时间分数阶对流—扩散方程反问题研究
|
摘要
时间分数阶对流-扩散方程(简称TFADE)可以用来模拟与时间相关的反常扩散,它是由传统的对流-扩散方程(简称ADE)演变而来的。虽然对于反常扩散正问题已有不少研究,但对于反常扩散模型参数识别的反问题研究却鲜见于文献。
本文基于对时间分数阶扩散方程和时间分数阶对流-扩散方程正问题的数值求解方法研究,主要探讨扩散系数和分数微分阶数的数值反演问题。
论文前两章简要介绍分数阶微分算子的发展历史和现状及趋势,给出三种分数阶微积分的定义,分数阶导数的性质及其区别与联系,进一步推导了函数eλ1的Caputo分数阶导数的表达式。
第三章主要讨论有限域上时间分数阶扩散方程正问题的数值求解。基于对Caputo意义下时间导数的离散,对于两种形式的扩散系数分别提出了正问题求解的隐式差分格式,证明了差分格式的稳定性和收敛性,并给出了一个数值算例分析。
第四章着重探讨时间分数阶扩散方程的参数反演问题。应用最佳摄动量算法和同伦正则化算法,对确定分数微分阶数与扩散系数的反问题进行了数值反演模拟。通过数值算例,讨论了正则参数(同伦参数)、数值微分步长等参数选取对反演结果的影响。计算结果表明,一维时间分数阶扩散方程中同时确定分数微分阶数与扩散系数的反问题具有数值唯一性。
第五章讨论了时间分数阶对流-扩散方程的正问题及参数反演问题。对于多项式形式的扩散系数推导出了求解正问题的隐式差分格式,利用最佳摄动量算法对扩散系数与分数微分阶数的同时反演进行数值模拟,并讨论了对流系数、正则参数选取对反演结果的影响。
Time fractional advection-diffusion equation (TFADE in short), which is different from traditional convection-diffusion equation (ADE in short), can be utilized to simulate time-related abnormal diffusion process. There are quite a few researches on the anomalous diffusion, but it is still rare in the literature we have for studies on inverse problems of TFADE.
This thesis will deal with inverse problems of determining fractional order and diffusion coefficient based on numerical methods to the forward problem of time fractional diffusion equation (TFDE in short) and TFADE.
In the first two chapters, surveys of history of fractional calculus, research background and development, and three definitions of fractional derivative, and relationships of them are introduced, respectively.
Chapter 3 mainly deals with solving of the forward problem of TFDE in a finite domain by finite difference method, where the fractional derivative of time is discreted by Caputo formula. An implicit difference scheme is presented, and stability and convergence of the difference scheme is analyzed. Finally, a numerical example is given to support the numerical scheme.
In Chapter 4, numerical inversions for parameters of TFDE are carried out by applying optimal perturbation regularization algorithm and homotopy regularization algorithm, respectively. Several numerical simulations are performed, and impacts of regularization parameter (homotopy parameter), and numerical differential step on the inversion algorithms are discussed. Inversion results show that the inverse problem here is of numerical uniqueness.
Chapter 5 is devoted to study numerical method to TFADE and corresponding inverse problems of determining fractional order and diffusion coefficient. An implicit difference scheme is presented in the case of the diffusion coefficient taking on polynomials. Numerical simulations are carried out also by the optimal perturbation regularization algorithm, and impacts of choices of regularization parameter, and advection velocity on the inversion algorithm are discussed.
