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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 109340, 9 pages
A Mollification Regularization Method for a Fractional-Diffusion Inverse Heat Conduction Problem
1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China
2School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
3Department of Mathematical Sciences, Xidian University, Xi'an 710071, China
Received 12 June 2012; Revised 19 December 2012; Accepted 20 December 2012
Academic Editor: Fatih Yaman
Copyright © 2013 Zhi-Liang Deng 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.
- V. V. Kulish and J. L. Lage, “Fractional-diffusion solution for transient local temperature and heat flux,” Transactions of ASME, vol. 122, pp. 372–376, 2000.
- K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differential and Integration to Arbitrary Order, Academic Press, 1974.
- K. B. Oldham and J. Spanier, “A general solution of the diffusion equation for semiinfinite geometries,” Journal of Mathematical Analysis and Applications, vol. 39, pp. 655–669, 1972.
- K. B. Oldham and J. Spanier, “The replacement of Fick’s law by a formulation involving semidifferentiation,” Journal of Electroanalytical Chemistry, vol. 26, pp. 331–341, 1970.
- R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, “Time fractional diffusion: a discrete random walk approach,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 129–143, 2002.
- V. V. Anh and N. N. Leonenko, “Non-Gaussian scenarios for the heat equation with singular initial conditions,” Stochastic Processes and their Applications, vol. 84, no. 1, pp. 91–114, 1999.
- I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
- M. M. El-Borai, “The fundamental solutions for fractional evolution equations of parabolic type,” Boletín de la Asociación Matemática Venezolana, vol. 6, no. 1, pp. 29–43, 2004.
- F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001.
- F. Liu, V. Anhb, and I. Turnerb, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, pp. 209–219, 2004.
- 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.
- S. B. Yuste, “Weighted average finite difference methods for fractional diffusion equations,” Journal of Computational Physics, vol. 216, no. 1, pp. 264–274, 2006.
- G. J. Fix and J. P. Roop, “Least squares finite-element solution of a fractional order two-point boundary value problem,” Computers & Mathematics with Applications, vol. 48, no. 7-8, pp. 1017–1033, 2004.
- J. P. Roop, “Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in ,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 243–268, 2006.
- Z. M. Odibat and S. Momani, “Approximate solutions for boundary value problems of time-fractional wave equation,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 767–774, 2006.
- S. S. Ray and R. K. Bera, “Analytical solution of a fractional diffusion equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 329–336, 2006.
- 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.
- W. McLean, “Regularity of solutions to a time-fractional diffusion equation,” The ANZIAM Journal, vol. 52, no. 2, pp. 123–138, 2010.
- J. Prüss, Evolutionary Integral Equations and Applications, vol. 87, Birkhäuser, Basel, Switzerland, 1993.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Drivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
- D. A. Murio, “Stable numerical solution of a fractional-diffusion inverse heat conduction problem,” Computers & Mathematics with Applications, vol. 53, no. 10, pp. 1492–1501, 2007.
- H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, 1996.
- L. Yang, Z.-C. Deng, J.-N. Yu, and G.-W. Luo, “Two regularization strategies for an evolutional type inverse heat source problem,” Journal of Physics A, vol. 42, no. 36, Article ID 365203, 16 pages, 2009.
- L. Yang, Z.-C. Deng, J.-N. Yu, and G.-W. Luo, “Optimization method for the inverse problem of reconstructing the source term in a parabolic equation,” Mathematics and Computers in Simulation, vol. 80, no. 2, pp. 314–326, 2009.
- L. Yang, M. Dehghan, J.-N. Yu, and G.-W. Luo, “Inverse problem of time-dependent heat sources numerical reconstruction,” Mathematics and Computers in Simulation, vol. 81, no. 8, pp. 1656–1672, 2011.
- L. Yang, J. N. Yu, G. W. Luo, and Z. C. Deng, “Reconstruction of a space and time dependent heat source from finite measurement data,” International Journal of Heat and Mass Transfer, vol. 55, pp. 6573–6581, 2012.
- L. Yang, J. N. Yu, G. W. Luo, and Z. C. Deng, “Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate system,” Applied Mathematical Modelling, vol. 37, pp. 939–957, 2013.
- F. F. Dou and Y. C. Hon, “Kernel-based approximation for Cauchy problem of the time-fractional diffusion equation,” Engineering Analysis with Boundary Elements, vol. 36, no. 9, pp. 1344–1352, 2012.
- Y. C. Hon and T. Wei, “A fundamental solution method for inverse heat conduction problem,” Engineering Analysis With Boundary Elements, vol. 28, no. 5, pp. 489–495, 2004.
- 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, pp. 1–16, 2009.
- B. Jin and W. Rundell, “An inverse problem for a one-dimensional time-fractional diffusion problem,” Inverse Problems, vol. 28, no. 7, Article ID 075010, 19 pages, 2012.
- D. N. Hào, “A mollification method for ill-posed problems,” Numerische Mathematik, vol. 68, no. 4, pp. 469–506, 1994.
- U. Tautenhahn, “Optimality for ill-posed problems under general source conditions,” Numerical Functional Analysis and Optimization, vol. 19, no. 3-4, pp. 377–398, 1998.