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Determination of a time-dependent diffusivity in a nonlinear parabolic problem

Karel Van Bockstal (UGent) and Marian Slodicka (UGent)
(2015) INVERSE PROBLEMS IN SCIENCE AND ENGINEERING. 23(2). p.307-330
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Abstract
In this article, an inverse nonlinear convection-diffusion problem is considered for the identification of an unknown solely time-dependent diffusion coefficient in a subregion of a bounded domain in . The missing data are compensated by boundary observations on a part of the surface of the subdomain: the total flux through that surface or the values of the solution at that surface are measured. Two solution methods are discussed. In both cases, the solvability of the problem is proved using coefficient to data mappings. More specific, a nonlinear numerical algorithm based on Rothe's method is designed and the convergence of approximations towards the weak solution in suitable function spaces is shown. In the proofs, also the monotonicity methods and the Minty-Browder argument are employed. The results of numerical experiments are discussed.
Keywords
RESISTIVITY IDENTIFICATION, INVERSE PROBLEM, COEFFICIENT, 35B30, 35K61, 65M32, 35R30, parameter identification, nonlocal boundary condition, convergence, Minty-Browder, time discretization, inverse problems, nonlinear parabolic initial boundary value problem

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Citation

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MLA
Van Bockstal, Karel, and Marian Slodicka. “Determination of a Time-Dependent Diffusivity in a Nonlinear Parabolic Problem.” INVERSE PROBLEMS IN SCIENCE AND ENGINEERING, vol. 23, no. 2, 2015, pp. 307–30, doi:10.1080/17415977.2014.900615.
APA
Van Bockstal, K., & Slodicka, M. (2015). Determination of a time-dependent diffusivity in a nonlinear parabolic problem. INVERSE PROBLEMS IN SCIENCE AND ENGINEERING, 23(2), 307–330. https://doi.org/10.1080/17415977.2014.900615
Chicago author-date
Van Bockstal, Karel, and Marian Slodicka. 2015. “Determination of a Time-Dependent Diffusivity in a Nonlinear Parabolic Problem.” INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 23 (2): 307–30. https://doi.org/10.1080/17415977.2014.900615.
Chicago author-date (all authors)
Van Bockstal, Karel, and Marian Slodicka. 2015. “Determination of a Time-Dependent Diffusivity in a Nonlinear Parabolic Problem.” INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 23 (2): 307–330. doi:10.1080/17415977.2014.900615.
Vancouver
1.
Van Bockstal K, Slodicka M. Determination of a time-dependent diffusivity in a nonlinear parabolic problem. INVERSE PROBLEMS IN SCIENCE AND ENGINEERING. 2015;23(2):307–30.
IEEE
[1]
K. Van Bockstal and M. Slodicka, “Determination of a time-dependent diffusivity in a nonlinear parabolic problem,” INVERSE PROBLEMS IN SCIENCE AND ENGINEERING, vol. 23, no. 2, pp. 307–330, 2015.
@article{5759539,
  abstract     = {{In this article, an inverse nonlinear convection-diffusion problem is considered for the identification of an unknown solely time-dependent diffusion coefficient in a subregion of a bounded domain in . The missing data are compensated by boundary observations on a part of the surface of the subdomain: the total flux through that surface or the values of the solution at that surface are measured. Two solution methods are discussed. In both cases, the solvability of the problem is proved using coefficient to data mappings. More specific, a nonlinear numerical algorithm based on Rothe's method is designed and the convergence of approximations towards the weak solution in suitable function spaces is shown. In the proofs, also the monotonicity methods and the Minty-Browder argument are employed. The results of numerical experiments are discussed.}},
  author       = {{Van Bockstal, Karel and Slodicka, Marian}},
  issn         = {{1741-5977}},
  journal      = {{INVERSE PROBLEMS IN SCIENCE AND ENGINEERING}},
  keywords     = {{RESISTIVITY IDENTIFICATION,INVERSE PROBLEM,COEFFICIENT,35B30,35K61,65M32,35R30,parameter identification,nonlocal boundary condition,convergence,Minty-Browder,time discretization,inverse problems,nonlinear parabolic initial boundary value problem}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{307--330}},
  title        = {{Determination of a time-dependent diffusivity in a nonlinear parabolic problem}},
  url          = {{http://doi.org/10.1080/17415977.2014.900615}},
  volume       = {{23}},
  year         = {{2015}},
}
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