本文基于对时间分数阶扩散方程和时间分数阶对流-扩散方程正问题的数值求解方法研究,主要探讨扩散系数和分数微分阶数的数值反演问题。
论文前两章简要介绍分数阶微分算子的发展历史和现状及趋势,给出三种分数阶微积分的定义,分数阶导数的性质及其区别与联系,进一步推导了函数eλ1的Caputo分数阶导数的表达式。
第三章主要讨论有限域上时间分数阶扩散方程正问题的数值求解。基于对Caputo意义下时间导数的离散,对于两种形式的扩散系数分别提出了正问题求解的隐式差分格式,证明了差分格式的稳定性和收敛性,并给出了一个数值算例分析。
第四章着重探讨时间分数阶扩散方程的参数反演问题。应用最佳摄动量算法和同伦正则化算法,对确定分数微分阶数与扩散系数的反问题进行了数值反演模拟。通过数值算例,讨论了正则参数(同伦参数)、数值微分步长等参数选取对反演结果的影响。计算结果表明,一维时间分数阶扩散方程中同时确定分数微分阶数与扩散系数的反问题具有数值唯一性。
第五章讨论了时间分数阶对流-扩散方程的正问题及参数反演问题。对于多项式形式的扩散系数推导出了求解正问题的隐式差分格式,利用最佳摄动量算法对扩散系数与分数微分阶数的同时反演进行数值模拟,并讨论了对流系数、正则参数选取对反演结果的影响。
Time fractional advection-diffusion equation (TFADE in short), which is different from traditional convection-diffusion equation (ADE in short), can be utilized to simulate time-related abnormal diffusion process. There are quite a few researches on the anomalous diffusion, but it is still rare in the literature we have for studies on inverse problems of TFADE.
This thesis will deal with inverse problems of determining fractional order and diffusion coefficient based on numerical methods to the forward problem of time fractional diffusion equation (TFDE in short) and TFADE.
In the first two chapters, surveys of history of fractional calculus, research background and development, and three definitions of fractional derivative, and relationships of them are introduced, respectively.
Chapter 3 mainly deals with solving of the forward problem of TFDE in a finite domain by finite difference method, where the fractional derivative of time is discreted by Caputo formula. An implicit difference scheme is presented, and stability and convergence of the difference scheme is analyzed. Finally, a numerical example is given to support the numerical scheme.
In Chapter 4, numerical inversions for parameters of TFDE are carried out by applying optimal perturbation regularization algorithm and homotopy regularization algorithm, respectively. Several numerical simulations are performed, and impacts of regularization parameter (homotopy parameter), and numerical differential step on the inversion algorithms are discussed. Inversion results show that the inverse problem here is of numerical uniqueness.
Chapter 5 is devoted to study numerical method to TFADE and corresponding inverse problems of determining fractional order and diffusion coefficient. An implicit difference scheme is presented in the case of the diffusion coefficient taking on polynomials. Numerical simulations are carried out also by the optimal perturbation regularization algorithm, and impacts of choices of regularization parameter, and advection velocity on the inversion algorithm are discussed.
引文
[1]E.E.Adams and L.W.Gelhar. Field study of dispersion in a heterogeneous aquifer 2. Spatial moments analysis[J]. Water Resour. Res,1992,28:3293-307.
[2]I.Podlubny. Fractional Differential Equations. Academic Press,1999.
[3]F.Liu,V.Anh and I.Turner. Numerical solution of the fractional-order advection dispersion. Proceedings of the International Conference on Boundary and Interior Layers, Perth, Australia, 2002,159-164.
[4]F.Mainardi et al. The fundamental solution of the space-time fractional diffusion Equation[J]. Fractional Calculus and Applied Analysis 2001,4:153-192.
[5]F.Huang, F.Liu.The time fractional diffusion and advection-dispersion equation[J]. ANZIAM J, 2005,46:317-330.
[6]F.huang and F.Liu. The fundamental solution of the space-time fractional advection-dispersion equation[J]. Journal of applied mathematics and computing,2005,18:339-350.
[7]V.Daftardar-Gejji, Hossein Jafari. Positive solutions of a system of non-autonomous fractional differential equations[J]. Journal of Mathematical Analysis and Applications,2005,302:5-64.
[8]J.Chen, F.Liu. Analytical solution for the time-fractional telegraph equation by the method of separating variables[J].Journal of Mathematical Analysis and Applications,2008,338:1364-1377.
[9]Diego A.Murio.Stable numerical solution of fractional-diffusion inverse heat conduction problem. Computers and Mathematics with Applications,2007,53:1492-1501.
[10]Diego A. Murio. Implicit finite difference approximation for time fractional diffusion equations. Computers and Mathematics with Applications,2008,56:1138-1145.
