Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 276080, 26 pages
http://dx.doi.org/10.1155/2012/276080
Research Article

A Note on the Inverse Problem for a Fractional Parabolic Equation

Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

Received 15 May 2012; Accepted 8 July 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Abdullah Said Erdogan and Hulya Uygun. 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

  1. A. Hasanov, “Identification of unknown diffusion and convection coefficients in ion transport problems from flux data: an analytical approach,” Journal of Mathematical Chemistry, vol. 48, no. 2, pp. 413–423, 2010. View at Publisher · View at Google Scholar
  2. A. Hasanov and S. Tatar, “An inversion method for identification of elastoplastic properties of a beam from torsional experiment,” International Journal of Non-Linear Mechanics, vol. 45, pp. 562–571, 2010. View at Google Scholar
  3. G. Di Blasio and A. Lorenzi, “Identification problems for parabolic delay differential equations with measurement on the boundary,” Journal of Inverse and Ill-Posed Problems, vol. 15, no. 7, pp. 709–734, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. D. Orlovsky and S. Piskarev, “On approximation of inverse problems for abstract elliptic problems,” Journal of Inverse and Ill-Posed Problems, vol. 17, no. 8, pp. 765–782, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. Y. S. Eidelman, “A boundary value problem for a differential equation with a parameter,” Differentsia'nye Uravneniya, vol. 14, no. 7, pp. 1335–1337, 1978. View at Google Scholar
  6. A. Ashyralyev, “On a problem of determining the parameter of a parabolic equation,” Ukranian Mathematical Journal, vol. 62, no. 9, pp. 1200–1210, 2010. View at Google Scholar
  7. V. Serov and L. Päivärinta, “Inverse scattering problem for two-dimensional Schrödinger operator,” Journal of Inverse and Ill-Posed Problems, vol. 14, no. 3, pp. 295–305, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. V. T. Borukhov and P. N. Vabishchevich, “Numerical solution of the inverse problem of reconstructing a distributed right-hand side of a parabolic equation,” Computer Physics Communications, vol. 126, no. 1-2, pp. 32–36, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-Posed Problems Series, Walter de Gruyter, Berlin, Germany, 2007. View at Publisher · View at Google Scholar
  10. A. I. Prilepko and A. B. Kostin, “Some inverse problems for parabolic equations with final and integral observation,” Matematicheskiĭ Sbornik, vol. 183, no. 4, pp. 49–68, 1992. View at Publisher · View at Google Scholar
  11. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  12. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
  13. A. A. Kilbas, H. M. Sristava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science, 2006.
  14. J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,” SIAM Review, vol. 18, no. 2, pp. 240–268, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. V. E. Tarasov, “Fractional derivative as fractional power of derivative,” International Journal of Mathematics, vol. 18, no. 3, pp. 281–299, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. 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., vol. 378 of CISM Courses and Lectures, pp. 223–276, Springer, Vienna, Austria, 1997. View at Google Scholar
  17. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in System Application, vol. 2, Lille, France, 1996. View at Google Scholar
  18. A. B. Basset, “On the descent of a sphere in a viscous liquid,” Quarterly Journal of Mathematics, vol. 42, pp. 369–381, 1910. View at Google Scholar
  19. F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 291–348, Springer, New York, NY, USA, 1997. View at Google Scholar · View at Zentralblatt MATH
  20. A. Ashyralyev, F. Dal, and Z. Pinar, “On the numerical solution of fractional hyperbolic partial differential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 730465, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736–750, 2011. View at Publisher · View at Google Scholar
  23. Y. Zhang, “A finite difference method for fractional partial differential equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 524–529, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. E. Cuesta, C. Lubich, and C. Palencia, “Convolution quadrature time discretization of fractional diffusion-wave equations,” Mathematics of Computation, vol. 75, no. 254, pp. 673–696, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. P. E. Sobolevskii, “Some properties of the solutions of differential equations in fractional spaces,” Trudy Naucno-Issledovatel'skogi Instituta Matematiki VGU, vol. 14, pp. 68–74, 1975 (Russian). View at Google Scholar
  26. G. Da Prato and P. Grisvard, “Sommes d'opérateurs linéaires et équations différentielles opérationnelles,” Journal de Mathématiques Pures et Appliquées, vol. 54, no. 3, pp. 305–387, 1975. View at Google Scholar · View at Zentralblatt MATH
  27. A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differential equations with the Dirichlet condition,” in Proceedings of the 2nd International Symposium on Computing in Science and Engineering (ISCSE '11), M. Gunes, Ed., pp. 529–530, Kusadasi, Aydın, Turkey, June 2011.
  28. A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1176–1180, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. D. A. Murio and C. E. Mejía, “Generalized time fractional IHCP with Caputo fractional derivatives,” in Proceedings of the 6th International Conference on Inverse Problems in Engineering: Theory and Practice, vol. 135 of Journal of Physics: Conference Series, pp. 1–8, Dourdan, Paris, France, 2008.
  30. 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
  31. J. Nakagawa, K. Sakamoto, and M. Yamamoto, “Overview to mathematical analysis for fractional diffusion equations—new mathematical aspects motivated by industrial collaboration,” Journal of Math-for-Industry, vol. 2A, pp. 99–108, 2010. View at Google Scholar · View at Zentralblatt MATH
  32. Y. Zhang and X. Xu, “Inverse source problem for a fractional diffusion equation,” Inverse Problems, vol. 27, no. 3, pp. 1–12, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. K. Sakamoto and M. Yamamoto, “Inverse source problem with a final overdetermination for a fractional diffusion equation,” Mathematical Control and Related Fields, vol. 1, no. 4, pp. 509–518, 2011. View at Publisher · View at Google Scholar
  34. A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser Verlag, Berlin, Germany, 2004. View at Publisher · View at Google Scholar
  36. A. Ashyralyev, “Fractional spaces generated by the positive differential and difference operators in a Banach space,” in Mathematical Methods in Engineering, K. Taş, J. A. Tenreiro Machado, and D. Baleanu, Eds., pp. 13–22, Springer, Dordrecht, The Netherlands, 2007. View at Google Scholar