Journal of Applied Mathematics
Volume 2012 (2012), Article ID 740385, 14 pages
http://dx.doi.org/10.1155/2012/740385
Research Article
Numerical Identification of Multiparameters in the Space Fractional Advection Dispersion Equation by Final Observations
1Institute of Applied Mathematics, Shandong University of Technology, Zibo 255049, China
2Department of Basic Courses, Shandong Kaiwen College of Science and Technology, Jinan 250020, China
Received 6 September 2012; Accepted 13 November 2012
Academic Editor: Bo Han
Copyright © 2012 Dali Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- E. E. Adams and L. W. Gelhar, “Field study of dispersion in a heterogeneous aquifer: 2,” Spatial Moments Analysis, Water Resources Research, vol. 28, no. 12, pp. 3293–3307, 1992. View at Google Scholar
- D. A. Benson, The fractional advection-dispersion equation: development and application [Ph.D. thesis], University of Nevada, Reno, Nev, USA, 1998.
- Y. Hatano and N. Hatano, “Dispersive transport of ions in column experiments: an explanation of long-tailed profiles,” Water Resources Research, vol. 34, no. 5, pp. 1027–1033, 1998. View at Google Scholar
- Y. Pachepsky, D. Benson, and W. Rawls, “Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation,” Soil Science Society of American Journal, vol. 64, no. 4, pp. 1234–1243, 2000. View at Google Scholar
- B. Berkowitz, H. Scher, and S. E. Silliman, “Anomalous transport in laboratory-scale heterogeneous porous media,” Water Resources Research, vol. 36, no. 1, pp. 149–158, 2000. View at Google Scholar
- D. A. Benson, R. Schumer, M. M. Meerschaert, and S. W. Wheatcraft, “Fractional dispersion, Lévy motion, and the MADE tracer tests,” Transport in Porous Media, vol. 42, no. 1-2, pp. 211–240, 2001. View at Publisher · View at Google Scholar
- L. Z. Zhou and H. M. Selim, “Application of the fractional advection-dispersion equations in porous media,” Soil Science Society of American Journal, vol. 67, no. 4, pp. 1079–1084, 2003. View at Google Scholar
- Y. W. Xiong, G. H. Huang, and Q. Z. Huang, “Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk,” Journal of Contaminant Hydrology, vol. 86, no. 3-4, pp. 163–175, 2006. View at Google Scholar
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
- M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 65–77, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- F. Liu, V. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, no. 1, pp. 209–219, 2004. View at Google Scholar
- F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, “Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 12–20, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Q. Liu, F. Liu, I. Turner, and V. Anh, “Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method,” Journal of Computational Physics, vol. 222, no. 1, pp. 57–70, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- B. Baeumer, M. Kovács, and M. M. Meerschaert, “Numerical solutions for fractional reaction-diffusion equations,” Computers & Mathematics with Applications, vol. 55, no. 10, pp. 2212–2226, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Q. Z. Huang, G. H. Huang, and H. B. Zhan, “A finite element solution for the fractional advection dispersion equation,” Advances in Water Resources, vol. 31, no. 12, pp. 1578–1589, 2008. View at Google Scholar
- Q. Yang, F. Liu, and I. Turner, “Numerical methods for fractional partial differential equations with Riesz space fractional derivatives,” Applied Mathematical Modelling, vol. 34, no. 1, pp. 200–218, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- J. Cheng, J. Nakagawa, M. Yamamoto, and T. Yamazaki, “Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation,” Inverse Problems, vol. 25, no. 11, Article ID 115002, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- J. J. Liu and M. Yamamoto, “A backward problem for the time-fractional diffusion equation,” Applicable Analysis, vol. 89, no. 11, pp. 1769–1788, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Y. Zhang and X. Xu, “Inverse source problem for a fractional diffusion equation,” Inverse Problems, vol. 27, no. 3, Article ID 035010, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- V. K. Tuan, “Inverse problem for fractional diffusion equation,” Fractional Calculus and Applied Analysis, vol. 14, no. 1, pp. 31–55, 2011. View at Publisher · View at Google Scholar
- B. T. Jin and W. Rundell, “An inverse problem for a one-dimensional time-fractional diffusion problem,” Inverse Problems, vol. 28, no. 7, Article ID 075010, 2012. View at Google Scholar
- G. S. Li, W. J. Gu, and X. Z. Jia, “Numerical inversions for space-dependent diffusion coefficient in the time fractional diffusion equation,” Journal of Inverse and Ill-Posed Problems, vol. 20, no. 3, pp. 339–366, 2012. View at Google Scholar
- H. Wei, W. Chen, H. Sun, and X. Li, “A coupled method for inverse source problem of spatial fractional anomalous diffusion equations,” Inverse Problems in Science and Engineering, vol. 18, no. 7, pp. 945–956, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- G. Chi, G. Li, and X. Jia, “Numerical inversions of a source term in the FADE with a Dirichlet boundary condition using final observations,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1619–1626, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- F. A. Rodrigues, H. R. B. Orlande, and G. S. Dulikravich, “Simultaneous estimation of spatially dependent diffusion coefficient and source term in a nonlinear 1D diffusion problem,” Mathematics and Computers in Simulation, vol. 66, no. 4-5, pp. 409–424, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- G. S. Li, J. Cheng, D. Yao, H. L. Liu, and J. J. Liu, “One-dimensional equilibrium model and source parameter determination for soil-column experiment,” Applied Mathematics and Computation, vol. 190, pp. 1365–1374, 2007. View at Google Scholar
- G. S. Li, Y. J. Tan, D. Yao, X. Q. Wang, and H. L. Liu, “A nonlinear mathematical model for undisturbed soil-column experiment and source parameter identification,” Inverse Problems in Science and Engineering, vol. 16, no. 7, pp. 885–901, 2008. View at Google Scholar
- G. S. Li, D. Yao, H. Y. Jiang, and X. Z. Jia, “Numerical inversion of a time-dependent reaction coefficient in a soil-column infiltrating experiment,” Computer Modeling in Engineering & Sciences, vol. 74, no. 2, pp. 83–107, 2011. View at Google Scholar · View at Zentralblatt MATH
- G. S. Li, D. Yao, and Y. Z. Wang, “Data reconstruction for a disturbed soil-column experimentusing an optimal perturbation regularization algorithm,” Journal of Applied Mathematics, vol. 2012, Article ID Article ID 732791, 16 pages, 2012. View at Google Scholar
- J. R. Cannon and P. DuChateau, “Structural identification of an unknown source term in a heat equation,” Inverse Problems, vol. 14, no. 3, pp. 535–551, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- B. Han, G. F. Feng, and J. Q. Liu, “A widely convergent generalized pulse-spectrum technique for the inversion of two-dimensional acoustic wave equation,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 406–420, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- W. Chen, J. Cheng, J. Lin, and L. Wang, “A level set method to reconstruct the discontinuity of the conductivity in EIT,” Science in China A, vol. 52, no. 1, pp. 29–44, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- J. Cheng, S. Lu, and M. Yamamoto, “Reconstruction of the Stefan-Boltzmann coefficients in a heat-transfer process,” Inverse Problems, vol. 28, no. 4, Article ID 045007, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- C. W. Su, Numerical Methods and Applications of Inverse Problems in PDE, Northwestern Polytechnical University Press, Xi’an, China, 1995, (in Chinese).
- A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, NY, USA, 1996. View at Publisher · View at Google Scholar