Affiliation:
1. School of Mathematics and Statistics, Shandong University of Technology , Zibo, 255049 , China
Abstract
Abstract
In this article, we consider the identification of an unknown steady source in a class of fractional diffusion equations. A modified Tikhonov regularization method based on Hermite expansion is presented to deal with the ill-posedness of the problem. By using the properties of Hermitian functions, we construct a modified penalty term for the Tikhonov functional. It can be proved that the method can adaptively achieve the order optimal results when we choose the regularization parameter by the discrepancy principle. Some examples are also provided to verify the effectiveness of the method.
Reference23 articles.
1. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2001), no. 4, 230–237.
2. R. Metzler and J. Klafter, The random walkas guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 1–77.
3. R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), no. 31, R161.
4. W. Y. Tian, C. Li, W. H. Deng, and Y. J. Wu, Regularization methods for unknown source in space fractional diffusion equation, Math. Comput. Simulation 85 (2012), 45–56.
5. J. R. Cannon and P. Duchateau, Structural identification of an unknown source term in a heat equation, Inverse Problems 14 (1999), no. 3, 535–551.