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Advances in Mathematical Physics

Volume 2013 (2013), Article ID 485273, 8 pages

http://dx.doi.org/10.1155/2013/485273

Research Article

## A Point Source Identification Problem for a Time Fractional Diffusion Equation

^{1}School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China^{2}School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China

Received 19 September 2013; Revised 20 October 2013; Accepted 21 October 2013

Academic Editor: Ming Li

Copyright © 2013 Xiao-Mei Yang and Zhi-Liang Deng. 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

- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science, Amsterdam, The Netherlands, 2006. View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet - J. Sabatier, O. P. Agrawal, and J. A. T. Machado,
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives*, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at MathSciNet - V. V. Anh and N. N. Leonenko, “Spectral analysis of fractional kinetic equations with random data,”
*Journal of Statistical Physics*, vol. 104, no. 5-6, pp. 1349–1387, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,”
*Physics Reports*, vol. 339, no. 1, p. 77, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Cattani, A. Ciancio, and B. Lods, “On a mathematical model of immune competition,”
*Applied Mathematics Letters*, vol. 19, no. 7, pp. 678–683, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Li, “Fractal time series—a tutorial review,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 157264, 26 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Li, Y. Q. Chen, J. Y. Li, and W. Zhao, “Holder scales of sea level,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 863707, 22 pages, 2012. View at Publisher · View at Google Scholar - M. Li, W. Zhao, and C. Cattani, “Delay bound: fractal traffic passes through servers,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 157636, 15 pages, 2013. View at Publisher · View at Google Scholar - M. Li and W. Zhao, “On $1/f$ noise,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 673648, 23 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. M. Khader, “On the numerical solutions for the fractional diffusion equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 6, pp. 2535–2542, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Luchko, “Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation,”
*Computers & Mathematics with Applications*, vol. 59, no. 5, pp. 1766–1772, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Isakov,
*Inverse Problems for Partial Differential Equations*, vol. 127 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1998. View at MathSciNet - E. C. Baran and A. G. Fatullayev, “Determination of an unknown source parameter in two-dimensional heat equation,”
*Applied Mathematics and Computation*, vol. 159, no. 3, pp. 881–886, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. de Cezaro and B. T. Johansson, “A note on uniqueness in the identification of a spacewise dependent source and diffusion coefficient for the heat equation,” http://arxiv.org/abs/1210.7346.
- A. de Cezaro and F. T. de Cezaro, “Uniqueness and regularization for unknown spacewise lower-order coefficient and source for the heat type equation,” http://arxiv.org/abs/1210.7348.
- S. D'haeyer, B. T. Johansson, and M. Slodička, “Reconstruction of a spacewise-dependent heat source in a time-dependent heat diffusion process,”
*IMA Journal of Applied Mathematics*, 2012. View at Publisher · View at Google Scholar - V. Isakov, “Inverse parabolic problems with the final overdetermination,”
*Communications on Pure and Applied Mathematics*, vol. 44, no. 2, pp. 185–209, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Johansson and D. Lesnic, “Determination of a spacewise dependent heat source,”
*Journal of Computational and Applied Mathematics*, vol. 209, no. 1, pp. 66–80, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. T. Johansson and D. Lesnic, “A procedure for determining a spacewise dependent heat source and the initial temperature,”
*Applicable Analysis*, vol. 87, no. 3, pp. 265–276, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. A. Kaliev and M. M. Sabitova, “Problems of the determination of the temperature and density of heat sources from the initial and final temperatures,”
*Journal of Applied and Industrial Mathematics*, vol. 4, no. 3, pp. 332–339, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - G. A. Kriegsmann and W. E. Olmstead, “Source identification for the heat equation,”
*Applied Mathematics Letters*, vol. 1, no. 3, pp. 241–245, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Rundell, “The determination of a parabolic equation from initial and final data,”
*Proceedings of the American Mathematical Society*, vol. 99, no. 4, pp. 637–642, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Yan, C.-L. Fu, and F.-L. Yang, “The method of fundamental solutions for the inverse heat source problem,”
*Engineering Analysis with Boundary Elements*, vol. 32, no. 3, pp. 216–222, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - L. Yan, F.-L. Yang, and C.-L. Fu, “A meshless method for solving an inverse spacewise-dependent heat source problem,”
*Journal of Computational Physics*, vol. 228, no. 1, pp. 123–136, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. C. Hon, M. Li, and Y. A. Melnikov, “Inverse source identification by Green's function,”
*Engineering Analysis with Boundary Elements*, vol. 34, no. 4, pp. 352–358, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. F. M. Martins, “An iterative shape reconstruction of source functions in a potential problem using the MFS,”
*Inverse Problems in Science and Engineering*, vol. 20, no. 8, pp. 1175–1193, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - L. Ling, Y. C. Hon, and M. Yamamoto, “Inverse source identification for Poisson equation,”
*Inverse Problems in Science and Engineering*, vol. 13, no. 4, pp. 433–447, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Kirane and S. A. Malik, “Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time,”
*Applied Mathematics and Computation*, vol. 218, no. 1, pp. 163–170, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. A. Murio and C. E. Mejía, “Source terms identification for time fractional diffusion equation,”
*Revista Colombiana de Matemáticas*, vol. 42, no. 1, pp. 25–46, 2008. View at Zentralblatt MATH · View at MathSciNet - J. G. Wang, Y. B. Zhou, and T. Wei, “Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation,”
*Applied Numerical Mathematics*, vol. 68, pp. 39–57, 2013. 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, 12 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. W. Engl, M. Hanke, and A. Neubauer,
*Regularization of Inverse Problems*, vol. 375 of*Mathematics and its Applications*, Kluwer Academic, Dordrecht, The Netherlands, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - P. C. Hansen,
*Rank-Deficient and Discrete Ill-Posed Problems*, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - P. C. Hansen, “Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems,”
*Numerical Algorithms*, vol. 6, no. 1-2, pp. 1–35, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. C. Hansen and D. P. O'Leary, “The use of the $L$-curve in the regularization of discrete ill-posed problems,”
*SIAM Journal on Scientific Computing*, vol. 14, no. 6, pp. 1487–1503, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, A. Carpinteri and F. Mainardi, Eds., pp. 223–276, Springer, New York, NY, USA, 1997.