Journal Menu

- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 109340, 9 pages

http://dx.doi.org/10.1155/2013/109340

Research Article

## A Mollification Regularization Method for a Fractional-Diffusion Inverse Heat Conduction Problem

^{1}School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China^{2}School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China^{3}Department 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.

#### Linked References

- 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. View at Google Scholar - K. B. Oldham and J. Spanier,
*The Fractional Calculus: Theory and Application of Differential and Integration to Arbitrary Order*, Academic Press, 1974. View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. B. Yuste, “Weighted average finite difference methods for fractional diffusion equations,”
*Journal of Computational Physics*, vol. 216, no. 1, pp. 264–274, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. P. Roop, “Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in ${R}^{2}$,”
*Journal of Computational and Applied Mathematics*, vol. 193, no. 1, pp. 243–268, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. McLean, “Regularity of solutions to a time-fractional diffusion equation,”
*The ANZIAM Journal*, vol. 52, no. 2, pp. 123–138, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Prüss,
*Evolutionary Integral Equations and Applications*, vol. 87, Birkhäuser, Basel, Switzerland, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, 2006. View at MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Drivatives*, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at MathSciNet - 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. 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*, Kluwer Academic, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. N. Hào, “A mollification method for ill-posed problems,”
*Numerische Mathematik*, vol. 68, no. 4, pp. 469–506, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet