双层复合介质中时间分数阶热传导正问题及参数估计研究

发布时间：2018-10-31 06:32

【摘要】：近些年,在力学和工程建模中,分数阶微积分理论的应用十分广泛.分数阶微积分能够很好地刻画复杂现象或材料中的记忆性和非局部性等特殊性质.本文主要研究的是两类在双层复合介质中的时间分数阶热传导模型:给出半无限区域内分数阶热传导正问题的解析解;构造有界区域内模型的有限差分格式;讨论时间分数阶传热模型中的参数估计问题.具体地:第一章为预备知识:在第一节中,主要介绍了分数阶微积分的起源及发展史,给出了常用的Riemann-Liouville型和Caputo型分数阶微积分的定义及一些简单性质;在第二小节中,我们介绍了几种特殊函数(Mittag-Leffler函数、Wright函数、Mainardi函数等);第三节则给出了分数阶微积分中一些特殊函数的Laplace变换公式及其相互关系;在最后一小节中则介绍了一些分数阶微积分的典型应用.第二章,我们研究的是一类在双层半无限复合介质的分数阶热传导问题.在前两节中,简要介绍了时间分数阶传热问题的背景,并提出了一类在双层复合介质中时间分数阶热传导问题的数学模型,给出了热传导方程:(?)(?)及初始条件和边界条件.在第三节中给出了该方程的解析解,其解依赖于分数阶阶数α,β的关系:当αβ时在αβ时,解有一定的相似性,同时也给出了α = β这一特殊情况时的解析解表达式.第四节中,研究了分数阶阶数α、β的相关性问题,并利用最小二乘法和共轭梯度法对两参数进行了估计.在最后一节中比较了两种估计方法,并给了该章结论.第三章中,我们研究了在有界双层复合介质中的一类分数阶热传导问题.第一节简要介绍了数学模型.在第二节中,给出了基于有限差分方法得到的数值解.第三节则借助共轭梯度方法和Levenberg-Marquardt方法分别对其中阶数进行了估计.同时给出了参数的灵敏度分析.本章结论则在第四节给出.第四章,我们给出本文总结和未来工作的展望.
[Abstract]:In recent years, fractional calculus theory has been widely used in mechanics and engineering modeling. Fractional calculus can well describe some special properties such as memory and nonlocality in complex phenomena or materials. In this paper, two kinds of time fractional heat conduction models in two-layer composite media are studied: the analytical solution of fractional heat conduction positive problem in semi-infinite region, the finite difference scheme of bounded domain model and the finite difference scheme are given. The problem of parameter estimation in time fractional heat transfer model is discussed. In the first section, the origin and history of fractional calculus are introduced, and the definitions and some simple properties of Riemann-Liouville type and Caputo type fractional calculus are given. In the second section, we introduce several special functions (Mittag-Leffler function, Wright function, Mainardi function, etc.), in the third section, we give the Laplace transformation formulas and their relations of some special functions in fractional calculus. In the last section, some typical applications of fractional calculus are introduced. In chapter 2, we study a class of fractional heat conduction problems in double-layer semi-infinite composite medium. In the first two sections, the background of time fractional heat transfer problem is briefly introduced, and a mathematical model of time fractional heat transfer problem in bilayer composite medium is proposed. The heat conduction equation is given as follows: (?) And initial conditions and boundary conditions. In the third section, the analytic solution of the equation is given. The solution depends on the relation of fractional order 伪, 尾: when 伪尾 is in 伪尾, the solution has certain similarity. At the same time, the analytic solution expression of 伪 = 尾 is also given. In the fourth section, the correlation of fractional order 伪, 尾 is studied, and the two parameters are estimated by least square method and conjugate gradient method. In the last section, two estimation methods are compared and the conclusion is given. In chapter 3, we study a class of fractional heat conduction problems in a bounded bilayer composite medium. The first section briefly introduces the mathematical model. In the second section, the numerical solution based on the finite difference method is given. In the third section, the order is estimated by conjugate gradient method and Levenberg-Marquardt method. The sensitivity analysis of the parameters is also given. The conclusion of this chapter is given in the fourth section. In the fourth chapter, we give the summary of this paper and the prospect of future work.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O411

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