PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS

doi:10.1016/S0252-9602(13)60045-4

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (3): 855-864.doi: 10.1016/S0252-9602(13)60045-4

PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS

李景|郭柏灵

School of Mathematics and Computational Science, Changsha University of Science and Technology, Changsha 410004, China； Institute of Applied Physics and Computational Mathematics, Beijing 100088, China； Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

出版日期:2013-05-20 发布日期:2013-05-20
基金资助:
The first author is supported by the National Natural Science Foundation of China (11226166 and 11001033) and Scientific Research Fund of Hunan Provinical Education (11C0052).

PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS

LI Jing, GUO Bo-Ling

School of Mathematics and Computational Science, Changsha University of Science and Technology, Changsha 410004, China； Institute of Applied Physics and Computational Mathematics, Beijing 100088, China； Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Online:2013-05-20 Published:2013-05-20
Supported by:
The first author is supported by the National Natural Science Foundation of China (11226166 and 11001033) and Scientific Research Fund of Hunan Provinical Education (11C0052).

摘要/Abstract

摘要：

This article investigates the fractional derivative order identification, the coef-ficient identification, and the source identification in the fractional diffusion problems. If 1 < < 2, we prove the unique determination of the fractional derivative order and the dif-fusion coefficient p(x) by R t0 u(0, s)ds, 0 < t < T for one-dimensional fractional diffusion-wave equations. Besides, if 0 < < 1, we show the unique determination of the source term f(x, y) by U(0, 0, t), 0 < t < T for two-dimensional fractional diffusion equations. Here, denotes the fractional derivative order over t.

关键词: Fractional differential equation, inverse problems, parameter identification

Key words: Fractional differential equation, inverse problems, parameter identification

中图分类号:

35R30

李景, 郭柏灵. PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2013, 33(3): 855-864.

LI Jing, GUO Bo-Ling. PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia,Series B, 2013, 33(3): 855-864.

参考文献

[1] Berkowitz B, Scher H, Silliman S E. Anomalous transport in laboratory-scale heterogeneous porusmedia. Water Resour Res, 2002, 36: 149–158

[2] Scher H, Montroll E W. Anomalous transit-time dispersion in amorphous solids. Phys Rev B, 1975, 12: 455–477

[3] Adams E E, Gelhar L W. Field study of dispersion in heterogeneous aquifer. Water Resour Res, 1992, 28: 3293–3307

[4] Chechkin A V, Gonchar V Y, Szydlowski M. Fractional kinetics for relaxation and superdiffusion in a magnetic field. Phys Plasma, 2002, 9: 78–88

[5] Mainardi F. Fractional diffusive waves in viscoelastic solids//Wegner J L, Norwood F R. Nonlinear Waves in Solids. Fairfield: ASME/AMR, 1995: 93–97

[6] Mandelbrojt S. The Fractal Geometry of Nature. New York: Freeman, 1982

[7] Metzler R, Klafter J. The random walk´s guide to anomalous diffusion: A fractional dynamics approach. Phys Rep, 2000, 339: 1–77

[8] Shen G J, Chen C, Yan L T. Remarks on sub-fractional Bessel processes. Acta Mathematica Scientia, 2011, 31: 1860–1876

[9] Nigmatullin R R. The realization of the generalized transfer equation in a medium with fractal geometry. Phy Stat Sol B, 1986, 133: 425–430

[10] Klibas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006

[11] Podlubny I. Fractional Differential Equations. San Diego: Academic, 1999

[12] Guo B L, Pu X K, Huang F H. Fractional Partial Differential Equations and their Numerical Solutions (in Chinese). Beijing: Science Press, 2011

[13] Sakamoto K, Yamamoto M. Initial value/boundary value problems for fractional diffusion-wave equa-tions and applications to some inverse problems. http://www.sciencedirect.com/science/article/pii/S0022247X11003970

[14] Cheng J, et al. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl, 2009, 25. http://iopscience.iop.org/0266-5611/25/11/115002/

[15] Liu J J, Yamamoto M. A backward problem for the time-fractional diffusion equation. Appl Anal, 2010, 89: 1769–1788

[16] Murio D A,Mej´?a C E. Source terms identification for time fractional diffusion equation. Revista Colombiana de Matem´as, 2008, 42: 25–46

[17] Nakagawa J, Sakamoto K, Yamamoto M. Overview to mathematical analysis for fractional diffusion equations-new mathematical aspects motivated by industrial collaboration. J Math Ind, 2010, 2: 99–108

[18] Tuan V K. Inverse problem for fractional diffusion equation [J]. Frac Calc Appl Anal, 2011, 14: 31–55

[19] Zhang Y, Xu X. Inverse source problem for a fractional diffusion equation. Inverse Probl, 2011, 27.
http://iopscience.iop.org/0266-5611/27/3/035010

[20] Zelttl A. Mathematical surveys and monographs. Vol 121. Sturm-Liouville Theory. Americal Mathematical Society, 2005

[21] Umarov S R, Saidamatov E M. A generalization of Duhamels principle for differential equations of fractional order. Dokl Math, 2007, 75: 94–96

PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS

PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS

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