分数阶偏微分方程的理论和数值研究
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摘要
近年来，分数阶偏微分方程(FPDEs)在数学模型中的应用受到越来越广泛的关注。不同的FPDEs模型已被应用到越来越多的领域中，包括：材料，力学，以及生物系统等，并且发现FPDEs在研究一些具有记忆过程、遗传性质以及异质材料时比整数阶方程模型更有优势。FPDEs在数学建模上取得的进展，激发了人们研究数值算法的兴趣。
本文从理论和数值计算两方面对分数阶扩散方程(FDEs)及其相关问题进行深入研究，主要内容包括以下三个方面：
我们引进了一类新的利用分数阶导数定义的分数阶空间，并证明了此类空间与传统的分数阶Sobolev空间在范数意义下是等价的。利用这些结果我们导出了FDEs初边值问题的弱形式，并借助椭圆型问题的经典理论证明了弱解的存在唯一性。上述研究结果表明在Riemann-Liouville分数阶导数定义的情况下，分数阶扩散方程与弱形式的等价性证明不需要添加初值条件。相反地，在Caputo导数定义的情况下，该等价性则需要加初值条件来保证。
基于上述弱解理论，我们计算时间分数阶扩散方程(TFDE)的数值解。TFDE与传统的扩散方程有本质的不同。对于前者，时间上的一阶导数被分数阶导数所代替，使得问题在时间上是全局的。我们提出将谱方法应用于TFDE时间和空间上的离散，给出最优误差估计证明该方法的收敛性，并用数值结果验证理论估计。归功于该方法在时间和空间方向上所具有的谱精度，我们能够有效地减少由全局时间依赖性所引起的对存储量的要求，从而可以计算长时间的解。
我们考察用以描述神经细胞中离子反常扩散现象的分数阶Nernst-Planck方程。我们提出了一种时间有限差分／空间谱元法对该方程进行数值求解，并给出了数值方法的详细构造过程以及实现方法。数值结果表明数值解在空间方向上具有指数阶收敛精度，在时间方向上具有2-α(0＜α＜1)阶精度。最后，通过计算一个具有实际背景参数的问题说明所提方法的潜在应用。
The use of fractional partial differential equations (FPDEs) in mathematical modelshas become increasingly popular in recent years. Different models using FPDEs have beenproposed in more and more fields, covering materials, mechanical, and biological systems,and it's found that FPDEs gain the advantage over the classical one in modeling somematerials with memory, heterogeneity or inheritable character. The modeling progresson using FPDEs has led to increasing interest in developing numerical schemes for theirsolutions.
In this paper, our work is focused on the theoretical investigation and numericalcomputation of the fractional diffusion equations (FDEs), which are of interest not onlyin their own right, but also in that they constitute the principal parts in many otherFPDEs. The main contribution of this work is threefold:
First, we introduce a new family of functional spaces defined by using fractionalderivatives, and prove that these spaces are equivalent to usual Sobolev spaces in thesense that their norms are equivalent. Based on these spaces the variational formulationof the initial boundary value problems of FDEs are developed, and the existence anduniqueness of the weak solution are established by using classical theory for elliptic problems.The obtained results indicate that in the case of Riemann-Liouville definition, theequivalence between FDEs and weak formulation does not require any initial conditions.This contrasts with the case of Caputo definition, in which the initial condition has to beintegrated into the weak formulation in order to establish the equivalence.
Second, based on the proposed weak formulation, we investigate the numerical solutionsof the time fractional diffusion equation (TFDE). Essentially, the TFDE differs fromthe standard diffusion equation in the time derivative term. In TFDE, the first-order timederivative is replaced by a fractional derivative, making the problem global in time. Wepropose a spectral method in both temporal and spatial discretizations for this equation.The convergence of the method is proven by providing a priori error estimate. Numericaltests are carried out to confirm the theoretical results. Thanks to the spectral accuracy inboth space and time of the proposed method, the storage requirement due to the "globaltime dependence" can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.
Third, we consider the fractional Nernst-Planck equation, which describes theanomalous diffusion in the movement of the ions in neuronal system. A methodcombining finite differences in time and spectral element methods in space is proposed tonumerically solve the underlying problem. The detailed construction and implementationof the method are presented. Our numerical experiences show that the convergence of theproposed method is exponential in space and (2-α)-order (0＜α＜1) in time. Finally,a practical problem with realistic physical parameters is simulated to demonstrate thepotential applicability of the method.
本文从理论和数值计算两方面对分数阶扩散方程(FDEs)及其相关问题进行深入研究，主要内容包括以下三个方面：
我们引进了一类新的利用分数阶导数定义的分数阶空间，并证明了此类空间与传统的分数阶Sobolev空间在范数意义下是等价的。利用这些结果我们导出了FDEs初边值问题的弱形式，并借助椭圆型问题的经典理论证明了弱解的存在唯一性。上述研究结果表明在Riemann-Liouville分数阶导数定义的情况下，分数阶扩散方程与弱形式的等价性证明不需要添加初值条件。相反地，在Caputo导数定义的情况下，该等价性则需要加初值条件来保证。
基于上述弱解理论，我们计算时间分数阶扩散方程(TFDE)的数值解。TFDE与传统的扩散方程有本质的不同。对于前者，时间上的一阶导数被分数阶导数所代替，使得问题在时间上是全局的。我们提出将谱方法应用于TFDE时间和空间上的离散，给出最优误差估计证明该方法的收敛性，并用数值结果验证理论估计。归功于该方法在时间和空间方向上所具有的谱精度，我们能够有效地减少由全局时间依赖性所引起的对存储量的要求，从而可以计算长时间的解。
我们考察用以描述神经细胞中离子反常扩散现象的分数阶Nernst-Planck方程。我们提出了一种时间有限差分／空间谱元法对该方程进行数值求解，并给出了数值方法的详细构造过程以及实现方法。数值结果表明数值解在空间方向上具有指数阶收敛精度，在时间方向上具有2-α(0＜α＜1)阶精度。最后，通过计算一个具有实际背景参数的问题说明所提方法的潜在应用。
The use of fractional partial differential equations (FPDEs) in mathematical modelshas become increasingly popular in recent years. Different models using FPDEs have beenproposed in more and more fields, covering materials, mechanical, and biological systems,and it's found that FPDEs gain the advantage over the classical one in modeling somematerials with memory, heterogeneity or inheritable character. The modeling progresson using FPDEs has led to increasing interest in developing numerical schemes for theirsolutions.
