一类时间分数阶偏微分方程数值解的有限元逼近
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摘要
本文讨论了一类Caputo意义下的时间分数阶扩散方程的初边值问题的数值逼近.分数阶偏微分方程是一类将经典整数阶偏微分方程中的导数定义用分数阶导数替换而得到的微分方程.和整数阶偏微分方程相比,分数阶偏微分方程更能准确地描述某些自然现象和物理过程.本文将有限元方法应用于分数阶偏微分方程,采用了一种不同的方法来近似Caputo分数阶导数并且讨论了迭代格式的稳定性,得到了较为满意的结果.
首先,文章阐述了分数阶偏微分方程的相关历史背景和研究现状,并提出了本文将要解决的问题.其次,我们利用Caputo分数导数和Riemann - Liouville分数导数的一个关系式,得到了与原问题相等价的一个系统.接着在时间层上作了半离散,并利用Gru¨wald - Letnikov导数定义提出了一个差分算子,用其来近似分数阶微分算子,从而建立起了相应的变分方程,并得到了在α-范数意义下的误差估计.然后,又在空间维上对方程进一步作全离散,得到了全离散方程.对于全离散方程,我们都得到了0-范数意义和α-范数意义的误差估计.为了能应用于实际计算,我们进一步推出了有限元方程的具体表达形式,然后得到了相应刚度矩阵和代数方程.同时证明了迭代格式关于初值的稳定性.在论文的最后,我们给出了两个数值计算的例子和一些图表,数据表明计算结果与理论分析相吻合,进而证明了该方法的可行性.
A numerical approximation for a Caputo’s time-fractional diffusion equation withinitial and boundary conditions is discussed in the paper. Fractional differential equa-tions are a type of differential equations in which the definition of classical integer orderderivative is replaced by that of fractional derivative. It is better to model some natu-ral phenomenon and physical processes by fractional differential equations than integerorder equations. The finite element method is applied to fractional partial equations. Adifferent method is proposed for approximating the Caputo’s fractional derivative,andthe stability of iterative scheme is proved.
First,recent researches of fractional partial differential equations ,as well as histo-ry background,and the problem to be solved are stated.Secondly,a system equivalentto the original problem is obtained by using a relationship between Riemann-Liouvillefractional derivative and Caputo’s.After that,semiffdiscretization is executed at the timedirection.With the help of Gru¨wald-Letnikov derivative,a variational equation is de-duced by approximating the differential operator with difference operator.The error isestimated in theα-norm sense.Then,the full-discrete equation is derived,and both errorestimation in theα-norm and 0-norm sense are obtained. The finite element equationform is presented for the practical computation.Then, the corresponding stiffness matrixand algebraic equations are derived.Also the stability about initial values of iterationis proved.Last,two numerical examples and some diagrams are given to illustrate thefeasibility of this method.It is indicated that results coincide with theoretical analysis.
首先,文章阐述了分数阶偏微分方程的相关历史背景和研究现状,并提出了本文将要解决的问题.其次,我们利用Caputo分数导数和Riemann - Liouville分数导数的一个关系式,得到了与原问题相等价的一个系统.接着在时间层上作了半离散,并利用Gru¨wald - Letnikov导数定义提出了一个差分算子,用其来近似分数阶微分算子,从而建立起了相应的变分方程,并得到了在α-范数意义下的误差估计.然后,又在空间维上对方程进一步作全离散,得到了全离散方程.对于全离散方程,我们都得到了0-范数意义和α-范数意义的误差估计.为了能应用于实际计算,我们进一步推出了有限元方程的具体表达形式,然后得到了相应刚度矩阵和代数方程.同时证明了迭代格式关于初值的稳定性.在论文的最后,我们给出了两个数值计算的例子和一些图表,数据表明计算结果与理论分析相吻合,进而证明了该方法的可行性.
A numerical approximation for a Caputo’s time-fractional diffusion equation withinitial and boundary conditions is discussed in the paper. Fractional differential equa-tions are a type of differential equations in which the definition of classical integer orderderivative is replaced by that of fractional derivative. It is better to model some natu-ral phenomenon and physical processes by fractional differential equations than integerorder equations. The finite element method is applied to fractional partial equations. Adifferent method is proposed for approximating the Caputo’s fractional derivative,andthe stability of iterative scheme is proved.
