An inverse source time-fractional diffusion problem via an input-output mapping

• Autores: Rahima Atmania, Loubna Settara
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 42, Nº. 5, 2023, págs. 1105-1127
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-4401
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• Resumen

  • In this paper, we investigate an inverse source problem involving a one-dimensional diffusion equation of a time-fractional RiemannLiouville derivative with 0 < α < 1. First, results on the existence and regularity of the weak solution of the direct problem are obtained. For the determination of the unknown time-dependent source term, we use a monotone and distinguishable input-output mapping defined by the additional over-determination integral data for the considered sub-diffusion problem. Finally, the uniqueness of the solution of the inverse problem is proved.

• Referencias bibliográficas
  • J. R. Cannon and Y. Lin, “Determination of source parameter in parabolic equations”, Meccanica, vol. 27, pp. 85-94, 1992. https://doi.org/10.1007/BF00420586
  • J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, “Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation”,...
  • M. Choulli and M. Yamamoto, “Some stability estimates in determining sources and coefficients”, Journal of Inverse and Ill-Posed Problems,...
  • A. Erdem, D. Lesnic and A. Hasanov, “Identification of a spacewise dependent heat source”, Applied Mathematical Modelling, vol. 37, pp. 10231-10244,...
  • R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, “Time fractional diffusion: a discrete random walk approach”, Nonlinear Dynamics, vol....
  • A. Hasanov, A. Demir and A. Erdem, “Monotonicity of input—output mappings in inverse coefficient and source problems for parabolic equations”,...
  • A. Hazanee, D. Lesnic, M. I. Ismailov and N. B. Kerimov, “An inverse timedependent source problem for the heat equation with a non-classical...
  • R. Hilfer, Application of fractional in physics. Singapore: World scientific publishing company, 2000.
  • V. Isakov, Inverse Problems for Partial Differential Equations. New York: Springer, 1998.
  • B. Jin and W. Rundell, “An inverse problem for a one-dimensional time-fractional diffusion problem”, Inverse Problems, vol. 28, 2012. ID 075010....
  • B. Kaltenbacher and W. Rundell, “On an inverse potential problem for a fractional reaction-diffusion equation”, Inverse Problems, vol. 35,...
  • A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006.
  • A. A. Kilbas, J. J. Trujillo and A. A. Voroshilov, “Cauchy type problem for diffusion- wave equations with the Riemann-Liouville derivative”,...
  • Z. Li, Y. Liu and M. Yamamoto, “Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant...
  • J. J. Liu and M. Yamamoto, “A backward problem for the time-fractional diffusion equation”, Applicable Analysis, vol. 89, pp. 1769-1788, 2010....
  • Yu. Luchko, “Some uniqueness and existence results for the initial boundary-value problems for the generalized time-fractional diffusion equation”,...
  • J. Nakagawa, K. Sakamoto and M. Yamamoto, “Overview to mathematical analysis for fractional diffusion equations-new mathematical aspects motivated...
  • E. Ozbilge and A. Demir, “Inverse problem for a time-fractional parabolic equation”, Journal of Inequalities and Applications, vol. 81, pp....
  • B. Ozbilge, A. Demir, F. Kanca and E. Ozbilge, “Determination of the unknown source function in time fractional parabolic equation with Dirichlet...
  • W. Rundell, X. Xu and L. Zuo, “The determination of an unknown boundary condition in a fractional diffusion equation”, Applicable Analysis,...
  • K. Sakamoto and M. Yamamoto, “Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse...
  • K. Sakamoto and M. Yamamoto, “Inverse source problem with a final overdetermination for a fractional diffusion equation”, Mathematical Control...
  • S. G. Samko, A. A. Kilbas and D. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers,...
  • W.R. Schneider, Fractional diffusion. In: R. Lima, L. Streit, R. Vilela Mendes (eds) Dynamics and Stochastic Processes Theory and Applications....
  • L. Settara and R. Atmania, “An inverse coefficient-source problem for a time-fractional diffusion equation”, International Journal of Applied...
  • S. Umarov, “On fractional Duhamels principle and its applications”, J. D. Equations, vol. 252, pp. 5217-5234, 2012. https://doi.org/10.48550/arXiv.1004.2098
  • S. Wang, M. Zhang and X. Li, “Radial anomalous diffusion in an annulus”, Physica A: Statistical Mechanics and its Applications, vol. 390,...
  • Y. Zhang and X. Xiang, “Inverse source problem for a fractional diffusion equation”, Inverse Problems, vol. 27, 2011, ID 035010. https://doi.org/10.1088/0266-5611/27/3/035010