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A note on the inverse problem for a fractional parabolic equation. (English) Zbl 1253.35217
Summary: 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.

35R30Inverse problems for PDE
35R11Fractional partial differential equations
Full Text: DOI
[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. · Zbl 1302.93075 · doi:10.1007/s10910-010-9683-5
[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. · Zbl 05713105
[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. · Zbl 1132.35388 · doi:10.1515/jiip.2007.039
[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. · Zbl 1195.65159 · doi:10.1515/JIIP.2009.045
[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. · Zbl 0423.34026
[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. · Zbl 1240.35305
[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. · Zbl 1111.35126 · doi:10.1163/156939406777340946
[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. · Zbl 0966.65067 · doi:10.1016/S0010-4655(99)00416-6
[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. · Zbl 1136.65105 · doi:10.1515/9783110205794
[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. · Zbl 0802.35158 · doi:10.1070/SM1993v075n02ABEH003394
[11] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[12] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. · Zbl 0818.26003
[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. · Zbl 0324.44002 · doi:10.1137/1018042
[15] V. E. Tarasov, “Fractional derivative as fractional power of derivative,” International Journal of Mathematics, vol. 18, no. 3, pp. 281-299, 2007. · Zbl 1119.26011 · doi:10.1142/S0129167X07004102
[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.
[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.
[18] A. B. Basset, “On the descent of a sphere in a viscous liquid,” Quarterly Journal of Mathematics, vol. 42, pp. 369-381, 1910. · Zbl 41.0826.01
[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. · Zbl 0917.73004
[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. · Zbl 1184.65083 · doi:10.1155/2009/730465 · eudml:45895
[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. · Zbl 1175.26004 · doi:10.1016/j.jmaa.2009.04.012
[22] A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736-750, 2011. · doi:10.1108/03684921111142287
[23] Y. Zhang, “A finite difference method for fractional partial differential equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 524-529, 2009. · Zbl 1177.65198 · doi:10.1016/j.amc.2009.05.018
[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. · Zbl 1090.65147 · doi:10.1090/S0025-5718-06-01788-1
[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).
[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. · Zbl 0315.47009
[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. · Zbl 1217.34006 · doi:10.1016/j.aml.2011.02.002
[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. · Zbl 1165.65386
[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. · Zbl 1181.35322 · doi:10.1088/0266-5611/25/11/115002
[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. · Zbl 1206.35247 · http://gcoe-mi.jp/english/./temp/publish/a3e3b09f4bedd92df6c8ef07b57d5568.pdf
[32] Y. Zhang and X. Xu, “Inverse source problem for a fractional diffusion equation,” Inverse Problems, vol. 27, no. 3, pp. 1-12, 2011. · Zbl 1211.35280 · doi:10.1088/0266-5611/27/3/035010
[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. · Zbl 1241.35220 · doi:10.3934/mcrf.2011.1.509
[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. · Zbl 1175.26004 · doi:10.1016/j.jmaa.2009.04.012
[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. · Zbl 1060.65055 · doi:10.1007/978-3-0348-7922-4
[36] A. Ashyralyev, “Fractional spaces generated by the positive differential and difference operators in a Banach space,” in Mathematical Methods in Engineering, K. Ta\cs, J. A. Tenreiro Machado, and D. Baleanu, Eds., pp. 13-22, Springer, Dordrecht, The Netherlands, 2007. · Zbl 1130.46303 · doi:10.1007/978-1-4020-5678-9_2