An inverse source identification problem for a time fractional diffusion equation is discussed. The unknown heat source is supposed to be space dependent only. Based on the use of Green’s function, an effective numerical algorithm is developed to recover both the intensities and locations of unknown point sources from final measurements. Numerical results indicate that the proposed method is efficient and accurate. 1. Introduction Let be a bounded domain in and let be the boundary of . Consider the following time fractional diffusion process: where is the uniformly elliptic operator, is the outward normal at the boundary , and , are known constants which are not simultaneously zero. Here, stands for the Caputo fractional derivative operator of order defined by where is the standard -function and the prime denotes the general derivative. From the last few decades, fractional calculus grabbed great attention of not only mathematicians and engineers but also many scientists from all fields (e.g., see [1–4]). Fractional diffusion equations describe anomalous diffusions on fractals (physical objects of fractional dimension, like some amorphous semiconductors or strongly porous materials; see [5, 6] and references therein). Indeed, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main superiority of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. For the detailed theory and application of fractional calculus, one can refer to [1–4] and references therein. Not only have differential equations of fractional order attracted people’s attention, but also theories and applications related to physics and geometry of fractal dimension have been well studied (e.g., [7–11]). If the initial condition is nonhomogeneous, that is, , we are always able to simplify the system (1) into two components; that is, , where solves the homogeneous equation with nonhomogeneous initial condition and satisfies the nonhomogeneous equation with homogeneous initial condition. As we know, the initial value/boundary value problem associated with is well-posed and there exist many works on such forward problem, for example, [12, 13]. In the following, instead of nonhomogeneous initial condition, we only focus on the system (1) with homogeneous initial condition. Ordinarily, when is a known function, we are asked to determine the solution function so as to satisfy (1). So posed, this is a direct problem. However, the
References
[1]
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.
[2]
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
[3]
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.
[4]
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
[5]
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.
[6]
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.
[7]
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.
[8]
M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010.
[9]
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.
[10]
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.
[11]
M. Li and W. Zhao, “On noise,” Mathematical Problems in Engineering, vol. 2012, Article ID 673648, 23 pages, 2012.
[12]
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.
[13]
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.
[14]
V. Isakov, Inverse Problems for Partial Differential Equations, vol. 127 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1998.
[15]
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.
[16]
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.
[17]
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.
[18]
S. D'haeyer, B. T. Johansson, and M. Slodi？ka, “Reconstruction of a spacewise-dependent heat source in a time-dependent heat diffusion process,” IMA Journal of Applied Mathematics, 2012.
[19]
V. Isakov, “Inverse parabolic problems with the final overdetermination,” Communications on Pure and Applied Mathematics, vol. 44, no. 2, pp. 185–209, 1991.
[20]
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.
[21]
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.
[22]
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.
[23]
G. A. Kriegsmann and W. E. Olmstead, “Source identification for the heat equation,” Applied Mathematics Letters, vol. 1, no. 3, pp. 241–245, 1988.
[24]
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.
[25]
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.
[26]
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.
[27]
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.
[28]
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.
[29]
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.
[30]
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.
[31]
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.
[32]
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.
[33]
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.
[34]
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.
[35]
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.
[36]
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.
[37]
P. C. Hansen and D. P. O'Leary, “The use of the -curve in the regularization of discrete ill-posed problems,” SIAM Journal on Scientific Computing, vol. 14, no. 6, pp. 1487–1503, 1993.
[38]
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.
