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Reconstruction of the Space-dependent Source from Partial Neumann Data for Slow Diffusion System

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Abstract

Consider a linear inverse problem of determining the space-dependent source term in a diffusion equation with time fractional order derivative from the flux measurement specified in partial boundary. Based on the analysis on the forward problem and the adjoint problem with inhomogeneous boundary condition, a variational identity connecting the inversion input data with the unknown source function is established. The uniqueness and the conditional stability for the inverse problem are proven by weak unique continuation and the variational identity in some norm. The inversion scheme minimizing the regularizing cost functional is implemented by using conjugate gradient method, with numerical examples showing the validity of the proposed reconstruction scheme.

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References

  1. Adams, E.E., Gelhar, L.W. Field study of dispersion in a heterogeneous aquifer 2: spatial moments analysis. Water Res. Res., 28(12): 3293–3307 (1992)

    Article  Google Scholar 

  2. Banuelos, R. Shape estimates for Dirichlet eigenfunctions in simply connected domain. J. Diff. Eqns., 125(1): 282–298 (1996)

    Article  MathSciNet  Google Scholar 

  3. Benson, D.A. The Fractional Advection-Dispersion Equation: Development and Application. University of Nevada, Reno, 1998

    Google Scholar 

  4. Cheng, J., Nakagawa, J., Yamamoto, M., Yamazaki, T. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl., 25(11): 115002 (2009)

    Article  MathSciNet  Google Scholar 

  5. Fujishiro, K. Approximate controllability for fractional diffusion equations by Dirichlet boundary control. arXiv:1404.0207v3 [math.OC] 6 Jan 2015, 2015

  6. Gorenflo, R., Luchko, Y., Yamamoto, M. Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal., 18(3): 799–820 (2015)

    Article  MathSciNet  Google Scholar 

  7. Gorenflo, R., Yamamoto, M. Operator theoretic treatment of linear Abel integral equations of first kind. Jpn. J. Ind. Appl. Math., 16(1): 137–161 (1999)

    Article  MathSciNet  Google Scholar 

  8. Jiang, D.J., Li, Z.Y., Liu, Y.K., Yamamoto, M. Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations. Inverse Probl., 33(5): 055013 (2017)

    Article  MathSciNet  Google Scholar 

  9. Jin, B.T., Lazarov, R., Liu, Y.K., Zhou, Z. The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys., 281: 825–843 (2015)

    Article  MathSciNet  Google Scholar 

  10. Jin, B.T., Rundell, W. A tutorial on inverse problems for anomalous diffusion processes. Inverse Probl., 31(3): 035003 (2015)

    Article  MathSciNet  Google Scholar 

  11. Kemppainen, J., Ruotsalainen, K. Boundary integral solution of the time-fractional diffusion equation. Int. Eqns. Oper. Theory, 64(2): 239–249 (2009)

    Article  MathSciNet  Google Scholar 

  12. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006

    MATH  Google Scholar 

  13. Li, G.S., Yamamoto, M. Stability analysis for determining a source term in a 1-D advection-dispersion equation. J. Inver. Ill-posed Probl., 14(2): 147–155 (2006)

    Article  MathSciNet  Google Scholar 

  14. Li, G.S., Jia, X.Z., Sun, C.L. A conditional Lipschitz stability for determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation. J. Inver. Ill-Posed Probl., 25(2): 221–236 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Liu, J.J., Yamamoto, M. A backward problem for the time-fractional diffusion equation. Appl. Anal., 89(11): 1769–1788 (2010)

    Article  MathSciNet  Google Scholar 

  16. Luchko, Y. Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl., 351(1): 218–223 (2009)

    Article  MathSciNet  Google Scholar 

  17. Luchko, Y., Rundell, W., Yamamoto, M., Zuo, L.H. Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation. Inverse Probl., 29(6): 065019 (2013)

    Article  MathSciNet  Google Scholar 

  18. Miller, L., Yamamoto, M. Coefficient inverse problem for a fractional diffusion equation. Inverse Probl., 29(7): 075013 (2013)

    Article  MathSciNet  Google Scholar 

  19. Murio, D.A. Stable numerical solution of fractional-diffusion inverse heat conduction problem. Comput. Math. Appl., 53(10): 1492–501 (2007)

    Article  MathSciNet  Google Scholar 

  20. Podlubny, I. Fractional Differential Equations. Academic Press, San Diego, 1999

    MATH  Google Scholar 

  21. Sakamoto, K., Yamamoto, M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl., 382(1): 426–447 (2011)

    Article  MathSciNet  Google Scholar 

  22. Wei, T., Li, X.L., Li, Y.S. An inverse time-dependent source problem for a time-fractional diffusion equation. Inverse Probl., 32(8): 085003 (2016)

    Article  MathSciNet  Google Scholar 

  23. Yamamoto, M., Zhang, Y. Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate. Inverse Probl., 28(10): 105010 (2012)

    Article  MathSciNet  Google Scholar 

  24. Yamamoto, M. Weak solutions to non-homogeneous boundary value problems for time-fractional diffusion equations. J. Math. Anal. Appl., 460(1): 365–381 (2018)

    Article  MathSciNet  Google Scholar 

  25. Zhang, Y., Xu, X. Inverse source problem for a fractional diffusion equation. Inverse Probl., 27(3): 035010 (2011)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, S.T. Yau Center of Southeast University, Southeast University, Nanjing, 210096, China

    Chun-long Sun & Ji-jun Liu

Authors
  1. Chun-long Sun
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  2. Ji-jun Liu
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Correspondence to Ji-jun Liu.

Additional information

This work is supported by NSFC (Nos. 11971104, 11871149). The first author is also supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX18_0051).

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Sun, Cl., Liu, Jj. Reconstruction of the Space-dependent Source from Partial Neumann Data for Slow Diffusion System. Acta Math. Appl. Sin. Engl. Ser. 36, 166–182 (2020). https://doi.org/10.1007/s10255-020-0919-2

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  • DOI: https://doi.org/10.1007/s10255-020-0919-2

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