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Error Estimates for the Time Discretization of a Semilinear Integrodifferential Parabolic Problem with Unknown Memory Kernel

  • Marijke Grimmonprez (a1), Karel Van Bockstal (a1) and Marián Slodička (a1)

This paper is devoted to the study of an inverse problem containing a semilinear integrodifferential parabolic equation with an unknown memory kernel. This equation is accompanied by a Robin boundary condition. The missing kernel can be recovered from an additional global measurement in integral form. In this contribution, an error analysis is performed for a time-discrete numerical scheme based on Backward Euler's Method. The theoretical results are supported by some numerical experiments.

Corresponding author
*Corresponding author. Email addresses: (Marijke Grimmonprez), (Karel Van Bockstal), (Marián Slodička)
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[1] Lorenzi A. and Sinestrari E.. An inverse problem in the theory of materials with memory. Nonlinear Anal.-Theor., 12(12):13171335, 1988.
[2] Lorenzi A.. Identification problems for integrodifferential equations. In Ill-posed problems in natural sciences. Proceedings of the international conference held in Moscow (Russia), August 19-25, 1991, pages 342366. Utrecht: VSP; Moscow: TVP Science Publishers, 1992.
[3] Lorenzi A. and Paparoni E.. Direct and inverse problems in the theory of materials with memory. Rendiconti del Seminario Matematico della Università di Padova, 87:105138, 1992.
[4] Grasselli M.. An inverse problem in three-dimensional linear thermoviscoelasticity of boltzmann type. In Ill-posed problems in natural sciences. Proceedings of the international conference held in Moscow (Russia), August 19-25, 1991, pages 284299. Utrecht: VSP; Moscow: TVP Science Publishers, 1992.
[5] Grasselli M. and Lorenzi A.. An inverse problem for an abstract nonlinear parabolic integrodifferential equation. Differ. Integral Equ., 6(1):6381, 1993.
[6] von Wolfersdorf L.. Inverse problems for memory kernels in heat flow and viscoelasticity. J. Inverse Ill-Posed Probl., 4(4):341354, 1996.
[7] Janno J., Kiss E.M., and von Wolfersdorf L.. On tikhonov regularization for identifying memory kernels in heat conduction and viscoelasticity. ZAMM - J. Appl. Math. Mech. / Z. Angew. Math. Me., 80(4):259272, 2000.
[8] Janno J. and von Wolfersdorf L.. Inverse problems for memory kernels by laplace transform methods. Z. Anal. Anwend., 19(2):489510, 2000.
[9] Bukhgeǐm A. L. and Dyatlov G. V.. Inverse problems for equations with memory. In Inverse problems and related topics. Proceedings of a seminar, Kobe Institute, Kobe, Japan, February 6–10, 1998, pages 1935. Boca Raton, FL: Chapman & Hall/CRC, 2000.
[10] Colombo F., Guidetti D., and Lorenzi A.. On applications of maximal regularity to inverse problems for integrodifferential equations of parabolic type. Goldstein Ruiz, Gisèle (ed.) et al., Evolution equations. Proceedings of the conference, Blaubeuren, Germany, June 11–17, 2001 in honor of the 60th birthdays of Philippe Bénilan, Goldstein Jerome A. and Nagel Rainer. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 234, 77-89, 2003.
[11] Colombo F., Guidetti D., and Vespri V.. Some global in time results for integrodifferential parabolic inverse problems. Favini , Angelo (ed.) et al., Differential equations. Inverse and direct problems. Papers of the meeting, Cortona, Italy, June 21–25, 2004. Boca Raton, FL: CRC Press. Lecture Notes in Pure and Applied Mathematics 251, 35-58, 2006.
[12] Colombo F. and Guidetti D.. A global in time existence and uniqueness result for a semilinear integrodifferential parabolic inverse problem in Sobolev spaces. Math. Models Methods Appl. Sci., 17(4):537565, 2007.
[13] Guidetti D.. Convergence to a stationary state for solutions to parabolic inverse problems of reconstruction of convolution kernels. Differ. Integral Equ., 20(9):961990, 2007.
[14] Colombo F. and Guidetti D.. Some results on the identification of memory kernels. In Ruzhansky M. and Wirth J., editors, Modern aspects of the theory of partial differential equations. Including mainly selected papers based on the presentations at the 7th international ISAAC congress, London, UK, July 13–18, 2009., pages 121138. Oper. Theor. 216. Basel: Birkhäuser, 2011.
[15] De Staelen R.H. and Slodička M.. Reconstruction of a convolution kernel in a semilinear parabolic problem based on a global measurement. Nonlinear Anal.-Theor., 112(0):4357, 2015.
[16] De Staelen R.H., Van Bockstal K., and Slodička M.. Error analysis in the reconstruction of a convolution kernel in a semilinear parabolic problem with integral overdetermination. J. of Comput. Appl. Math., 275(0):382391, 2015.
[17] Delleur J. W.. The Handbook of Groundwater Engineering. CRS Press, 1999.
[18] Van Bockstal K., De Staelen R.H., and Slodička M.. Identification of a memory kernel in a semilinear integrodifferential parabolic problem with applications in heat conduction with memory. J. Comput. Appl. Math., 289(0):196207, 2015.
[19] Kačur J.. Method of Rothe in evolution equations, volume 80 of Teub. Text. M. Teubner, Leipzig, 1985.
[20] Guenther R. B. and Lee L. W.. Partial Differential Equations of Mathematical Physics and Integral Equations. Dover Publications, 1988.
[21] Nečas J.. Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967.
[22] Nochetto R.H. and Verdi C.. An efficient scheme to approximate parabolic free boundary problems: error estimates and implementation. Math. Comput., 51(183):2753, 1988.
[23] Slodička M. and Dehilis S.. A nonlinear parabolic equation with a nonlocal boundary term. J. Comput. Appl. Math., 233(12):31303138, 2010.
[24] Kufner A., John O., and Fučík S.. Function Spaces. Monographs and textbooks on mechanics of solids and fluids. Noordhoff International Publishing, Leyden, 1977.
[25] Logg Anders and Wells Garth N.. Dolfin: Automated finite element computing. ACM T. Math. Software, 37(2), 2010.
[26] Logg Anders, Wells Garth N., and Hake Johan. DOLFIN: a C++/Python Finite Element Library, chapter 10. Springer, 2012.
[27] Logg A., Mardal K-A., Wells G.N., et al. Automated Solution of Differential Equations by the Finite Element Method. Springer, 2012.
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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
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