First,recent researches of fractional partial differential equations ,as well as histo-ry background,and the problem to be solved are stated.Secondly,a system equivalentto the original problem is obtained by using a relationship between Riemann-Liouvillefractional derivative and Caputo’s.After that,semiffdiscretization is executed at the timedirection.With the help of Gru¨wald-Letnikov derivative,a variational equation is de-duced by approximating the differential operator with difference operator.The error isestimated in theα-norm sense.Then,the full-discrete equation is derived,and both errorestimation in theα-norm and 0-norm sense are obtained. The finite element equationform is presented for the practical computation.Then, the corresponding stiffness matrixand algebraic equations are derived.Also the stability about initial values of iterationis proved.Last,two numerical examples and some diagrams are given to illustrate thefeasibility of this method.It is indicated that results coincide with theoretical analysis.
引文
[1] Bagley R.L., Torvik P.J. A theoretical basis for the application of fractional calculus toviscoelasticity. J. Rheology, 1983, 27(3): 201-210.
[2] Bagley R.L., Torvik P.J. Fractional calculus in the transient analysis of viscoelasticallydamped structures. AIAA J., 1985, 23(6): 918-925.
[3] Baillie R.T. Long memory processes and fractional integration in econometrics. Econo-metrics, 1996, 73: 5-59
[4] Yuste S.B., Acedo L., Lindenberg K. Reaction front in an A+B→C reaction-subdiffusionprocess. Phys. Rev. E, 2004, 69(3):036126
[5] Rossikhin Y.A., Shitikova M.V. Applications of fractional calculus to dynamic problemsof liner and nolinear hereditary mechanics of solids. Appl. Mech. Rev., 1997, 50: 15-67.
[6] Barkia E., Metzler R., Klafter J. From continuous time random walks to the fractionalFokker-Planck equation. Phys. Rev. E., 2000, 61(1): 132-138.
[7] Magin R.L. Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng., 2004, 32(1):1-104.
[8] Magin R.L. Fractional calculus in bioengineering-part2. Crit. Rev. Biomed. Eng., 2004,32(2): 105-193.
[9] Magin R.L. Fractional calculus in bioengineering-part3. Crit. Rev. Biomed. Eng., 2004,32(3/4): 194-377.
[10] Metzler R., Klafter J. The random walk’s guide to anomalous diffusion: a fractionaldynamics approach. Phys. Rep., 2000, 339(1): 1-77.
[11] Liu F., Anh V., Turner I., et al. Time fractional advection dispersion equation. J. Appl.Math. Comput., 2003, 13: 233-245.
[12] Schneider W.R., Wyss W. Fractional diffusion and wave equations. J. Math. Phys., 1989,30: 34-144.
[13] Wyss W. The fractional diffusion equation. J. Math. Phys., 1986, 27: 2782-2785.
[14] Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl.Math. Lett., 1996, 9(6): 23-28.
[15] Sousa Erc′ilia. Finite difference approximations for a fractional advection diffusion prob-lem. Journal of Computational Physics, 2009, 228: 4038-4054.
[16] Wang Hong, Wang Kaixin, Sircar Treena. A direct O(N log2 N) ?nite difference methodor fractional diffusion equations. Journal of Computational Physics, 2010, 229: 8095-8104.
[17] Cui Mingrong. Compact ?nite difference method for the fractional diffusion equation.Journal of Computational Physics, 2009, 228: 7792-7804.
[18] Sweilam N.H., Khader M.M., Nagy A.M. Numerical solution of two-sided space-fractionalwave equation using ?nite difference method. Journal of Computational and AppliedMathematics, 2011, 235: 2832-2841.
[19] Chen Chang-ming, Liu F., Burrage K. Finite difference methods and a fourier analysis forthe fractional reaction–subdiffusion equation. Applied Mathematics and Computation,2008, 198: 754-769.
[20] Su Lijulsatan, Wang Wenqia, Yang Zhaoxia. Finite difference approximations for thefractional advection–diffusion equation. Physics Letters A, 2009, 373: 4405-4408.
[21] Murio D.A. Implicit ?nite difference approximation for time fractional diffusion equa-tions. Computers and Mathematics with Applications, 2008, 56: 1138-1145.
[22] Richtmyer R.D. Morton K.W. Difference Methods for Initial-Value Problems. New York:Interscience Publishers, 1967.
[23] Zhang Yang. A ?nite difference method for fractional partial differential equation. AppliedMathematics and Computation, 2009, 215: 524-529.
[24] Meerschaert Mark M., Tadjeran Charles. Finite difference approximations for two-sidedspace-fractional partial differential equations. Applied Numerical Mathematics, 2006,56: 80-90.
[25] Liu F., Shen S., Anh V., et al. Analysis of a discrete non-Markovian random walk approx-imation for the time fractional diffusion equation. ANZIAM J., 2005, 46(E): 488-504.
[26] Saadatmandi Abbas, Dehghan Mehdi. A new operational matrix for solving fractional-order differential equations. Computers and Mathematics with Applications, 2010, 59:1326-1336.
