On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process |
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| Institution: | 1. Department of Mathematics, Southeast University, Nanjing, 210096, PR China;2. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan;1. Department of Mathematics, Penza State University, 40, Krasnaya Str., Penza, 440026, Russia;2. Department of Engineering Science and Mechanics, The Pennsylvania State University, 203 B Earth and Engineering Science Building, University Park, PA 16802-1401, USA;3. St-Petersburg State University, 198504, St-Petersburg, 1 Ulianovskaya Str., Russia;1. Department of Applied Chemistry, Indian School of Mines, Dhanbad, Jharkhand 826004, India;2. Smart Materials and Biodevices, Biosensors and Bioelectronics Centre, IFM-Linköpings Universitet, 581 83 Linköping, Sweden;3. Functional Nanomaterials Research Laboratory, Department of Applied Physics, Indian School of Mines, Dhanbad, Jharkhand 826004, India;1. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland;2. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland;3. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan;1. College of Chemical Science and Engineering, Laboratory of Fiber Materials and Modern Textile, The Growing Base for State Key Laboratory, Qingdao University, Qingdao, China;2. Centre for Chemistry and Biotechnology, Deakin University, Geelong, VIC 3217, Australia |
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| Abstract: | Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme. |
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| Keywords: | Inverse problem Fractional derivative Uniqueness Regularization Convergence Numerics |
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