## Abstract and Applied Analysis

### The Difference Problem of Obtaining the Parameter of a Parabolic Equation

#### Abstract

The boundary value problem of determining the parameter $p$ of a parabolic equation ${\upsilon }^{\prime }(t)+A\upsilon (t)=f(t)+\mathrm{p }(0\le t\le 1),\mathrm{ \upsilon }(0)=\phi ,\mathrm{ \upsilon }(1)=\psi$ in an arbitrary Banach space $E$ with the strongly positive operator $A$ is considered. The first order of accuracy stable difference scheme for the approximate solution of this problemis investigated. The well-posedness of this difference scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 603018, 14 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174052

Digital Object Identifier
doi:10.1155/2012/603018

Mathematical Reviews number (MathSciNet)
MR2947682

Zentralblatt MATH identifier
1250.35180

#### Citation

Ashyralyyev, Charyyar; Demirdag, Oznur. The Difference Problem of Obtaining the Parameter of a Parabolic Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 603018, 14 pages. doi:10.1155/2012/603018. https://projecteuclid.org/euclid.aaa/1365174052

#### References

• 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\textcurrency 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\textcurrency 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, “ čommentComment on ref. [10?]: Please provide the title of the journal.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\textcurrency 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\textcurrency 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\textcurrency 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).