In this paper, our work is focused on the theoretical investigation and numericalcomputation of the fractional diffusion equations (FDEs), which are of interest not onlyin their own right, but also in that they constitute the principal parts in many otherFPDEs. The main contribution of this work is threefold:
First, we introduce a new family of functional spaces defined by using fractionalderivatives, and prove that these spaces are equivalent to usual Sobolev spaces in thesense that their norms are equivalent. Based on these spaces the variational formulationof the initial boundary value problems of FDEs are developed, and the existence anduniqueness of the weak solution are established by using classical theory for elliptic problems.The obtained results indicate that in the case of Riemann-Liouville definition, theequivalence between FDEs and weak formulation does not require any initial conditions.This contrasts with the case of Caputo definition, in which the initial condition has to beintegrated into the weak formulation in order to establish the equivalence.
Second, based on the proposed weak formulation, we investigate the numerical solutionsof the time fractional diffusion equation (TFDE). Essentially, the TFDE differs fromthe standard diffusion equation in the time derivative term. In TFDE, the first-order timederivative is replaced by a fractional derivative, making the problem global in time. Wepropose a spectral method in both temporal and spatial discretizations for this equation.The convergence of the method is proven by providing a priori error estimate. Numericaltests are carried out to confirm the theoretical results. Thanks to the spectral accuracy inboth space and time of the proposed method, the storage requirement due to the "globaltime dependence" can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.
Third, we consider the fractional Nernst-Planck equation, which describes theanomalous diffusion in the movement of the ions in neuronal system. A methodcombining finite differences in time and spectral element methods in space is proposed tonumerically solve the underlying problem. The detailed construction and implementationof the method are presented. Our numerical experiences show that the convergence of theproposed method is exponential in space and (2-α)-order (0＜α＜1) in time. Finally,a practical problem with realistic physical parameters is simulated to demonstrate thepotential applicability of the method.
引文
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[2] Stratton J. Electromagnetic theory. Wiley-IEEE Press, 2007
[3] Sugimoto N. Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. Journal of Fluid Mechanics Digital Archive, 2006, 225:631-653
[4] Oustaloup A, Coiffet P. Syst(?)mes asservis lin(?)aires d'ordre fractionnaire: th(?)orie et pratique. Masson, 1983
[5] Oustaloup A, Mathieu B. La commande CRONE. HERMES science publ. Paris, 1999
[6] Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics. COURSES AND LECTURES-INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES, 1997. 291-348
[7] Rossikhin Y, Shitikova M. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Applied Mechanics Reviews, 1997, 50:15-67
[8] Bode H. Network analysis and feedback amplifier design. Van Nostrand Reinhold Princeton, NJ, 1956
[9] Ichise M, Nagayanagi Y, Kojima T. An analog simulation of non-integer order transfer functions for analysis of electrode processes. J. Electroanal. Chem. Interfacial Electrochem, 1971, 33:253
[10] Sun H, Abdelwahab A, Onaral B. Linear approximation of transfer function with a pole of fractional power. IEEE Transactions on Automatic Control, 1984, 29(5):441-444
[11] Cole K. Electric conductance of biological systems. Proceedings of Cold Spring Harbor symposia on quantitative biology. Cold Spring Harbor Laboratory Press, 1933. 107
[12] Knsnezov D, Bulgac A, Dang G. Quantum levy processes and fractional kinetics. Physical Review Letters, 1999, 82(6):1136-1139
[13] Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals, 1996, 7(9):20-31
[14] Deng W. Generalized synchronization in fractional order systems. Physical Review E, 2007, 75(5)
[15] Langlands T, Henry B, Wearne S. Solution of a Fractional Cable Equation: Finite case.
[16] Hughes B. Random walks and random environments: Volume 1: Random Walks. Oxford University Press, USA, 1995
[17] Scher H, Michael M, Shlesinger F, et al. New particle acceleration techniques. Phys. Today41, 1991, 26
[18] Gu Q, Schiff E, Grebner S, et al. Non-Ganssian transport measurements and the Einstein relation in amorphous silicon. Physical Review Letters, 1996, 76(17):3196-3199
[19] Klemm A, Mu|¨ller H, Kimmich R. NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects. Magn. Reson. Imaging Phys Rev E, 1996, 55:4413
[20] Klammler F, Kimmich R. Geometrical restrictions of incoherent transport of water by diffusion in protein of silica fine-particle systems and by flow in a sponge-a study of anomalous properties using an NMR field-gradient technique. Croat. Chem. Acta, 1992, 62:455
[21] Weber H, Kimmich R. Anomalous segment diffusion in polymers and NMR relaxation spectroscopy. Macromolecules, 1993, 26(10):2597-2606
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