[11]F.Liu.The fractional advection-diffusion equation[J]. Appl. Math.&Computing,2003,13:233-245.
[12]N.Sanin, G.Montero.A finite difference model for air pollution simulation.Advances in Enginee-ring Software,2007,38:358-365.
[13]陈春华,一种时间分数阶扩散方程初边值问题的隐式有限差分格式.东华理工学院学报,2006,3：289-292.
[14]沈淑君,分数阶对流-扩散方程的基本解和数值方法[D].厦门大学博士论文,2008.
[15]林玉闽,时间分数阶偏微分方程的解及其应用[D].厦门大学博士论文,2008.
[16]Berkowitz B,Scher H and Silliman S E.Anomalous transport in laboratory-scale heterogeneous porus media [J]. Water Resour. Res,2000,36:149-58.
[17]Hatano Y and Hatano N. Dispersive transport of ions in column experiments:an explanation of long-tailed profiles[J]. Water Resour. Res,1998,34:1027-33.
[18]D.A.Benson. The Fractional Reaction-Diffusion Equation:Development and Application[D]. Dissertation of Doctorial Degree, University of Nevada, Reno,1998.
[19]Xiong Y, Huang G and Huang Q. Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk[J]. Journal of Contaminant Hydrology,2006,86:163-75.
[20]Berkowitz B, Cortis A, Dentz M,et al. Modeling non-Fickian transport in geological formations as a continuous time random walk [J]. Rev. Geophys,2006,44:RG2003/2006,2005RG0001780.
[21]Jin Cheng, Junichi Nakagawa, Masahiro Yamamoto, et al. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation[J]. Inverse Problems,2009,25,115002 (16pp).
[22]Anh,et al.Scaling laws for fractional diffusion.wave equations with singular data. Statistics Probability Letters,2000.
[23]Sixdeniers, Penson, Solomon. Mittag-Leffler coherent states[J]. phys.A:Math. Gen.,1999,32.
[24]Kiryakova. Multiple(multiindex)Mittag-Leffler functions and relations to gen-eralized fractional calculus[J].Compu.and App. Math.,118,2000.
[25]于强,刘发旺.时间分数阶反应-扩散方程的隐式差分近似[J].厦门大学学报,2006,45(3)：315-319.
[26]苏超伟.偏微分方程逆问题的数值解法及其应用[M].西安：西北工业大学出版社,1995.
[27]魏镇,李功胜,池光胜.一个扰动土柱试验模型及参数反演的数值模拟[J].山东理工大学学报,2009,23(2)：1-6.
[28]崔凯,李兴斯,李宝元等.求解非线性反问题的大范围收敛梯度正则化算法.计算力学学报,2005,22(4).
[29]Lin Zhen hua, Yu Bo, Feng Guo-chen.. A combined homotopy interior point method for convex nonlinear programming problems[J].Appl mathematics and computation,1997,84:193-211.
[30]刘庆怀,林止华.求解多目标规划最小弱有效解的同伦内点算法[J].应用数学学报,2000,23(2)：188-195.
[31]王则柯,高堂安.同伦算法引论[M].重庆：重庆出版社,1990.1-24.
[32]范江华.求解非线性互补问题的单纯同伦算法[J].工程数学学报,2008,23(4)：619-624.
[33]薛齐文,杨海天,林秀云.同伦正则化算法求解多总量瞬态热传导反问题[J].计算物理,2006, 23(2)：151-157.
[34]于波,林正华.一类非凸Brouwer不动点问题的同伦算法[J].吉林大学自然科学学报,1994,2：37-38.
[35]阳明盛,熊西文,林建华Matlab基础及数学软件[M].大连：大连理工大学出版社,2003,37-90.
[36]彭芳麟.数学物理方程的Matlab解法与可视化[M].北京：清华大学出版社,2004,276-292.
[37]姜健飞,胡良剑.数值分析及其Matlab实验[M].科学出版社,2004.
[38]李立康,於崇华,朱政华等.微分方程数值解法[M].上海,复旦大学出版社,1998.
[39]李庆扬,王能超,易大义等.数值分析[M],北京,清华大学出版社,2001.
[40]丁丽娟.数值计算方法[M],北京,北京理工大学出版社,1997.8.
[41]龚纯,王正林Matlab语言常用算法程序集[M].北京电子工业出版社,2008.6.
[42]阳明盛,熊西文,林建华Matlab基础及数学软件[M].大连：大连理工大学出版社,2003,37-90.
[43]彭芳麟.数学物理方程的Matlab解法与可视化[M].北京：清华大学出版社,2004,276-292.
[44]B.Baeumer, M. Kovacs and M. M. Meerschaert. Numerical solutions for fractional reaction-diffusion equations[J]. Computers and Mathematics with Applications,2008,55:2212-2226.
[45]S.Lu,F.J.Molz and G.J.Fix. Possible problems of scale dependency in applications of the three dimensional fractional advection dispersion equation to natural porous media[J]. Water Resource Research,2002,38(9):1165.
[46]Q.Liu et al.Approximation of the levy-feller advection-dispersion process by random walk and finite difference method[J]. Journal of Computational Physics,2006.
[47]S.Lu, F.J.Molz and G.J.Fix. Possible problems of scale dependency in applications of the three dimensional fractional advection dispersion equation to natural porous media[J]. Water Resource Research,2002,38(9):1165.
[2]I.Podlubny. Fractional Differential Equations. Academic Press,1999.
[3]F.Liu,V.Anh and I.Turner. Numerical solution of the fractional-order advection dispersion. Proceedings of the International Conference on Boundary and Interior Layers, Perth, Australia, 2002,159-164.
[4]F.Mainardi et al. The fundamental solution of the space-time fractional diffusion Equation[J]. Fractional Calculus and Applied Analysis 2001,4:153-192.
[5]F.Huang, F.Liu.The time fractional diffusion and advection-dispersion equation[J]. ANZIAM J, 2005,46:317-330.
[6]F.huang and F.Liu. The fundamental solution of the space-time fractional advection-dispersion equation[J]. Journal of applied mathematics and computing,2005,18:339-350.
[7]V.Daftardar-Gejji, Hossein Jafari. Positive solutions of a system of non-autonomous fractional differential equations[J]. Journal of Mathematical Analysis and Applications,2005,302:5-64.
[8]J.Chen, F.Liu. Analytical solution for the time-fractional telegraph equation by the method of separating variables[J].Journal of Mathematical Analysis and Applications,2008,338:1364-1377.
[9]Diego A.Murio.Stable numerical solution of fractional-diffusion inverse heat conduction problem. Computers and Mathematics with Applications,2007,53:1492-1501.
[10]Diego A. Murio. Implicit finite difference approximation for time fractional diffusion equations. Computers and Mathematics with Applications,2008,56:1138-1145.
[11]F.Liu.The fractional advection-diffusion equation[J]. Appl. Math.&Computing,2003,13:233-245.
[12]N.Sanin, G.Montero.A finite difference model for air pollution simulation.Advances in Enginee-ring Software,2007,38:358-365.
[13]陈春华,一种时间分数阶扩散方程初边值问题的隐式有限差分格式.东华理工学院学报,2006,3：289-292.
[14]沈淑君,分数阶对流-扩散方程的基本解和数值方法[D].厦门大学博士论文,2008.
[15]林玉闽,时间分数阶偏微分方程的解及其应用[D].厦门大学博士论文,2008.
[16]Berkowitz B,Scher H and Silliman S E.Anomalous transport in laboratory-scale heterogeneous porus media [J]. Water Resour. Res,2000,36:149-58.
[17]Hatano Y and Hatano N. Dispersive transport of ions in column experiments:an explanation of long-tailed profiles[J]. Water Resour. Res,1998,34:1027-33.
[18]D.A.Benson. The Fractional Reaction-Diffusion Equation:Development and Application[D]. Dissertation of Doctorial Degree, University of Nevada, Reno,1998.
[19]Xiong Y, Huang G and Huang Q. Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk[J]. Journal of Contaminant Hydrology,2006,86:163-75.
[20]Berkowitz B, Cortis A, Dentz M,et al. Modeling non-Fickian transport in geological formations as a continuous time random walk [J]. Rev. Geophys,2006,44:RG2003/2006,2005RG0001780.
[21]Jin Cheng, Junichi Nakagawa, Masahiro Yamamoto, et al. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation[J]. Inverse Problems,2009,25,115002 (16pp).
[22]Anh,et al.Scaling laws for fractional diffusion.wave equations with singular data. Statistics Probability Letters,2000.
[23]Sixdeniers, Penson, Solomon. Mittag-Leffler coherent states[J]. phys.A:Math. Gen.,1999,32.
[24]Kiryakova. Multiple(multiindex)Mittag-Leffler functions and relations to gen-eralized fractional calculus[J].Compu.and App. Math.,118,2000.
[25]于强,刘发旺.时间分数阶反应-扩散方程的隐式差分近似[J].厦门大学学报,2006,45(3)：315-319.
[26]苏超伟.偏微分方程逆问题的数值解法及其应用[M].西安：西北工业大学出版社,1995.
[27]魏镇,李功胜,池光胜.一个扰动土柱试验模型及参数反演的数值模拟[J].山东理工大学学报,2009,23(2)：1-6.
[28]崔凯,李兴斯,李宝元等.求解非线性反问题的大范围收敛梯度正则化算法.计算力学学报,2005,22(4).
[29]Lin Zhen hua, Yu Bo, Feng Guo-chen.. A combined homotopy interior point method for convex nonlinear programming problems[J].Appl mathematics and computation,1997,84:193-211.
[30]刘庆怀,林止华.求解多目标规划最小弱有效解的同伦内点算法[J].应用数学学报,2000,23(2)：188-195.
[31]王则柯,高堂安.同伦算法引论[M].重庆：重庆出版社,1990.1-24.
[32]范江华.求解非线性互补问题的单纯同伦算法[J].工程数学学报,2008,23(4)：619-624.
[33]薛齐文,杨海天,林秀云.同伦正则化算法求解多总量瞬态热传导反问题[J].计算物理,2006, 23(2)：151-157.
[34]于波,林正华.一类非凸Brouwer不动点问题的同伦算法[J].吉林大学自然科学学报,1994,2：37-38.
[35]阳明盛,熊西文,林建华Matlab基础及数学软件[M].大连：大连理工大学出版社,2003,37-90.
[36]彭芳麟.数学物理方程的Matlab解法与可视化[M].北京：清华大学出版社,2004,276-292.
[37]姜健飞,胡良剑.数值分析及其Matlab实验[M].科学出版社,2004.
[38]李立康,於崇华,朱政华等.微分方程数值解法[M].上海,复旦大学出版社,1998.
[39]李庆扬,王能超,易大义等.数值分析[M],北京,清华大学出版社,2001.
[40]丁丽娟.数值计算方法[M],北京,北京理工大学出版社,1997.8.
[41]龚纯,王正林Matlab语言常用算法程序集[M].北京电子工业出版社,2008.6.
[42]阳明盛,熊西文,林建华Matlab基础及数学软件[M].大连：大连理工大学出版社,2003,37-90.
[43]彭芳麟.数学物理方程的Matlab解法与可视化[M].北京：清华大学出版社,2004,276-292.
[44]B.Baeumer, M. Kovacs and M. M. Meerschaert. Numerical solutions for fractional reaction-diffusion equations[J]. Computers and Mathematics with Applications,2008,55:2212-2226.
[45]S.Lu,F.J.Molz and G.J.Fix. Possible problems of scale dependency in applications of the three dimensional fractional advection dispersion equation to natural porous media[J]. Water Resource Research,2002,38(9):1165.
[46]Q.Liu et al.Approximation of the levy-feller advection-dispersion process by random walk and finite difference method[J]. Journal of Computational Physics,2006.
[47]S.Lu, F.J.Molz and G.J.Fix. Possible problems of scale dependency in applications of the three dimensional fractional advection dispersion equation to natural porous media[J]. Water Resource Research,2002,38(9):1165.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |