# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2022036
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## The enclosure method for the detection of variable order in fractional diffusion equations

 1 Laboratory of Mathematics, Graduate School of Advanced Science and Engineering, Hiroshima University, Higashihiroshima 739-8527, Emeritus Professor at Gunma University, Maebashi 371-8510, Japan 2 Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

* Corresponding author: Yavar Kian

Received  February 2022 Revised  May 2022 Early access June 2022

This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space dependent variable-order of fractional time derivative of the governing equation deviates from a known homogeneous background one. The observation data is given by the Neumann data of the solution of the governing equation for a specially designed Dirichlet data. Under a suitable jump condition on the deviation, it is shown that the most recent version of the time domain enclosure method enables one to extract information about the geometry of the obstacle and a qualitative nature of the jump, from the observation data.

Citation: Masaru Ikehata, Yavar Kian. The enclosure method for the detection of variable order in fractional diffusion equations. Inverse Problems and Imaging, doi: 10.3934/ipi.2022036
##### References:
 [1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer 2. Spatial moments analysis, Water Resources Res., 285 (1992), 3293-3307. [2] S. Alimov and R. Ashurov, Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation, J. Inverse Ill-Posed Probl., 28 (2020), 651-658.  doi: 10.1515/jiip-2020-0072. [3] A. Atangana and S. C. Oukouomi Noutchie, Stability and convergence of a time-fractional variable order Hantush aquation for a deformable aquifer, Abstract and Applied Analysis, (2013), Art. ID 691060, 8 pp. doi: 10.1155/2013/691060. [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [5] J. M. Carcione, F. J. Sanchez-Sesma, F. Luzón and J. J. Perez Gavilán, Theory and simulation of time-fractional fluid diffusion in porous media, Journal of Physics A: Mathematical and Theoretical, 46 (2013), 345501, 23 pp. doi: 10.1088/1751-8113/46/34/345501. [6] W. Chen, J. Zhang and J. Zhang, Variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures, Fract. Calc. Appl. Anal., 16 (2013), 76-92.  doi: 10.2478/s13540-013-0006-y. [7] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002, 16 pp. doi: 10.1088/0266-5611/25/11/115002. [8] S. Fedotov and S. Falconer, Subdiffusive master equation with space-dependent anomalous exponent and structural instability, Phys. Rev. E, 85 (2012), 031132. [9] S. Fedotov and D. Han, Asymptotic behavior of the solution of the space dependent variable order fractional diffusion equation: ultraslow anomalous aggregation, Physical Review Letters, 123 (2019), 050602, 5 pp. doi: 10.1103/PhysRevLett.123.050602. [10] W. G. Glöckle and T. F. Nonnenmacher, A Fractional Calculus Approach to Self-Similar Protein Dynamics, Biophys. J., 68 (1995), 46-53. [11] Y. Hatano, J. Nakagawa, S. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, J. Math-for-Ind., 5A (2013), 51-57. [12] M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Problems, 6 (1998), 127-140.  doi: 10.1515/jiip.1998.6.2.127. [13] M. Ikehata, Reconstruction of a source domain drom the Cauhy data, Inverse Problems, 15 (1999), 637-645.  doi: 10.1088/0266-5611/15/2/019. [14] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308. [15] M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inverse Ill-Posed Problems, 8 (2000), 367-378.  doi: 10.1515/jiip.2000.8.4.367. [16] M. Ikehata, On finding an obstacle embedded in the rough background medium via the enclosure method in the time domain, Inverse Problems, 31 (2015), 085011, 21 pp. doi: 10.1088/0266-5611/31/8/085011. [17] M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: Ⅳ. Extraction from a single point on the graph of the response operator, J. Inverse Ill-Posed Probl., 25 (2017), 747-761.  doi: 10.1515/jiip-2016-0023. [18] M. Ikehata, On finding a cavity in a thermoelastic body using a single displacement measurement over a finite time interval on the surface of the body, J. Inverse Ill-Posed Probl., 26 (2018), 369-394.  doi: 10.1515/jiip-2017-0066. [19] M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: Ⅵ. Using shell-type initial data, J. Inverse Ill-Posed Probl., 28 (2020), 349-366.  doi: 10.1515/jiip-2019-0039. [20] M. Ikehata, The enclosure method using a single point on the graph of the response operator for the Stokes system, arXiv: 2010.02435, math.AP, 2020 [21] J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electron. J. Differential Equations, (2016), Paper No. 199, 28 pp. [22] J. Janno and N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements, Inverse Problems, 34 (2018), 025007, 19 pp. doi: 10.1088/1361-6420/aaa0f0. [23] B. Jin and Y. Kian, Recovering multiple fractional orders in time-fractional diffusion in an unknown medium, Proceedings of the Royal Society A, 477 (2021), Paper No. 20210468, 21 pp. doi: 10.1098/rspa.2021.0468. [24] B. Jin and Y. Kian, Recovery of the order of derivation for fractional diffusion equations in an unknown medium, to appear in SIAM J. Appl. Math.. [25] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse problems, 31 (2015), 035003, 40 pp. doi: 10.1088/0266-5611/31/3/035003. [26] Y. Kian, Simultaneous determination of coefficients and internal source of a diffusion equation from a single measurement, to appear in Inverse Problems. [27] Y. Kian, Equivalence of definitions of solutions for some class of fractional diffusion equations, preprint, arXiv: 2111.06168. [28] Y. Kian, E. Soccorsi and M. Yamamoto, On time-fractional diffusion equations with space-dependent variable order, Ann. Henri Poincaré, 19 (2018), 3855-3881.  doi: 10.1007/s00023-018-0734-y. [29] Y. Kian and M. Yamamoto, Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations, Fract. Calc. Appl. Anal., 24 (2021), 168–201. doi: 10.1515/fca-2021-0008. [30] Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems, 32 (2016), 015004, 16 pp. doi: 10.1088/0266-5611/32/1/015004. [31] Z. Li, Y. Liu and M. Yamamoto, Inverse problems of determining parameters of the fractional partial differential equations, Handbook of Fractional Calculus with Applications., Vol. 2, pages 431–442. De Gruyter, Berlin, 2019. [32] Z. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Appl. Anal., 94 (2015), 570-579.  doi: 10.1080/00036811.2014.926335. [33] Z. Li and Z. Zhang, Unique determination of fractional order and source term in a fractional diffusion equation from sparse boundary data, Inverse Problems, 36 (2020), 115013, 20 pp. doi: 10.1088/1361-6420/abbc5d. [34] K. Liao and T. Wei, Identifying a fractional order and a space source term in a time-fractional diffusion wave equation simultaneously, Inverse Problems, 35 (2019), 115002, 23 pp. doi: 10.1088/1361-6420/ab383f. [35] F. W. Olver, Asymptotics and Special Functions, Academic Press, New York and London, 1974. [36] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [37] W. Rudin, Real and Complex Analysis, McGraw Hill, New York, 1974. [38] W. Smit and H. De Vries, Rheological models containing fractional derivatives, Rheol. Acta, 9 (1970), 525-534. [39] B. A. Stickler and E. Schachinger, Continuous time anomalous diffusion in a composite medium, J. Phys. E., 84 (2011), 021116. [40] H. Sun, W. Chen and Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388 (2009), 4586-4592. [41] M. Yamamoto, Uniqueness in determining fractional orders of derivatives and initial values, Inverse Probl., 37 (2021), 095006, 34 pp. doi: 10.1088/1361-6420/abf9e9. [42] H. Zhang, G.-H. Li and M.-K. Luo, Fractional Feynman-Kac equation with space-dependent anomalous exponent, J. Stat. Phys., 152 (2013), 1194-1206.  doi: 10.1007/s10955-013-0810-0.

