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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 294154, 17 pages
Finite Difference Method for the Reverse Parabolic Problem
1Department of Computer Technology of the Turkmen Agricultural University, Gerogly Street, 74400 Ashgabat, Turkmenistan
2Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey
3Gaziosmanpaşa Lisesi, 34245 Istanbul, Turkey
4Department of Mathematics, Fatih University, 34500 Istanbul, Turkey
Received 17 April 2012; Accepted 12 June 2012
Academic Editor: Valery Covachev
Copyright © 2012 Charyyar Ashyralyyev 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.
- O. A. Ladyzhenskaya, V.A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Providence, RI, USA, 1968.
- O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York, NY, USA, 1968.
- M. L. Vishik, A. D. Myshkis, and O. A. Oleinik, Partial Differential Equations in: Mathematics in USSR in the Last 40 Years, 1917–1957, Fizmatgiz, Moscow, Russia.
- A. Ashyralyev, “Nonlocal boundary-value problems for PDE: well-posedness,” in Global Analysis and Applied Mathematics: International Workshop on Global Analysis, K. Tas, D. Baleanu, O. Krupková, and D. Krupka, Eds., vol. 729 of AIP Conference Proceedings, pp. 325–331, American Institute of Physics, Melville, NY, USA, 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, “Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces,” Journal of Evolution Equations, vol. 6, no. 1, pp. 1–28, 2006.
- A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, vol. 69 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1994.
- P. Clément and S. Guerre-Delabrière, “On the regularity of abstract Cauchy problems and boundary value problems,” Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol. 9, no. 4, pp. 245–266, 1998.
- A. V. Gulin, N. I. Ionkin, and V. A. Morozova, “On the stability of a nonlocal two-dimensional difference problem,” Differentsia' nye Uravneniya, vol. 37, no. 7, pp. 926–932, 2001.
- X.-Z. Liu, X. Cui, and J.-G. Sun, “FDM for multi-dimensional nonlinear coupled system of parabolic and hyperbolic equations,” Journal of Computational and Applied Mathematics, vol. 186, no. 2, pp. 432–449, 2006.
- J. Martín-Vaquero and J. Vigo-Aguiar, “A note on efficient techniques for the second-order parabolic equation subject to non-local conditions,” Applied Numerical Mathematics, vol. 59, no. 6, pp. 1258–1264, 2009.
- J. Martín-Vaquero and J. Vigo-Aguiar, “On the numerical solution of the heat conduction equations subject to nonlocal conditions,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2507–2514, 2009.
- M. Sapagovas, “On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems,” Lithuanian Mathematical Journal, vol. 48, no. 3, pp. 339–356, 2008.
- Ž. Jesevičiūt\.e and M. Sapagovas, “On the stability of finite-difference schemes for parabolic equations subject to integral conditions with applications to thermoelasticity,” Computational Methods in Applied Mathematics, vol. 8, no. 4, pp. 360–373, 2008.
- V. B. Shakhmurov, “Coercive boundary value problems for regular degenerate differential-operator equations,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 605–620, 2004.
- K. Stewartson, “Multistructural boundary layers on flat plates and related bodies,” Advances in Applied Mechanics, vol. 14, pp. 145–239, 1974.
- K. Stewartson, “D'Alembert's paradox,” SIAM Review, vol. 23, no. 3, pp. 308–343, 1981.
- T. LaRosa, The propagation of an electron beam through the solar corona [Ph.D. thesis], Department of Physics and Astronomy, University of Maryland, 1986.
- J. Chabrowski, “On nonlocal problems for parabolic equations,” Nagoya Mathematical Journal, vol. 93, pp. 109–131, 1984.
- H. Lu, “Galerkin and weighted Galerkin methods for a forward-backward heat equation,” Numerische Mathematik, vol. 75, no. 3, pp. 339–356, 1997.
- G. N. Milstein and M. V. Tretyakov, “Numerical algorithms for forward-backward stochastic differential equations,” SIAM Journal on Scientific Computing, vol. 28, no. 2, pp. 561–582, 2006.
- T. Klimsiak, “Strong solutions of semilinear parabolic equations with measure data and generalized backward stochastic differential equation,” Potential Analysis, vol. 36, no. 2, pp. 373–404, 2012.
- A. Ashyralyev, A. Dural, and Y. S. Sözen, “Multipoint nonlocal boundary value problems for reverse parabolic equations: well-posedness,” Vestnik of Odessa National University. Mathematics and Mechanics, vol. 13, pp. 1–12, 2009.
- A. Ashyralyev, A. Dural, and Y. S. Sözen, “Well-posedness of the Rothe difference scheme for reverse parabolic equations,” Iranian Journal of Optimization, vol. 1, pp. 1–25, 2009.
- P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.
- P. E. Sobolevskii and M. F. Tiunčik, “The difference method of approximate solution for elliptic equations,” no. 4, pp. 117–127, 1970 (Russian).
- A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. II, Birkhäuser, Basel, Switzerland, 1989.