Stability of Implicit Difference Scheme for Solving the Identification Problem of a Parabolic Equation

  • Abdullah Said Erdogan
  • Allaberen Ashyralyev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


We consider the inverse problem of reconstructing the right side of a parabolic equation with an unknown time dependent source function. Numerical solution and well-posedness of this type problem with local boundary conditions considered previously by A.A. Samarskii, P.N. Vabishchevich and V.T. Borukhov. In this paper, we focus on studying the stability of the problem with nonlocal conditions. A stable algorithm for the approximate solution of the problem is presented.


Parabolic equations identification problem stability analysis implicit difference scheme 


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  1. 1.
    Hasanov, A.: Identification of unknown diffusion and convection coefficients in ion transport problems from flux data: an analytical approach. J. Math. Chem. 48(2), 413–423 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Shestopalov, Y., Smirnov, Y.: Existence and uniqueness of a solution to the inverse problem of the complex permittivity reconstruction of a dielectric body in a waveguide. Inverse Probl. 26(10), 105002 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Serov, V.S., Paivarinta, L.: Inverse scattering problem for two-dimensional Schrödinger operator. J. Inv. Ill-Pose. Probl. 14(3), 295–305 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cannon, J.R., Lin, Y.L., Xu, S.: Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations. Inverse Probl. 10, 227–243 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borukhov, V.T., Vabishchevich, P.N.: Numerical solution of the inverse problem of reconstructing a distributed right-hand side of a parabolic equation. Comput. Phys. Commun. 126, 32–36 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cannon, J.R., Yin, H.-M.: Numerical solutions of some parabolic inverse problems. Numer. Meth. Part. D. E. 2, 177–191 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Prilepko, A.I., Orlovsky, D.G., Vasin, I.A.: Methods for Solving Inverse Problems in Mathematical Physics. Marcel Dekker, New York (2000)Google Scholar
  8. 8.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences, vol. 127. Springer, New York (2006)zbMATHGoogle Scholar
  9. 9.
    Ivanchov, N.I.: On the determination of unknown source in the heat equation with nonlocal boundary conditions. Ukr. Math. J. 47(10), 1647–1652 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Orlovsky, D., Piskarev, S.: On approximation of inverse problems for abstract elliptic problems. J. Inverse Ill-Pose. P. 17(8), 765–782 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Wang, Y., Zheng, S.: The existence and behavior of solutions for nonlocal boundary problems. Value Probl. 2009, Article ID 484879 (2009)Google Scholar
  12. 12.
    Agarwal, R.P., Shakhmurov, V.B.: Multipoint problems for degenerate abstract differential equations. Acta Math. Hung. 123(1-2), 65–89 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Di Blasio, G.: Maximal L p regularity for nonautonomous parabolic equations in extrapolation spaces. J. Evol. Equ. 6(2), 229–245 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Eidelman, Y.S.: The boundary value problem for differential equations with a parameter. Differents. Uravneniya 14, 1335–1337 (1978)MathSciNetGoogle Scholar
  15. 15.
    Ashyralyev, A.: On the problem of determining the parameter of a parabolic equation. Ukr. Math. J. 62(9), 1200–1210 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Prilepko, A.I., Tikhonov, I.V.: Uniqueness of the solution of an inverse problem for an evolution equation and applications to the transfer equation. Mat. Zametki. 51(2), 77–87 (1992)MathSciNetGoogle Scholar
  17. 17.
    Choulli, M., Yamamoto, M.: Generic well-posedness of a linear inverse parabolic problem with diffusion parameter. J. Inverse Ill-Pose P. 7(3), 241–254 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Choulli, M., Yamamoto, M.: Generic well-posedness of an inverse parabolic problem-the Hölder-space approach. Inverse Probl. 12, 195–205 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Saitoh, S., Tuan, V.K., Yamamoto, M.: Reverse convolution inequalities and applications to inverse heat source problems. Journal of Inequalities in Pure and Applied Mathematics 3(5), Article 80 (2002) (electronic)Google Scholar
  20. 20.
    Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. Inverse and Ill-posed Problems Series. Walter de Gruyter, Berlin (2007)zbMATHGoogle Scholar
  21. 21.
    Ashyralyev, A.: A note on fractional derivatives and fractional powers of operators. J. Math. Anal. Appl. 357, 232–236 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ashyralyev, A., Sobolevskii, P.E.: Well-Posedness of Parabolic Difference Equations. Birkhäuser, Basel (1994)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Abdullah Said Erdogan
    • 1
  • Allaberen Ashyralyev
    • 1
  1. 1.Department of MathematicsFatih UniversityIstanbulTurkey

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