• 论文 •

### 时间分数次扩散方程反演源项问题的迭代正则化方法

1. 西北师范大学数学与统计学院, 计算数学研究所, 兰州 730070
• 收稿日期:2016-09-27 出版日期:2017-08-15 发布日期:2017-08-04
• 基金资助:

国家自然科学基金（11661072）和西北师范大学博士启动金（5002-577）资助项目

Cheng Qiang, Xiong Xiangtuan. AN ITERATIVE METHOD FOR AN INVERSE SOURCE PROBLEM OF A TIME-FRACTIONAL DIFFUSION EQUATION[J]. Mathematica Numerica Sinica, 2017, 39(3): 295-308.

### AN ITERATIVE METHOD FOR AN INVERSE SOURCE PROBLEM OF A TIME-FRACTIONAL DIFFUSION EQUATION

Cheng Qiang, Xiong Xiangtuan

1. Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
• Received:2016-09-27 Online:2017-08-15 Published:2017-08-04

Inverse source problems for time-fractional diffusion equation is a classical ill-posed inverse problem. A new iterative scheme is devised for solving this problem. Under the a-priori and post-priori parameter choice rules, the convergence rates are obtained. Some numerical tests are conducted for showing the effectiveness of the proposed method.

MR(2010)主题分类:

()
 [1] Barkai E, Metzler R, Klafter J. From continuous time random walks to the fractional Fokker-Planck equation[J]. Phys. Rev. E, 2000, 61: 132-138.[2] Chaves A S, A fractional diffusion equation to describe Levy flights[J]. Phys. Lett. A, 1998, 329: 13-16.[3] Zaslavsky G M. Fractional kinetic-equation for Hamiltonian chaos[J]. Physica D, 1994, 76: 110-122.[4] Meerschaert M M, Scheffer H P. Semi-stable Levy motion[J]. Fract. Calc. Appl. Anal., 2002: 5, 27-54.[5] Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach[J]. Physics Reports, 2000, 339(1): 1-77.[6] Roman H E, Alemany P A. Continuous-time random walks and the fractional diffusion equation[J]. Journal of Physics A: Mathematical and General, 1994, 27: 3407.[7] Zhang Y, Xu X. Inverse source problem for a fractional diffusion equation[J]. Inverse Problems, 2011, 27: 035010.[8] Nakagawa J, Sakamoto K, Yamamoto M. Overview to mathematical analysis for fractional diffusion equations: new mathematical aspects motivated by industrial collaboration[J]. Journal of Math-for-Industry, 2010, 2(A): 99-108.[9] Murio D A. Implicit finite difference approximation for time fractional diffusion equations[J]. Computers and Mathematics with Applications, 2008, 56(4): 1138-1145.[10] Ye H, Liu F, Turner I, Anh V, Burrage K. Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions[J]. Eur. Phys. J., 2013, 222: 1901-1914.[11] Luchko Y. Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation[J]. J. Math. Anal. Appl., 2011, 374: 538-548.[12] Stojanovic M. Numerical method for solving diffusion-wave problem[J]. J. Comput. Appl. Math., 2011, 235: 3121-3137.[13] Jiang H, Liu F, Turner I, Burrage K. Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain[J]. J. Math. Anal. Appl., 2012, 389: 1117-1127.[14] Ye H, Liu F, Turner I, Anh V, Burrage K. Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations[J]. Appl. Math. Comput., 2014, 227: 531-540.[15] Li X, Xu C. A space-time spectral method for the time fractional diffusion equation[J]. SIAM J. Numer. Anal., 2009, 47: 2108-2131.[16] Bondarenko A N, Ivaschenko D S. Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient[J]. J. Inverse Ill-Posed Probl., 2009, 17(5): 419-440.[17] Xu X, Cheng J, Yamamoto M. Carleman estimate for a fractional diffusion equation with half order and application[J]. Appl. Anal., 2011, 90(9): 1355-1371.[18] Cheng J, Nakagawa J, Yamamoto M, Yamazaki T. Uniqueness in an inverse problem for one-dimensional fractional diffusion equation[J]. Inverse Problems, 2009, 25: 115002. (16 pp).[19] Liu J J, Yamamoto M and Yan L. On the reconstruction of unknown time-dependent boundary sources for time fractional diffusion process by distributing measurement[J]. Inverse Problems, 2016, 32(1): 015009, 25 pp.