部分,我们研究带有空间Riesz导数的n维分数阶对流扩散方程利用Rothe方法(即所谓的半差方法),我们证明了上述方程弱解的存在性。
[Abstract]:Classical diffusion theory has been widely used in many fields of natural science, such as physics, biology, chemistry and so on. The diffusion on complex systems such as porous media indicates that the classical diffusion theory is no longer applicable. The fractional differential equation is an effective tool for characterizing anomalous diffusion. Due to the nonlocality of fractional differential operators, its numerical methods and analytical studies (such as the suitability of solutions, In this paper, we mainly study the bimolecular reaction under the condition of subdiffusion, the numerical method of the subdiffusion equation in fractal and the problem of the fitness of the solution of fractional differential equation. The main contents are as follows: chapter 1, This paper introduces the research background and contents of subdiffusion and fractional calculus. In chapter 2, the basic definition and properties of fractional calculus and the special functions such as Mittag-Leffler function and Fox H function are briefly introduced. The reversible bimolecular reactions under subdiffusion conditions are studied. The microscopic description of the reaction sub-diffusion process is given by using CTRWs. Based on the CTRWs model, the fractional differential equation macroscopic description of the reaction sub-diffusion process is established. Finally, the statistical properties of the particle and the steady state solution curve are analyzed. In chapter 4th, the first initial-boundary value problem of fractional order diffusion equation in fractal media is studied. The series form solution of the fractional order singular Sturm-Liouville problem is given, in addition, the first initial boundary value problem of fractional order diffusion equation in fractal medium is obtained. In this paper, we establish implicit numerical schemes for subdiffusion equations in fractal media, and prove their stability and convergence. Finally, we define the robustness index of the spectral dimension DS and the secondary diffusion index d 蠅. It is found that the spectral dimension DS has a greater effect on the maximum absolute error than the maximum absolute error. In Chapter 5th, in the first part, we study the local and nonlocal Cauchy problems of the abstract nonlinear differential equation 0CDT 伪 u ~ (t) = a ~ (t) t ~ (t)). The conditions for the existence of classical solution and moderate solution of the equation are given respectively, and the continuous dependence of the solution on the initial value is proved. In this paper, we study the n-dimensional fractional convection-diffusion equations with spatial Riesz derivatives. By using the Rothe method, we prove the existence of weak solutions of the above equations.
【学位授予单位】：中国科学院大学(中国科学院武汉物理与数学研究所)
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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