分数阶奇异扩散方程的几种解法及其应用
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摘要
分数阶微积分是传统的整数阶微积分概念的推广，尽管它的历史几乎和整数阶微积分一样长，但是由于缺乏相关应用，在其开始阶段发展十分缓慢.众所周知.对于解释和模拟许多应用科学领域的动力学过程，经典的微积分都是一个强有力的工具，但是实验和现实告诉我们.在自然界的反常动力学中有许多复杂系统无法用经典的整数阶微积分模型来刻画.最近，研究者们发现分数阶微积分算子与整数阶微积分算子不同，具有非局部性，非常适合用来描述现实世界中具有遗传性质的材料及随机模型.由此，分数阶微积分迎来了其蓬勃发展阶段.如今，它在物理.数学，机械工程，生物，电子工程，控制理论和金融等领域发挥越来越重要的作用.
由于分数阶微积分的非局部性，应用分数阶微积分理论将经典的整数阶扩散方程与波动方程推广到时间和空间的分数阶情形，建立分数阶扩散方程，这种方程能够很好地解释物理中的奇异扩散现象，因此分数阶微分方程成为描述奇异扩散模型的强有力工具,如粘弹性材料的本构关系,多孔介质渗流现象及分形媒介中的随机游走模型.从现实问题中抽象出分数阶微分方程之后，研究如何求解这类问题成为近几年分数阶微分方程应用的一个重要领域.
本文主要由以下几个部分组成：
第一章,我们概要地介绍了分数阶奇异扩散方程有关问题的历史发展、研究进展及前人在处理这类方程时所用到的方法,从而引出了本文所要解决的问题.同时陈述了本文的主要结果.
第二章,我们介绍了分数阶微积分的一些预备知识,给出了几种常用的分数阶微积分定义以及它们的一些基本性质,例如：Riemann-Liouville型分数阶微积分,Caputo型分数阶微积分和Grunwald-Letnikov型分数阶微积分等,同时还列出他们之间的关系.接着介绍了在求解分数阶微分方程时常用的两个函数Mittag-Leffler函数和Fox-H函数及它们的一些性质.最后介绍了连续时间随机游走模型与分数阶微分方程之间的关系.
第三章,我们介绍了一类分数阶微分方程解析解求法,运用Laplace变换,Fourier变换,Mellin变换及Green函数法,求得一类具有源汇项的广义分数阶奇异扩散方程的解析解.其中方程的扩散项、外力项、及源汇项具有不同阶的时间分数阶导数.我们发现这类方程的解与正态分布相比具有尖峰厚尾性质.
第四章，我们证明了由Jurnarie提出的分数阶泰勒公式，为求解分数阶微分方程的数值解及精度估计提供了非常有利的工具.
第五章，我们研究分数阶奇异扩散方程的随机表示.首先证明了用来描述非均匀土壤溶质运移模型的修正对流-弥散方程的随机表示是一个复合随机过程.其中次级过程为一个Levy过程的首达时，而主过程是由布朗运动驱动的带外力项的随机过程.然后，我们将由布朗运动驱动的随机过程推广到由Levy过程驱动的情形.证明了该随机过程是空间分数阶微分方程的随机表示.接着，我们部分解决了一个由Magdziarz在其文章中提出的问题，即给出了外力项和扩散项都与时间、空间都有关的的分数阶Fokker-Planck方程的随机表示.最后，利用随机表示，构建模拟算法，画出了这些分数阶微分方程解的图像.
第六章,由于与正态分布相比,分数阶扩散方程的解具有尖峰厚尾性，因此我们假设股票价格服从次扩散的几何布朗运动，在此基础上得到了期权定价公式，其价格高于经典Black-Scholes模型得到的价格，最后反推出“微笑的”隐含波动率.所得到的结果不仅比经典Black-Scholes的期权定价模型更符合实际历史数据，也很好地解释了一些经济现象.
Fractional calculus is an extension of the classical integer-order calculus. Just like the classical integer-order calculus, fractional calculus has a long history. How-ever, because of lack of application, fractional calculus developed slowly at its lie-ginning. Fortunately. It is known that the integer-order calculus is a powerful tool to describe the dynamic processes in applied science, but. experiments and reality teach us that there are many complex systems in nature with anomalous dynamics which can not be characterized by classical integer-order derivative model. In re-cent years, the fractional operator is different from the integer-order one. The former one is non-local, and has memory effect, which makes it be suitable to describe the hereditary viscoelastic materials and stochastic models with memory. After that, the fractional calculus developed very quickly. Now, fractional calculus plays an im-portant role in Physics, Mathematics, Mechanical Engineering, Biology. Electrical Engineering, Control theory and Finance, and so on.
Since the fractional calculus is non-local, then, by applying fractional calculus to classical diffusion equation and wave equation, a fractional differential equation with time and space fractional derivatives can be obtained, which can be used to describe the anomalous diffusion in Physics. So, the fractional calculus is becoming a useful tools to describe anomalous diffusion phenomenon, Such as constitutive relationships of viscoelastic materials, porous media seepage and random walk in fractal media. After getting the fractional differential equation, how to get its solution attracts many researchers. It also becomes a promising area.
This dissertation is divided into six chapters:
In Chapter1, we introduce; background of fractional anomalous diffusion equa-tion and the methods employed to deal with relevant problem, then present our questions and main results.
In Chapter2, we present the background of the fractional calculus and the relevant knowledge, including some definitions and properties of fractional calculus,such as Riemann-Liouville fractional calculus. Caputo fractional calculus and Grumuald-Letnikov fractional calculus. Some relations between these definitions are also listed. Then, we introduce two important functions for solving the fractional differential equation:Mittag-Lefner function and Fox-H function, as well as some of their properties. At last, we also give the relations between the continuous time random walk model and fractional differential equation.
In Chapter3, By using Laplace transform. Fourier transform. Mellin trans-form and Green function method, we get the solution of a generalized fractional differential equation with absorbent term, where the diffusion term, external term and absorbent term have different order of fractional derivatives. We find that its solution has heavier tail and higher peak, in contrast to Normal distribution.
In Chapter4, we prove the validity of the fractional Taylor's Formula pro-posed by Jumarie, which is an useful tool to get the numerical solution of fractional differential equations and its accuracy.
In Chapter5, we study the stochastic representation of the fractional differential equation. First, the stochastic representation of a modified advection dispersion equation is proved to be a subordinated process, where the parent process is a classical diffusion process driven by Brownian motion, and the subordinator is the inverse of a Levy motion, whose characteristic function is dependent on the function presented in the convolution. Then we extend the parent process to the one driven by Levy motion. A spacial fractional differential equation is obtained. After that, we also answer the question proposed by Magdziarz in his paper, which means we obtain the stochastic representation of a fractional Fokker-Planck equation with time and space dependent drift and diffusion. At last, taking advantage of stochastic representation, we picture the solution of these fractional differential equation.
In Chapter6, Comparing with the Normal distribution, the solution of frac- tional diffusion equation follows a stretched Gaussion distribution. i.e. has heavier tail and higher peak. So.we suppose that the stock price follows the stochastic pro-cess obtained in Chapter4.then, we obtain the option pricing formula. The "smirk" volatility is also obtained. The results got in this chapter extend the Black-Scholes Model, and more close to historical data.
由于分数阶微积分的非局部性，应用分数阶微积分理论将经典的整数阶扩散方程与波动方程推广到时间和空间的分数阶情形，建立分数阶扩散方程，这种方程能够很好地解释物理中的奇异扩散现象，因此分数阶微分方程成为描述奇异扩散模型的强有力工具,如粘弹性材料的本构关系,多孔介质渗流现象及分形媒介中的随机游走模型.从现实问题中抽象出分数阶微分方程之后，研究如何求解这类问题成为近几年分数阶微分方程应用的一个重要领域.
本文主要由以下几个部分组成：
第一章,我们概要地介绍了分数阶奇异扩散方程有关问题的历史发展、研究进展及前人在处理这类方程时所用到的方法,从而引出了本文所要解决的问题.同时陈述了本文的主要结果.
第二章,我们介绍了分数阶微积分的一些预备知识,给出了几种常用的分数阶微积分定义以及它们的一些基本性质,例如：Riemann-Liouville型分数阶微积分,Caputo型分数阶微积分和Grunwald-Letnikov型分数阶微积分等,同时还列出他们之间的关系.接着介绍了在求解分数阶微分方程时常用的两个函数Mittag-Leffler函数和Fox-H函数及它们的一些性质.最后介绍了连续时间随机游走模型与分数阶微分方程之间的关系.
第三章,我们介绍了一类分数阶微分方程解析解求法,运用Laplace变换,Fourier变换,Mellin变换及Green函数法,求得一类具有源汇项的广义分数阶奇异扩散方程的解析解.其中方程的扩散项、外力项、及源汇项具有不同阶的时间分数阶导数.我们发现这类方程的解与正态分布相比具有尖峰厚尾性质.
第四章，我们证明了由Jurnarie提出的分数阶泰勒公式，为求解分数阶微分方程的数值解及精度估计提供了非常有利的工具.