[27] Odibat Zaid M., Momani Shaher. Approximate solutions for boundary value problems oftime-fractional wave equation. Applied Mathematics and Computation, 2006, 181: 767-774.
[28] Ray S. Saha, Bera R.K. An approximate solution of a nonlinear fractional differentialequation by Adomian decomposition method. Applied Mathematics and Computation,2005, 167: 561-571.
[29] Li Changpin, Wang Yihong. Numerical algorithm based on Adomian decomposition forfractional differential equations. Computers and Mathematics with Applications, 2009,57: 1672-1681.
[30] El-Sayed A.M.A, Gaber M. The Adomian decomposition method for solving partial d-ifferential equations of fractal order in ?nite domains. Physics Letters A, 2006, 359:175-182.
[31] Shawagfeh Nabil T. Analytical approximate solutions for nonlinear fractional differentialequations. Applied Mathematics and Computation, 2002, 131:517-529.
[32] Al-Khaled Kamel, Momani Shaher. An approximate solution for a fractional diffusion-wave equation using the decomposition method. Applied Mathematics and Computation,2005, 165: 472-483.
[33] Diethelm Kai, Ford Judith M., Ford Neville J., et al. Pitfalls in fast numerical solversfor fractional differential equations. Journal of Computational and Applied Mathematics,2006, 186: 482-503.
[34] Lin Yumin, Xu Chuanju. Finite difference/spectral approximations for the time-fractionaldiffusion equation. Journal of Computational Physics, 2007, 225: 1533-1552.
[35] Roop John Paul. Computational aspects of FEM approximation of fractional advectiondispersion equations on bounded domains in R2. Journal of Computational and AppliedMathematics, 2006, 193: 243-268.
[36] Zhang H., Liu F., Anh V. Galerkin ?nite element approximation of symmetric space-fractional partial differential equations. Applied Mathematics and Computation, 2010,217: 2534-2545.
[37] Goren?o R., Mainardi F., Moretti D., et al. Time fractional diffusion: a discrete randomwalk approack. Nonliner Dynam., 2002, 29: 129-143.
[38] Podlubny I. Fractional Differential Equations. New York: Academic Press, 1999.
[39]应隆安.有限元方法讲义.北京:北京大学出版社, 1988. 71-73.
[40]张韵华,奚梅成,陈效群.数值计算方法与算法.(第二版).北京:科学出版社, 2006. 119-120.
[41]李荣华.偏微分方程数值解法.北京:高等教育出版社, 2005. 130-132.
[42]徐树方,高立,张平文.数值线性代数.北京:北京大学出版社, 2000. 48-51
[2] Bagley R.L., Torvik P.J. Fractional calculus in the transient analysis of viscoelasticallydamped structures. AIAA J., 1985, 23(6): 918-925.
[3] Baillie R.T. Long memory processes and fractional integration in econometrics. Econo-metrics, 1996, 73: 5-59
[4] Yuste S.B., Acedo L., Lindenberg K. Reaction front in an A+B→C reaction-subdiffusionprocess. Phys. Rev. E, 2004, 69(3):036126
[5] Rossikhin Y.A., Shitikova M.V. Applications of fractional calculus to dynamic problemsof liner and nolinear hereditary mechanics of solids. Appl. Mech. Rev., 1997, 50: 15-67.
[6] Barkia E., Metzler R., Klafter J. From continuous time random walks to the fractionalFokker-Planck equation. Phys. Rev. E., 2000, 61(1): 132-138.
[7] Magin R.L. Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng., 2004, 32(1):1-104.
[8] Magin R.L. Fractional calculus in bioengineering-part2. Crit. Rev. Biomed. Eng., 2004,32(2): 105-193.
[9] Magin R.L. Fractional calculus in bioengineering-part3. Crit. Rev. Biomed. Eng., 2004,32(3/4): 194-377.
[10] Metzler R., Klafter J. The random walk’s guide to anomalous diffusion: a fractionaldynamics approach. Phys. Rep., 2000, 339(1): 1-77.
[11] Liu F., Anh V., Turner I., et al. Time fractional advection dispersion equation. J. Appl.Math. Comput., 2003, 13: 233-245.
[12] Schneider W.R., Wyss W. Fractional diffusion and wave equations. J. Math. Phys., 1989,30: 34-144.
[13] Wyss W. The fractional diffusion equation. J. Math. Phys., 1986, 27: 2782-2785.
[14] Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl.Math. Lett., 1996, 9(6): 23-28.
[15] Sousa Erc′ilia. Finite difference approximations for a fractional advection diffusion prob-lem. Journal of Computational Physics, 2009, 228: 4038-4054.
