- About this Journal
- Abstracting and Indexing
- 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
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 603018, 14 pages
The Difference Problem of Obtaining the Parameter of a Parabolic Equation
1Department of Computer Technology, Turkmen Agricultural University, Gerology Street,74400 Asgabat, Turkmenistan
2Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey
3Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey
Received 4 April 2012; Accepted 23 April 2012
Academic Editor: Ravshan Ashurov
Copyright © 2012 Charyyar Ashyralyyev and Oznur Demirdag. 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.
- M. Dehghan, “Determination of a control parameter in the two-dimensional diffusion equation,” Applied Numerical Mathematics, vol. 37, no. 4, pp. 489–502, 2001.
- T. Kimura and T. Suzuki, “A parabolic inverse problem arising in a mathematical model for chromatography,” SIAM Journal on Applied Mathematics, vol. 53, no. 6, pp. 1747–1761, 1993.
- Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography,” Inverse Problems, vol. 2, no. 5, pp. 373–397, 1999.
- Y. S. Eidelman, Boundary value problems for di¤erential equations with parameters [Ph.D. thesis], Voronezh State University, Voronezh, Russia, 1984.
- Y. S. Eidelman, “Conditions for the solvability of inverse problems for evolution equations,” Doklady Akademii Nauk Ukrainskoj SSR Serija A, no. 7, pp. 28–31, 1990 (Russian).
- A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Di¤erential Equations, Operator Theory Advances and Applications, Birkhäuser, Boston, Berlin, 2004.
- R. P. Agarwal and V. B. Shakhmurov, “Multipoint problems for degenerate abstract differential equations,” Acta Mathematica Hungarica, vol. 123, no. 1-2, pp. 65–89, 2009.
- A. O. Ashyralyev and P. E. Sobolevskiĭ, “The linear operator interpolation theory and the stability of difference schemes,” Doklady Akademii Nauk SSSR, vol. 275, no. 6, pp. 1289–1291, 1984 (Russian).
- A. Ashyralyev, I. Karatay, and P. E. Sobolevskii, “Well-posedness of the nonlocal boundary value problem for parabolic difference equations,” Discrete Dynamics in Nature and Society, no. 2, pp. 273–286, 2004.
- A. Ashyralyev, A. Hanalyev, and P. E. Sobolevskii, “Coercive solvability of the nonlocal boundary value problem for parabolic differential equations,” Abstract and Applied Analysis, vol. 6, no. 1, pp. 53–61, 2001.
- A. Ashyralyev, “High-accuracy stable difference schemes for well-posed NBVP,” in Modern Analysis and Applications, vol. 191 of Operator Theory: Advances and Applications, pp. 229–252, 2009.
- A. Ashyralyev, “A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 557–573, 2008.
- Y. S. Eidelman, “Two-point boundary value problem for a differential equation with a parameter,” Dopovidi Akademii Nauk Ukrainskoi RSR Seriya A-Fiziko-Matematichni ta Technichni Nauki, no. 4, pp. 15–18, 1983 (Russian).
- A. I. Prilepko, “Inverse problems of potential theory,” Matematicheskie Zametki, vol. 14, pp. 755–767, 1973, English translation Mathematical Notes, vol. 14, 1973.
- A. D. Iskenderov and R. G. Tagiev, “ The inverse problem of determining the right-hand sides of evolution equations in Banach space,” Nauchnyye Trudy Azerbaidzhanskogo Gosudarstvennogo Universiteta, no. 1, pp. 51–56, 1979 (Russian).
- W. Rundell, “Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data,” Applicable Analysis, vol. 10, no. 3, pp. 231–242, 1980.
- A. I. Prilepko and I. A. Vasin, “Some time-dependent inverse problems of hydrodynamics with final observation,” Doklady Akademii Nauk SSSR, vol. 314, no. 5, pp. 1075–1078, 1990, English translation Soviet Mathematics. Doklady, vol. 42, 1991.
- A. I. Prilepko and A. B. Kostin, “On certain inverse problems for parabolic equations with final and integral observation,” Matematicheskii Sbornik, vol. 183, no. 4, pp. 49–68, 1992, English translation Russian Academy of Sciences. Sbornik Mathematics, vol. 75, 1993.
- A. I. Prilepko and I. V. Tikhonov, “Uniqueness of the solution of an inverse problem for an evolution equation and applications to the transfer equation,” Matematicheskie Zametki, vol. 51, no. 2, pp. 77–87, 1992, English translation Mathematical Notes, vol. 2, pp. 77–87, 1992.
- D. G. Orlovskii, “On a problem of determining the parameter of an evolution equation,” Differentsial'nye Uravneniya, vol. 26, no. 9, pp. 1614–1621, 1990, English translation Differential Equations, vol. 26, 1990.
- J. R. Cannon, Y. L. Lin, and S. Xu, “Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations,” Inverse Problems, vol. 10, no. 2, pp. 227–243, 1994.
- M. Dehghan, “Finding a control parameter in one-dimensional parabolic equations,” Applied Mathematics and Computation, vol. 135, no. 1-2, pp. 491–503, 2003.
- A. Ashyralyev, “On the problem of determining the parameter of a parabolic equation,” Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1397–1408, 2011.
- A. Ashyralyev and O. Demirdag, “A note on boundary value parabolic problems,” Vestnik of Odessa National University. Mathematics and Mechanics, vol. 16, no. 16, pp. 131–143, 2010.
- A. Ashyralyev and O. Demirdağ, “A note on a problem of obtaining the parameter of a parabolic equation,” International Journal of Mathematics and Computation, vol. 11, no. 11, pp. 10–20, 2011.
- C.-R. Ye and Z.-Z. Sun, “On the stability and convergence of a difference scheme for an one-dimensional parabolic inverse problem,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 214–225, 2007.
- A. Ashyralyev and O. Demirdağ, “On the numerical solution of parabolic equation with the Neumann condition arising in determination of a control parameter,” AIP Conference Proceedings, vol. 1389, pp. 613–616, 2011.
- A. Ashyralyev and A. S. Erdoğan, “On the numerical solution of a parabolic inverse problem with the Dirichlet condition,” International Journal of Mathematics and Computation, vol. 11, no. 11, pp. 73–81, 2011.
- A. Ashyralyev and A. S. Erdogan, “Well-posedness of the inverse problem of a multidimensional parabolic equation,” Vestnik of Odessa National University. Mathematicsand Mechanics, vol. 15, no. 18, pp. 129–135, 2010.
- A. Ashyralyev, “Fractional spaces generated by the positivite di¤erential and difference operator in a Banach space,” in Proceedings of the Mathematical Methods and Engineering, K. Taş, Ed., pp. 13–22, Springer, The Netherlands, 2007.
- A. Ashyralyev and P. E. Sobolevskiĭ, Well-Posedness of Parabolic Difference Equations, Operator Theory Advances and Applications, Birkhäuser, Boston, Berlin, 1994.
- Y. A. Smirnitskii and P. E. Sobolevskii, “Positivity of multidimensional di¤erence operators in the C-norm,” Uspekhi Matematicheskikh Nauk, vol. 36, no. 4, pp. 202–203, 1981 (Russian).
- Y. A. Smirnitskii, Fractional powers of elliptic di¤erence operators [Ph.D. thesis], Voronezh State University, Voronezh, Russia, 1983.
- P. E. Sobolevskiĭ, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971 (Russian).