第五章，我们研究分数阶奇异扩散方程的随机表示.首先证明了用来描述非均匀土壤溶质运移模型的修正对流-弥散方程的随机表示是一个复合随机过程.其中次级过程为一个Levy过程的首达时，而主过程是由布朗运动驱动的带外力项的随机过程.然后，我们将由布朗运动驱动的随机过程推广到由Levy过程驱动的情形.证明了该随机过程是空间分数阶微分方程的随机表示.接着，我们部分解决了一个由Magdziarz在其文章中提出的问题，即给出了外力项和扩散项都与时间、空间都有关的的分数阶Fokker-Planck方程的随机表示.最后，利用随机表示，构建模拟算法，画出了这些分数阶微分方程解的图像.
第六章,由于与正态分布相比,分数阶扩散方程的解具有尖峰厚尾性，因此我们假设股票价格服从次扩散的几何布朗运动，在此基础上得到了期权定价公式，其价格高于经典Black-Scholes模型得到的价格，最后反推出“微笑的”隐含波动率.所得到的结果不仅比经典Black-Scholes的期权定价模型更符合实际历史数据，也很好地解释了一些经济现象.
Fractional calculus is an extension of the classical integer-order calculus. Just like the classical integer-order calculus, fractional calculus has a long history. How-ever, because of lack of application, fractional calculus developed slowly at its lie-ginning. Fortunately. It is known that the integer-order calculus is a powerful tool to describe the dynamic processes in applied science, but. experiments and reality teach us that there are many complex systems in nature with anomalous dynamics which can not be characterized by classical integer-order derivative model. In re-cent years, the fractional operator is different from the integer-order one. The former one is non-local, and has memory effect, which makes it be suitable to describe the hereditary viscoelastic materials and stochastic models with memory. After that, the fractional calculus developed very quickly. Now, fractional calculus plays an im-portant role in Physics, Mathematics, Mechanical Engineering, Biology. Electrical Engineering, Control theory and Finance, and so on.
Since the fractional calculus is non-local, then, by applying fractional calculus to classical diffusion equation and wave equation, a fractional differential equation with time and space fractional derivatives can be obtained, which can be used to describe the anomalous diffusion in Physics. So, the fractional calculus is becoming a useful tools to describe anomalous diffusion phenomenon, Such as constitutive relationships of viscoelastic materials, porous media seepage and random walk in fractal media. After getting the fractional differential equation, how to get its solution attracts many researchers. It also becomes a promising area.
This dissertation is divided into six chapters:
In Chapter1, we introduce; background of fractional anomalous diffusion equa-tion and the methods employed to deal with relevant problem, then present our questions and main results.
In Chapter2, we present the background of the fractional calculus and the relevant knowledge, including some definitions and properties of fractional calculus,such as Riemann-Liouville fractional calculus. Caputo fractional calculus and Grumuald-Letnikov fractional calculus. Some relations between these definitions are also listed. Then, we introduce two important functions for solving the fractional differential equation:Mittag-Lefner function and Fox-H function, as well as some of their properties. At last, we also give the relations between the continuous time random walk model and fractional differential equation.
In Chapter3, By using Laplace transform. Fourier transform. Mellin trans-form and Green function method, we get the solution of a generalized fractional differential equation with absorbent term, where the diffusion term, external term and absorbent term have different order of fractional derivatives. We find that its solution has heavier tail and higher peak, in contrast to Normal distribution.
In Chapter4, we prove the validity of the fractional Taylor's Formula pro-posed by Jumarie, which is an useful tool to get the numerical solution of fractional differential equations and its accuracy.
In Chapter5, we study the stochastic representation of the fractional differential equation. First, the stochastic representation of a modified advection dispersion equation is proved to be a subordinated process, where the parent process is a classical diffusion process driven by Brownian motion, and the subordinator is the inverse of a Levy motion, whose characteristic function is dependent on the function presented in the convolution. Then we extend the parent process to the one driven by Levy motion. A spacial fractional differential equation is obtained. After that, we also answer the question proposed by Magdziarz in his paper, which means we obtain the stochastic representation of a fractional Fokker-Planck equation with time and space dependent drift and diffusion. At last, taking advantage of stochastic representation, we picture the solution of these fractional differential equation.
In Chapter6, Comparing with the Normal distribution, the solution of frac- tional diffusion equation follows a stretched Gaussion distribution. i.e. has heavier tail and higher peak. So.we suppose that the stock price follows the stochastic pro-cess obtained in Chapter4.then, we obtain the option pricing formula. The "smirk" volatility is also obtained. The results got in this chapter extend the Black-Scholes Model, and more close to historical data.
引文
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