[16] Wang Hong, Wang Kaixin, Sircar Treena. A direct O(N log2 N) ?nite difference methodor fractional diffusion equations. Journal of Computational Physics, 2010, 229: 8095-8104.
[17] Cui Mingrong. Compact ?nite difference method for the fractional diffusion equation.Journal of Computational Physics, 2009, 228: 7792-7804.
[18] Sweilam N.H., Khader M.M., Nagy A.M. Numerical solution of two-sided space-fractionalwave equation using ?nite difference method. Journal of Computational and AppliedMathematics, 2011, 235: 2832-2841.
[19] Chen Chang-ming, Liu F., Burrage K. Finite difference methods and a fourier analysis forthe fractional reaction–subdiffusion equation. Applied Mathematics and Computation,2008, 198: 754-769.
[20] Su Lijulsatan, Wang Wenqia, Yang Zhaoxia. Finite difference approximations for thefractional advection–diffusion equation. Physics Letters A, 2009, 373: 4405-4408.
[21] Murio D.A. Implicit ?nite difference approximation for time fractional diffusion equa-tions. Computers and Mathematics with Applications, 2008, 56: 1138-1145.
[22] Richtmyer R.D. Morton K.W. Difference Methods for Initial-Value Problems. New York:Interscience Publishers, 1967.
[23] Zhang Yang. A ?nite difference method for fractional partial differential equation. AppliedMathematics and Computation, 2009, 215: 524-529.
[24] Meerschaert Mark M., Tadjeran Charles. Finite difference approximations for two-sidedspace-fractional partial differential equations. Applied Numerical Mathematics, 2006,56: 80-90.
[25] Liu F., Shen S., Anh V., et al. Analysis of a discrete non-Markovian random walk approx-imation for the time fractional diffusion equation. ANZIAM J., 2005, 46(E): 488-504.
[26] Saadatmandi Abbas, Dehghan Mehdi. A new operational matrix for solving fractional-order differential equations. Computers and Mathematics with Applications, 2010, 59:1326-1336.
[27] Odibat Zaid M., Momani Shaher. Approximate solutions for boundary value problems oftime-fractional wave equation. Applied Mathematics and Computation, 2006, 181: 767-774.
[28] Ray S. Saha, Bera R.K. An approximate solution of a nonlinear fractional differentialequation by Adomian decomposition method. Applied Mathematics and Computation,2005, 167: 561-571.
[29] Li Changpin, Wang Yihong. Numerical algorithm based on Adomian decomposition forfractional differential equations. Computers and Mathematics with Applications, 2009,57: 1672-1681.
[30] El-Sayed A.M.A, Gaber M. The Adomian decomposition method for solving partial d-ifferential equations of fractal order in ?nite domains. Physics Letters A, 2006, 359:175-182.
[31] Shawagfeh Nabil T. Analytical approximate solutions for nonlinear fractional differentialequations. Applied Mathematics and Computation, 2002, 131:517-529.
[32] Al-Khaled Kamel, Momani Shaher. An approximate solution for a fractional diffusion-wave equation using the decomposition method. Applied Mathematics and Computation,2005, 165: 472-483.
[33] Diethelm Kai, Ford Judith M., Ford Neville J., et al. Pitfalls in fast numerical solversfor fractional differential equations. Journal of Computational and Applied Mathematics,2006, 186: 482-503.
[34] Lin Yumin, Xu Chuanju. Finite difference/spectral approximations for the time-fractionaldiffusion equation. Journal of Computational Physics, 2007, 225: 1533-1552.
[35] Roop John Paul. Computational aspects of FEM approximation of fractional advectiondispersion equations on bounded domains in R2. Journal of Computational and AppliedMathematics, 2006, 193: 243-268.
[36] Zhang H., Liu F., Anh V. Galerkin ?nite element approximation of symmetric space-fractional partial differential equations. Applied Mathematics and Computation, 2010,217: 2534-2545.
[37] Goren?o R., Mainardi F., Moretti D., et al. Time fractional diffusion: a discrete randomwalk approack. Nonliner Dynam., 2002, 29: 129-143.
[38] Podlubny I. Fractional Differential Equations. New York: Academic Press, 1999.
[39]应隆安.有限元方法讲义.北京:北京大学出版社, 1988. 71-73.
[40]张韵华,奚梅成,陈效群.数值计算方法与算法.(第二版).北京:科学出版社, 2006. 119-120.
[41]李荣华.偏微分方程数值解法.北京:高等教育出版社, 2005. 130-132.
[42]徐树方,高立,张平文.数值线性代数.北京:北京大学出版社, 2000. 48-51