分数阶扩散方程的几种数值解法
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摘要
尽管分数阶微积分的历史几乎和整数阶的一样长,但是由于缺少相关的实际应用背景,分数阶微积分在其初期发展十分缓慢.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,经典微积分都是一个强有力的工具.但是,越来越多的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,不能用经典的导数模型来描述.因此,在最近的十几年里,分数阶微积分已经被应用于几乎所有科学、工程和数学的领域中去.
物理中,反常扩散或许是一种最常研究的复杂问题.我们利用分数阶导数,可以将经典的整数阶扩散与波的偏微分方程,推广到时间和空间的分数阶上去.进而再扩展到各类非线性方程并给出其初边值问题的解,是近几年来分数阶微积分应用的一个主要领域.一般来讲,这些问题大都具非常重要的实际应用背景,如在分形和多孔介质中的弥散、半导体物理、湍流及凝聚态物理等.
本文主要研究一些分数阶扩散方程及其数值解法,共由四个彼此相关而又相互独立的章节构成.第一章简要介绍了分数阶微积分的历史、理论及其应用,以及文中将用到的一些基本知识,和相关数值解的现有研究成果；第二章和第三章研究的都是双边空间分数阶对流扩散方程,在这两章中我们分别给出了此类方程的几类不同有限差分法,主要有分数阶权平均法、改进型权平均法和特征有限差分法等；而在最后一章中,我们则是给出了时间分数阶扩散方程的一种高精度隐式数值解法.
第一章为序言.首先了介绍分数阶微积分的历史及其发展情况,并给出了几种常用的分数阶算子定义以及它们的一些基本性质,例如：Riemann-Liouville分数阶算子,Caputo分数阶算子和Grunwald-Letnikov分数阶算子等,同时还列出了几个相关的运算性质.然后,在§1.3中,我们对Mittag-Leffler型特殊函数和它的基本性质也进行了一定的叙述,这类特殊函数主要包括单参数的Mittag-Leffler函数和两参数的广义Mittag-Leffler函数.这类特殊函数常常是很多分数阶微分方程的基本解,其它具有类似性质的特殊函数还有Wright函数和H-fox函数,等等.
此外,在本章中的§1.4,我们还归纳叙述了目前为止,几类常见的分数阶微分方程的一些数值解法.例如,有限差分法,有限元法,微分变换法,Adomian区域分解法,变分迭代法,同伦摄动法,等等.同时对每种方法分别列举出了一些相关研究成果.最后,在本章的最后一节中,我们较详细的介绍了分数阶微积分在当前非线性物理复杂系统的各个领域中的应用.
在接下来的章节中,我们将研究两种不同的反常扩散模型.在第二章中,主要研究1维空间分数阶对流扩散方程.我们根据移位Grunwald公式离散Riemann-Liouville分数阶导数,从而提出了方程的分数阶权平均法.通过理论研究和算例分析,可以得知以前出现过的一些相关数值算法,它们大都是此方法的某些特例.在§2.3中,我们利用圆盘定理和矩阵法证明了分数阶权平均法的稳定性,具体理论结果由定理2.1详细给出.
然后,在§2.4中,我们又讨论了分数阶权平均法的一种新的改进格式,并再次给出了相关稳定性分析.最后,则是用数值例子来验证理论的正确性,同时又计算了分数阶权平均法的特例,分数阶Crank-Nicholson (FCN)法.显然,无条件稳定又拥有2阶时间精确度的FCN法更好一些.本章部分内容已经公开发表在Physics Letters A.
在第三章中,关于双边空间分数阶对流扩散方程,据我们所知,目前为止,它的数值解法全都是Eulerian法.结果,这些方法都具有和2阶对流扩散方程相同的数值局限性.在本章中,结合移位Griinwald-Letnikov有限差分过程以及Lagrangian法,我们在§3.3中首次提出了一种分数阶特征有限差分法(CFDM).此法保留了2阶对流扩散方程特征法和分数阶对流扩散方程有限差分法的所有数值优点.在§3.4中,我们证明了这种方法是无条件稳定、相容和收敛的,并且给出了本方法误差估计的最大值.
在§3.5中,我们给出了一个实际算例的数值模拟,并把分数阶特征有限差分法和其它的分数阶标准差分法相比较.算例结果表明,这种分数阶新CFDM在精度和稳定性上都大大优于其它已知方法,例如显式迎风差分法和隐式迎风差分法等.并且,此法对于对流占优问题,显得尤为高效、优越.本章内容已投到Journal of Computational Physics.
在第四章中,我们主要讨论的是一类时间分数阶扩散方程.反常次扩散运动是复杂系统中一个特别重要的内容,如在一些有机和无序材料中,它的运动路径被一些几何或能量因子约束着.对于反常次扩散随机游走过程的数学模型,一般扩散方程则会被Riemann-Liouville分数阶时间扩散方程所替代.分析表明,这些分数阶模型显然比经典的整数阶模型更加符合实际背景.
在本章中,首先我们我们利用移位Grunwald公式来逼近时间分数阶导数,并且使用中心差分格式去逼近1阶时间导数和2阶空间导数,从而提出了此类扩散方程的一种新的隐式差分法.它是一种三层差分格式,其中第一时间层的数值解可以由全隐式格式或Crank-Nicholson格式给出,这两种格式都是无条件稳定的.接着,我们利用广义化Fourier-Von Neumann分析法,证明了这种新方法的无条件稳定性,并导出此法关于时间的2阶精确度.最后,则是用数值试验和对比法,来验证和观察本算法的性质和特征.本章内容已投到Applied Mathematics and Computation.
Despite the history of fractional calculus is almost as long as integer-order cal-culus. However, because of lack of application background, fractional calculus was developed very slowly. It is known that the classical calculus provides a powerful tool for explaining and modeling important dynamic processes in many areas of the applied sciences. But experiments and reality teach us that there are many complex systems in nature with anomalous dynamics, which can not be characterized by classical deriva-tive models. Therefore, during the last decade fractional calculus has been applied to almost every field of science, engineering, and mathematics.
Anomalous diffusion is perhaps the most frequently studied complex problem. Classical partial differential equation of diffusion and wave has been extended to the equation with fractional time and/or space by means of fractional operator. Fur-thermore, it has been extended to the problems of every kind of nonlinear fractional differential equation. And to present the solutions to the problems of initial and bound-ary values subject to above equations is another rapidly developing field of fractional operator applications. In general, all of these equations have important background of practice applications, such as dispersion in fractals and porous media, semiconductor physics, turbulence and condensed matter physics.
The paper focuses on some fractional diffusion equations and their numerical meth-ods. It is composed of four chapters, which are independent and correlative to one another. The first chapter contains a brief introduction to fractional calculus and some elementary knowledge. The second and third chapters deal with the space-fractional advection-diffusion equations by some different finite difference methods, such as, the fractional weight average method and the characteristic finite difference method, etc. In the last chapter, we propose a new implicit numerical solution for the time-fractional diffusion equation.
The chapter 1 is introduction. Firstly, the history and the development of the fractional calculus and its applications is introduced. We also introduce some differ-ent kinds of fractional operators, such as Riemann-Liouville, Caputo and Griinwald-Letnikov fractional operators, and so on.At the same time, there are some properties of the fractional operators, too. Next, we present the Mittag-Leffler function, which usually contains Mittag-Leffler function and generalized Mittag-Leffler function in two parameters, etc. It is the elementary solution of many fractional differential equations. Similarly, there are still two different special functions, Wright function and H-fox function.
Additionally, in§1.4, we include some numerical methods relative researches of fractional differential equations, which are known so far. Such as, finite difference method, finite element method, differential transform method, Adomian decomposition method, variational iteration method, homotopy perturbation method, etc. In the last section of the chapter, we introduce in details the applications of the fractional calculus in almost every field of nonlinear complex physical systems.
In the following chapters, two different models of abnormal diffusion are studied. In chapter 2, we study the one-dimensional space fractional advection-diffusion equa-tion. Based on the shifted Grunwald approximation to the Riemann-Liouville fractional derivative, we propose the fractional weight average (FWA) method in this chapter. we can see that some results obtained previously are special cases of the method. In§2.3, stability of the FWA method is proved by Gerschgorin theorem and matrix analysis, and the conclusion is given in the form of Theorem 2.1.
Then, we also discuss a new improved FWA scheme and its stability in§2.4. At last, some numerical examples are carried out to confirm our theory. At the same time, as a special case of the FWA method, the fractional Crank-Nicholson (FCN) method is much better, which is not only unconditionally stable, but also second-order accurate in time. The result of this chapter has been published on Physics Letters A.
In chapter 3, to the best knowledge of the authors, the numerical methods devel-oped so far for two-sided space fractional advection-diffusion equations are all Eulerian methods. Consequently, these methods suffer from the same numerical limitations as their analogue for second-order advection-diffusion equations. In§3.3 of this chap-ter, we firstly develop a characteristic finite difference method (CFDM) for fractional advection-diffusion equations, by combining the shifted Griinwald-Letnikov fractional finite difference procedures with the Lagrangian treatment. The proposed method re- tains all the numerical advantages of characteristic methods for second-order advection-diffusion equations and the finite difference methods for fractional advection-diffusion equations. Then we prove that this method is unconditionally stability, consistent and convergence in§3.4. The maximum error estimate is derived, too.
Numerical solutions and exact solutions of a special fractional diffusion problem are shown in§3.5 and a comparison between the fractional CFDM and the standard finite difference methods (SFDM) is given. Finally, we draw our conclusions in§3.6. Obviously, the fractional CFDM is better in both the accuracy and the stability than the known fractional SFDM, included the fractional explicit upwind finite difference method and the fractional implicit upwind finite difference method. Especially for the convection-dominated problems, this new method is very efficient and superior. The result of this chapter has been submitted to Journal of Computational Physics.
In chapter 4, we study the time fractional diffusion equation. Subdiffusive mo-tion is particularly important in the context of complex systems such as glassy and disordered materials, in which pathways are constrained for geometric or energetic rea-sons. For anomalous subdiffusive random walkers, the continuum description via the ordinary diffusion equation is replaced by the Riemann-Liouville fractional diffusion equation. This fractional-order model tends to be more appropriate than the tradi-tional integer-order model.
In this chapter, firstly, we propose a new kind of implicit difference method based on the shifted Grunwald approximation to the time fractional derivative, and use the central difference scheme for the time first-order derivative and space second-order derivative, which is a three-layer difference scheme. The numerical solution of the first time layer can be calculated by the fully implicit scheme or the Crank-Nicholson scheme, which are both unconditionally stable. Next, stability and accuracy are dis-cussed by means of the generalized Fourier-Von Neumann method. Finally, numerical solutions and exact analytical solutions of a typical fractional diffusion problem are compared. The result of this chapter has been submitted to Applied Mathematics and Computation.
物理中,反常扩散或许是一种最常研究的复杂问题.我们利用分数阶导数,可以将经典的整数阶扩散与波的偏微分方程,推广到时间和空间的分数阶上去.进而再扩展到各类非线性方程并给出其初边值问题的解,是近几年来分数阶微积分应用的一个主要领域.一般来讲,这些问题大都具非常重要的实际应用背景,如在分形和多孔介质中的弥散、半导体物理、湍流及凝聚态物理等.
本文主要研究一些分数阶扩散方程及其数值解法,共由四个彼此相关而又相互独立的章节构成.第一章简要介绍了分数阶微积分的历史、理论及其应用,以及文中将用到的一些基本知识,和相关数值解的现有研究成果；第二章和第三章研究的都是双边空间分数阶对流扩散方程,在这两章中我们分别给出了此类方程的几类不同有限差分法,主要有分数阶权平均法、改进型权平均法和特征有限差分法等；而在最后一章中,我们则是给出了时间分数阶扩散方程的一种高精度隐式数值解法.
第一章为序言.首先了介绍分数阶微积分的历史及其发展情况,并给出了几种常用的分数阶算子定义以及它们的一些基本性质,例如：Riemann-Liouville分数阶算子,Caputo分数阶算子和Grunwald-Letnikov分数阶算子等,同时还列出了几个相关的运算性质.然后,在§1.3中,我们对Mittag-Leffler型特殊函数和它的基本性质也进行了一定的叙述,这类特殊函数主要包括单参数的Mittag-Leffler函数和两参数的广义Mittag-Leffler函数.这类特殊函数常常是很多分数阶微分方程的基本解,其它具有类似性质的特殊函数还有Wright函数和H-fox函数,等等.
此外,在本章中的§1.4,我们还归纳叙述了目前为止,几类常见的分数阶微分方程的一些数值解法.例如,有限差分法,有限元法,微分变换法,Adomian区域分解法,变分迭代法,同伦摄动法,等等.同时对每种方法分别列举出了一些相关研究成果.最后,在本章的最后一节中,我们较详细的介绍了分数阶微积分在当前非线性物理复杂系统的各个领域中的应用.
在接下来的章节中,我们将研究两种不同的反常扩散模型.在第二章中,主要研究1维空间分数阶对流扩散方程.我们根据移位Grunwald公式离散Riemann-Liouville分数阶导数,从而提出了方程的分数阶权平均法.通过理论研究和算例分析,可以得知以前出现过的一些相关数值算法,它们大都是此方法的某些特例.在§2.3中,我们利用圆盘定理和矩阵法证明了分数阶权平均法的稳定性,具体理论结果由定理2.1详细给出.
然后,在§2.4中,我们又讨论了分数阶权平均法的一种新的改进格式,并再次给出了相关稳定性分析.最后,则是用数值例子来验证理论的正确性,同时又计算了分数阶权平均法的特例,分数阶Crank-Nicholson (FCN)法.显然,无条件稳定又拥有2阶时间精确度的FCN法更好一些.本章部分内容已经公开发表在Physics Letters A.
在第三章中,关于双边空间分数阶对流扩散方程,据我们所知,目前为止,它的数值解法全都是Eulerian法.结果,这些方法都具有和2阶对流扩散方程相同的数值局限性.在本章中,结合移位Griinwald-Letnikov有限差分过程以及Lagrangian法,我们在§3.3中首次提出了一种分数阶特征有限差分法(CFDM).此法保留了2阶对流扩散方程特征法和分数阶对流扩散方程有限差分法的所有数值优点.在§3.4中,我们证明了这种方法是无条件稳定、相容和收敛的,并且给出了本方法误差估计的最大值.
在§3.5中,我们给出了一个实际算例的数值模拟,并把分数阶特征有限差分法和其它的分数阶标准差分法相比较.算例结果表明,这种分数阶新CFDM在精度和稳定性上都大大优于其它已知方法,例如显式迎风差分法和隐式迎风差分法等.并且,此法对于对流占优问题,显得尤为高效、优越.本章内容已投到Journal of Computational Physics.
在第四章中,我们主要讨论的是一类时间分数阶扩散方程.反常次扩散运动是复杂系统中一个特别重要的内容,如在一些有机和无序材料中,它的运动路径被一些几何或能量因子约束着.对于反常次扩散随机游走过程的数学模型,一般扩散方程则会被Riemann-Liouville分数阶时间扩散方程所替代.分析表明,这些分数阶模型显然比经典的整数阶模型更加符合实际背景.
在本章中,首先我们我们利用移位Grunwald公式来逼近时间分数阶导数,并且使用中心差分格式去逼近1阶时间导数和2阶空间导数,从而提出了此类扩散方程的一种新的隐式差分法.它是一种三层差分格式,其中第一时间层的数值解可以由全隐式格式或Crank-Nicholson格式给出,这两种格式都是无条件稳定的.接着,我们利用广义化Fourier-Von Neumann分析法,证明了这种新方法的无条件稳定性,并导出此法关于时间的2阶精确度.最后,则是用数值试验和对比法,来验证和观察本算法的性质和特征.本章内容已投到Applied Mathematics and Computation.
Despite the history of fractional calculus is almost as long as integer-order cal-culus. However, because of lack of application background, fractional calculus was developed very slowly. It is known that the classical calculus provides a powerful tool for explaining and modeling important dynamic processes in many areas of the applied sciences. But experiments and reality teach us that there are many complex systems in nature with anomalous dynamics, which can not be characterized by classical deriva-tive models. Therefore, during the last decade fractional calculus has been applied to almost every field of science, engineering, and mathematics.
Anomalous diffusion is perhaps the most frequently studied complex problem. Classical partial differential equation of diffusion and wave has been extended to the equation with fractional time and/or space by means of fractional operator. Fur-thermore, it has been extended to the problems of every kind of nonlinear fractional differential equation. And to present the solutions to the problems of initial and bound-ary values subject to above equations is another rapidly developing field of fractional operator applications. In general, all of these equations have important background of practice applications, such as dispersion in fractals and porous media, semiconductor physics, turbulence and condensed matter physics.
The paper focuses on some fractional diffusion equations and their numerical meth-ods. It is composed of four chapters, which are independent and correlative to one another. The first chapter contains a brief introduction to fractional calculus and some elementary knowledge. The second and third chapters deal with the space-fractional advection-diffusion equations by some different finite difference methods, such as, the fractional weight average method and the characteristic finite difference method, etc. In the last chapter, we propose a new implicit numerical solution for the time-fractional diffusion equation.
The chapter 1 is introduction. Firstly, the history and the development of the fractional calculus and its applications is introduced. We also introduce some differ-ent kinds of fractional operators, such as Riemann-Liouville, Caputo and Griinwald-Letnikov fractional operators, and so on.At the same time, there are some properties of the fractional operators, too. Next, we present the Mittag-Leffler function, which usually contains Mittag-Leffler function and generalized Mittag-Leffler function in two parameters, etc. It is the elementary solution of many fractional differential equations. Similarly, there are still two different special functions, Wright function and H-fox function.
Additionally, in§1.4, we include some numerical methods relative researches of fractional differential equations, which are known so far. Such as, finite difference method, finite element method, differential transform method, Adomian decomposition method, variational iteration method, homotopy perturbation method, etc. In the last section of the chapter, we introduce in details the applications of the fractional calculus in almost every field of nonlinear complex physical systems.
In the following chapters, two different models of abnormal diffusion are studied. In chapter 2, we study the one-dimensional space fractional advection-diffusion equa-tion. Based on the shifted Grunwald approximation to the Riemann-Liouville fractional derivative, we propose the fractional weight average (FWA) method in this chapter. we can see that some results obtained previously are special cases of the method. In§2.3, stability of the FWA method is proved by Gerschgorin theorem and matrix analysis, and the conclusion is given in the form of Theorem 2.1.
Then, we also discuss a new improved FWA scheme and its stability in§2.4. At last, some numerical examples are carried out to confirm our theory. At the same time, as a special case of the FWA method, the fractional Crank-Nicholson (FCN) method is much better, which is not only unconditionally stable, but also second-order accurate in time. The result of this chapter has been published on Physics Letters A.
In chapter 3, to the best knowledge of the authors, the numerical methods devel-oped so far for two-sided space fractional advection-diffusion equations are all Eulerian methods. Consequently, these methods suffer from the same numerical limitations as their analogue for second-order advection-diffusion equations. In§3.3 of this chap-ter, we firstly develop a characteristic finite difference method (CFDM) for fractional advection-diffusion equations, by combining the shifted Griinwald-Letnikov fractional finite difference procedures with the Lagrangian treatment. The proposed method re- tains all the numerical advantages of characteristic methods for second-order advection-diffusion equations and the finite difference methods for fractional advection-diffusion equations. Then we prove that this method is unconditionally stability, consistent and convergence in§3.4. The maximum error estimate is derived, too.
Numerical solutions and exact solutions of a special fractional diffusion problem are shown in§3.5 and a comparison between the fractional CFDM and the standard finite difference methods (SFDM) is given. Finally, we draw our conclusions in§3.6. Obviously, the fractional CFDM is better in both the accuracy and the stability than the known fractional SFDM, included the fractional explicit upwind finite difference method and the fractional implicit upwind finite difference method. Especially for the convection-dominated problems, this new method is very efficient and superior. The result of this chapter has been submitted to Journal of Computational Physics.
In chapter 4, we study the time fractional diffusion equation. Subdiffusive mo-tion is particularly important in the context of complex systems such as glassy and disordered materials, in which pathways are constrained for geometric or energetic rea-sons. For anomalous subdiffusive random walkers, the continuum description via the ordinary diffusion equation is replaced by the Riemann-Liouville fractional diffusion equation. This fractional-order model tends to be more appropriate than the tradi-tional integer-order model.
In this chapter, firstly, we propose a new kind of implicit difference method based on the shifted Grunwald approximation to the time fractional derivative, and use the central difference scheme for the time first-order derivative and space second-order derivative, which is a three-layer difference scheme. The numerical solution of the first time layer can be calculated by the fully implicit scheme or the Crank-Nicholson scheme, which are both unconditionally stable. Next, stability and accuracy are dis-cussed by means of the generalized Fourier-Von Neumann method. Finally, numerical solutions and exact analytical solutions of a typical fractional diffusion problem are compared. The result of this chapter has been submitted to Applied Mathematics and Computation.
引文
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