Abstract and Applied Analysis

A Note on the Inverse Problem for a Fractional Parabolic Equation

Abdullah Said Erdogan and Hulya Uygun

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Abstract

For a fractional inverse problem with an unknown time-dependent source term, stability estimates are obtained by using operator theory approach. For the approximate solutions of the problem, the stable difference schemes which have first and second orders of accuracy are presented. The algorithm is tested in a one-dimensional fractional inverse problem.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 276080, 26 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174073

Digital Object Identifier
doi:10.1155/2012/276080

Mathematical Reviews number (MathSciNet)
MR2969995

Zentralblatt MATH identifier
1253.35217

Citation

Erdogan, Abdullah Said; Uygun, Hulya. A Note on the Inverse Problem for a Fractional Parabolic Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 276080, 26 pages. doi:10.1155/2012/276080. https://projecteuclid.org/euclid.aaa/1365174073


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References

  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • A. Ashyralyev, “On a problem of determining the parameter of a parabolic equation,” Ukranian Mathematical Journal, vol. 62, no. 9, pp. 1200–1210, 2010.
  • 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.
  • 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.
  • 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.
  • 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.\setlengthemsep1.2pt
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
  • A. A. Kilbas, H. M. Sristava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science, 2006.
  • J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,” SIAM Review, vol. 18, no. 2, pp. 240–268, 1976.
  • V. E. Tarasov, “Fractional derivative as fractional power of derivative,” International Journal of Mathematics, vol. 18, no. 3, pp. 281–299, 2007.
  • 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.
  • D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in System Application, vol. 2, Lille, France, 1996.
  • A. B. Basset, “On the descent of a sphere in a viscous liquid,” Quarterly Journal of Mathematics, vol. 42, pp. 369–381, 1910.
  • 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.
  • 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.
  • 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.
  • A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736–750, 2011.
  • Y. Zhang, “A finite difference method for fractional partial differential equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 524–529, 2009.
  • 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.
  • 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).
  • 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.
  • 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, Ayd\in, Turkey, June 2011.
  • A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1176–1180, 2011.
  • 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.
  • 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.
  • 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.
  • Y. Zhang and X. Xu, “Inverse source problem for a fractional diffusion equation,” Inverse Problems, vol. 27, no. 3, pp. 1–12, 2011.
  • 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.
  • 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.
  • 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.
  • 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.