[20] Liu J J and Yamamoto M. A backward problem for the time-fractional diffusion equation[J]. Appl. Anal., 2010, 89(11): 1769-1788.[21] Sun Liangliang and Wei Ting. Identification of the zeroth-order coefficient in a time fractional diffusion equation[J]. Appl. Numer. Math., 2017, 111: 160-180.[22] Wei T, Li X L and Li Y S. An inverse time-dependent source problem for a time-fractional diffusion equation[J]. Inverse Problems, 2016, 32(8): 085003, 24 pp.[23] WeI Ting and Wang Jungang. Determination of Robin coefficient in a fractional diffusion problem[J]. Appl. Math. Model., 2016, 40(17-18): 7948-7961.[24] Li G, Zhang D, Jia X, Yamamoto M. Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation[J]. Inverse Problems, 2013, 29: 065014.[25] Bangti Jin and William Rundell. A tutorial on inverse problems for anomalous diffusion processes[J]. Inverse Problems, 2015, 31(3): 035003, 40 pp.[26] Zheng G H, Wei T. Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation[J]. J. Comput. Appl. Math., 2010, 233: 2631-2640.[27] Zheng G H, Wei T. A new regularization method for a Cauchy problem of the time fractional diffusion equation[J]. Adv. Comput. Math., 2012, 36: 377-398.[28] Murio D A. Stable numerical solution of a fractional-diffusion inverse heat conduction problem[J]. Comput. Math. Appl., 2007, 53: 1492-1501.[29] Zhang Y, Xu X. Inverse source problem for a fractional diffusion equation[J]. Inverse Prob., 27(2.11). 035010(12pp).[30] Wei H, Chen W, Sun H G, Li X C. A coupled method for inverse source problem of spatial fractional anomalous diffusion equations[J]. Inverse Prob. Sci. Eng., 2010, 18: 945-956.[31] Murio D A, Mejia C E. Source terms identification for time fractional diffusion equation[J]. Rev. Colomb. Mat., 2008, 42: 25-46.[32] Yang Liu, Deng Zuicha, Yu Jianning and Luo Guanwei. Two regularization strategies for an evolutional type inverse heat source problem[J]. J. Phys. A, 2009, 42(36): 365203, 16 pp.[33] Cheng H, Fu C L. An iteration regularization for a time-fractional inverse diffusion problem[J]. Appl. Anal., 2012, 36: 5642-5649.[34] Cheng H, Fu C L, Zhang Y X. An iteration method for stable analytic continuation[J]. Applied Mathematics and Computation, 2014, 233: 203-213.[35] Louis A K. Inverse und schlecht gestellte Probleme. Stuttgart, Teubner, 1989.[36] Vainikko G M, Veretennikov A Y. Iteration Procedures in Ill-Posed Problems. Moscow, Nauka, Mc-Cormick, S.F., 1986(in Russian)[37] 肖庭延, 于慎根, 王彦飞编著. 反问题的数值解法[M]. 北京: 科学出版社, 2003.
 [1] 高兴华, 李宏, 刘洋. 分布阶扩散—波动方程的有限元解的误差估计[J]. 计算数学, 2021, 43(4): 493-505. [2] 陈明卿, 谢小平. 随机平面线弹性问题的一类弱Galerkin方法[J]. 计算数学, 2021, 43(3): 279-300. [3] 董自明, 李宏, 赵智慧, 唐斯琴. 对流扩散反应方程的局部投影稳定化连续时空有限元方法[J]. 计算数学, 2021, 43(3): 367-387. [4] 曾玉平, 翁智峰, 胡汉章. 简化摩擦接触问题的对称弱超内罚间断Galerkin方法的先验和后验误差估计[J]. 计算数学, 2021, 43(2): 162-176. [5] 房明娟, 阳莺, 唐鸣. 稳态Poisson-Nernst-Planck方程的残量型后验误差估计[J]. 计算数学, 2021, 43(1): 17-32. [6] 王然, 张怀, 康彤. 求解带有非线性边界条件的涡流方程的A-φ解耦有限元格式[J]. 计算数学, 2021, 43(1): 33-55. [7] 唐斯琴, 李宏, 董自明, 赵智慧. 对流反应扩散方程的稳定化时间间断时空有限元解的误差估计[J]. 计算数学, 2020, 42(4): 472-486. [8] 关宏波, 洪亚鹏. 抛物型界面问题的变网格有限元方法[J]. 计算数学, 2020, 42(2): 196-206. [9] 何斯日古楞, 李宏, 刘洋, 方志朝. 非稳态奇异系数微分方程的时间间断时空有限元方法[J]. 计算数学, 2020, 42(1): 101-116. [10] 贾仲孝, 孙晓琳. 计算矩阵函数双线性形式的Krylov子空间算法的误差分析[J]. 计算数学, 2020, 42(1): 117-130. [11] 王芬玲, 樊明智, 赵艳敏, 史争光, 石东洋. 多项时间分数阶扩散方程各向异性线性三角元的高精度分析[J]. 计算数学, 2018, 40(3): 299-312. [12] 武海军. 高波数Helmholtz方程的有限元方法和连续内罚有限元方法[J]. 计算数学, 2018, 40(2): 191-213. [13] 葛志昊, 吴慧丽. 体积约束的非局部扩散问题的后验误差分析[J]. 计算数学, 2018, 40(1): 107-116. [14] 单炜琨, 李会元. 双调和算子特征值问题的混合三角谱元方法[J]. 计算数学, 2017, 39(1): 81-97. [15] 曹济伟. 求解二维时谐Maxwell方程的一种混合有限元新格式[J]. 计算数学, 2016, 38(4): 429-441.