时间分数阶偏微分方程的解及其应用
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摘要
分数阶微积分是专门研究任意阶积分和微分的数学性质及其应用的领域,是传统的整数阶微积分的推广,分数阶微分方程是含有非整数阶导数的方程。近几十年里,研究者们发现分数阶微分方程非常适合用来描述现实生活中具有记忆和遗传特性的问题,如:分形和多孔介质中的弥散,电容理论,电解化学,半导体物理、湍流、凝聚态物理,粘弹性系统,生物数学及统计力学等等,因此研究这类方程的性质和数值解法有现实的理论和应用意义。
本文主要讨论一类时间分数阶空间二阶偏微分方程,讨论其解析解,数值解。
第一章,给出本论文的研究背景和意义,总结了前人所做的工作,并叙述分数阶微积分的概念和分数阶微积分一些基本定义和性质,详列本论文的研究内容和结构。
第二章,从随机游走和一种随机过程的稳定分布推导出第三章所讨论的反常次扩散方程和第四章所讨论的时间分数阶扩散方程。
第三章,讨论非齐次时间分数阶反常次扩散方程的解析解,利用分离变量方法和Laplace变换分别导出在Dirichlet,Neumann和Robin三种边界条件下的非齐次反常次扩散方程的解析解,这些解析解以Mittag-Lettter函数的形式给出。本章最后说明这个技巧可以推广到其它类型的反常次扩散方程中。
第四章,考虑时间分数阶扩散方程的数值解。利用关于时间的有限差分格式及空间的Legendre谱方法构造了一种高阶稳定格式,并给出了此方法的稳定性与收敛性分析,证明了该全离散格式是无条件稳定的,其收敛阶为O(△t~(2-α)-N~(-m)),这里△t,N和m分别为时间步长,多项式阶数和精确解的正则度。这是目前已知的最好估计。最后的数值实验结果说明了理论分析的正确性。
在第五章,我们将第四章中数值求解时间分数阶扩散方程的方法推广到生物细胞学中研究离子运动的时间分数阶Cable方程,同样利用关于时间的有限差分格式及空间的谱方法构造一种高阶格式,利用测试的数值例子说明了我们方法的可行性。并用具体例子说明其应用。
第六章,讨论有限区间上具有初边值条件的非线性时间分数阶Fokker-Plabck方程,利用隐式差分方法构造离散格式,并用能量方法证明了所提出的差分格式的收敛性和稳定性。最后给出数值例子。
Fractional calculus is a branch of studying the property of any order integral or derivative.Fractional order differential equation is the equation containing the noninteger order derivative,raising from the standard differential equations by replacing the integer-order derivatives with fractional-order derivatives.Its application is very broad, many researchers find that the fractional differential equations more precisely describe the property of some materials with memory and heredity.Fractional order differential equations are playing an increasingly important role in engineering,physics and other fields,such as the fractal theory and the diffusion in porous media,fractional capacitance theory,electrolysis chemical,fractional biological neurons,condensate physics,vibration control of viscoelastic system,statistical mechanics and so on.
In this paper,we mainly consider the time-fractional anomalous diffusion equation, discuss its analytic solution,numerical solution and its application.
In Chapter 1,the developmental history of fractional calculus and the existing work about fractional calculus are reviewed.We also recall some definitions and properties of the fractional derivatives used in this paper.
In Chapter 2,two time-fractional anomalous diffusion equations are deduced from the random walk and a stable law.These two equations will be investigated numerically in the next two chapters.
In Chapter 3,the solution of time fractional anomalous diffusion equation is discussed. Using separation of variable methods and Laplace transform,the analytical solutions of a non-homogeneous anomalous sub-diffusion equation with Dirichlet,Neumann and Robin boundary conditions are derived respectively.The solution is expressed in terms of the Mittag-Leffler function.These techniques can be applied to solve other kinds of anomalous diffusion problems.
In Chapter 4,we consider a time fractional anomalous diffusion equation on a finite domain.We propose an efficient finite difference/spectral method to solve the time fractional diffusion equation.Stability and convergence of the method are rigourously established.We prove that the full discretization is unconditionally stable,and the numerical solution converges to the exact one with order O(△t~(2-α)+N~(-m)),where△t,N and m are the time step size,polynomial degree,and regularity of the exact solution respectively.Numerical experiments are carried out to support the theoretical claims.
In Chapter 5,we generalize the method that we have proposed in the Chapter 4 to the time fractional Cable equation for modeling neuronal dynamics.Numerical results are presented to show the applicability of the method.
In Chapter 6,we discuss one class of nonlinear time fractional Fokker-Planck equation with initial-boundary value on a finite domain.The stability and convergence of a finite difference method are discussed by energy methods.A numerical example is presented to compare with the exact analytical solution.
本文主要讨论一类时间分数阶空间二阶偏微分方程,讨论其解析解,数值解。
第一章,给出本论文的研究背景和意义,总结了前人所做的工作,并叙述分数阶微积分的概念和分数阶微积分一些基本定义和性质,详列本论文的研究内容和结构。
第二章,从随机游走和一种随机过程的稳定分布推导出第三章所讨论的反常次扩散方程和第四章所讨论的时间分数阶扩散方程。
第三章,讨论非齐次时间分数阶反常次扩散方程的解析解,利用分离变量方法和Laplace变换分别导出在Dirichlet,Neumann和Robin三种边界条件下的非齐次反常次扩散方程的解析解,这些解析解以Mittag-Lettter函数的形式给出。本章最后说明这个技巧可以推广到其它类型的反常次扩散方程中。
第四章,考虑时间分数阶扩散方程的数值解。利用关于时间的有限差分格式及空间的Legendre谱方法构造了一种高阶稳定格式,并给出了此方法的稳定性与收敛性分析,证明了该全离散格式是无条件稳定的,其收敛阶为O(△t~(2-α)-N~(-m)),这里△t,N和m分别为时间步长,多项式阶数和精确解的正则度。这是目前已知的最好估计。最后的数值实验结果说明了理论分析的正确性。
在第五章,我们将第四章中数值求解时间分数阶扩散方程的方法推广到生物细胞学中研究离子运动的时间分数阶Cable方程,同样利用关于时间的有限差分格式及空间的谱方法构造一种高阶格式,利用测试的数值例子说明了我们方法的可行性。并用具体例子说明其应用。
第六章,讨论有限区间上具有初边值条件的非线性时间分数阶Fokker-Plabck方程,利用隐式差分方法构造离散格式,并用能量方法证明了所提出的差分格式的收敛性和稳定性。最后给出数值例子。
Fractional calculus is a branch of studying the property of any order integral or derivative.Fractional order differential equation is the equation containing the noninteger order derivative,raising from the standard differential equations by replacing the integer-order derivatives with fractional-order derivatives.Its application is very broad, many researchers find that the fractional differential equations more precisely describe the property of some materials with memory and heredity.Fractional order differential equations are playing an increasingly important role in engineering,physics and other fields,such as the fractal theory and the diffusion in porous media,fractional capacitance theory,electrolysis chemical,fractional biological neurons,condensate physics,vibration control of viscoelastic system,statistical mechanics and so on.
In this paper,we mainly consider the time-fractional anomalous diffusion equation, discuss its analytic solution,numerical solution and its application.
In Chapter 1,the developmental history of fractional calculus and the existing work about fractional calculus are reviewed.We also recall some definitions and properties of the fractional derivatives used in this paper.
In Chapter 2,two time-fractional anomalous diffusion equations are deduced from the random walk and a stable law.These two equations will be investigated numerically in the next two chapters.
In Chapter 3,the solution of time fractional anomalous diffusion equation is discussed. Using separation of variable methods and Laplace transform,the analytical solutions of a non-homogeneous anomalous sub-diffusion equation with Dirichlet,Neumann and Robin boundary conditions are derived respectively.The solution is expressed in terms of the Mittag-Leffler function.These techniques can be applied to solve other kinds of anomalous diffusion problems.
In Chapter 4,we consider a time fractional anomalous diffusion equation on a finite domain.We propose an efficient finite difference/spectral method to solve the time fractional diffusion equation.Stability and convergence of the method are rigourously established.We prove that the full discretization is unconditionally stable,and the numerical solution converges to the exact one with order O(△t~(2-α)+N~(-m)),where△t,N and m are the time step size,polynomial degree,and regularity of the exact solution respectively.Numerical experiments are carried out to support the theoretical claims.
In Chapter 5,we generalize the method that we have proposed in the Chapter 4 to the time fractional Cable equation for modeling neuronal dynamics.Numerical results are presented to show the applicability of the method.
In Chapter 6,we discuss one class of nonlinear time fractional Fokker-Planck equation with initial-boundary value on a finite domain.The stability and convergence of a finite difference method are discussed by energy methods.A numerical example is presented to compare with the exact analytical solution.
引文
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[14]汪富泉,曹叔尤,刘兴年.悬移颗粒运动特征值的分形分布[J].水利学报,2000,09:65-69.
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