show all references

##### References:
 [1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer 2. Spatial moments analysis, Water Resources Res., 285 (1992), 3293-3307. [2] S. Alimov and R. Ashurov, Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation, J. Inverse Ill-Posed Probl., 28 (2020), 651-658.  doi: 10.1515/jiip-2020-0072. [3] A. Atangana and S. C. Oukouomi Noutchie, Stability and convergence of a time-fractional variable order Hantush aquation for a deformable aquifer, Abstract and Applied Analysis, (2013), Art. ID 691060, 8 pp. doi: 10.1155/2013/691060. [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [5] J. M. Carcione, F. J. Sanchez-Sesma, F. Luzón and J. J. Perez Gavilán, Theory and simulation of time-fractional fluid diffusion in porous media, Journal of Physics A: Mathematical and Theoretical, 46 (2013), 345501, 23 pp. doi: 10.1088/1751-8113/46/34/345501. [6] W. Chen, J. Zhang and J. Zhang, Variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures, Fract. Calc. Appl. Anal., 16 (2013), 76-92.  doi: 10.2478/s13540-013-0006-y. [7] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002, 16 pp. doi: 10.1088/0266-5611/25/11/115002. [8] S. Fedotov and S. Falconer, Subdiffusive master equation with space-dependent anomalous exponent and structural instability, Phys. Rev. E, 85 (2012), 031132. [9] S. Fedotov and D. Han, Asymptotic behavior of the solution of the space dependent variable order fractional diffusion equation: ultraslow anomalous aggregation, Physical Review Letters, 123 (2019), 050602, 5 pp. doi: 10.1103/PhysRevLett.123.050602. [10] W. G. Glöckle and T. F. Nonnenmacher, A Fractional Calculus Approach to Self-Similar Protein Dynamics, Biophys. J., 68 (1995), 46-53. [11] Y. Hatano, J. Nakagawa, S. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, J. Math-for-Ind., 5A (2013), 51-57. [12] M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Problems, 6 (1998), 127-140.  doi: 10.1515/jiip.1998.6.2.127. [13] M. Ikehata, Reconstruction of a source domain drom the Cauhy data, Inverse Problems, 15 (1999), 637-645.  doi: 10.1088/0266-5611/15/2/019. [14] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308. [15] M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inverse Ill-Posed Problems, 8 (2000), 367-378.  doi: 10.1515/jiip.2000.8.4.367. [16] M. Ikehata, On finding an obstacle embedded in the rough background medium via the enclosure method in the time domain, Inverse Problems, 31 (2015), 085011, 21 pp. doi: 10.1088/0266-5611/31/8/085011. [17] M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: Ⅳ. Extraction from a single point on the graph of the response operator, J. Inverse Ill-Posed Probl., 25 (2017), 747-761.  doi: 10.1515/jiip-2016-0023. [18] M. Ikehata, On finding a cavity in a thermoelastic body using a single displacement measurement over a finite time interval on the surface of the body, J. Inverse Ill-Posed Probl., 26 (2018), 369-394.  doi: 10.1515/jiip-2017-0066. [19] M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: Ⅵ. Using shell-type initial data, J. Inverse Ill-Posed Probl., 28 (2020), 349-366.  doi: 10.1515/jiip-2019-0039. [20] M. Ikehata, The enclosure method using a single point on the graph of the response operator for the Stokes system, arXiv: 2010.02435, math.AP, 2020 [21] J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electron. J. Differential Equations, (2016), Paper No. 199, 28 pp. [22] J. Janno and N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements, Inverse Problems, 34 (2018), 025007, 19 pp. doi: 10.1088/1361-6420/aaa0f0. [23] B. Jin and Y. Kian, Recovering multiple fractional orders in time-fractional diffusion in an unknown medium, Proceedings of the Royal Society A, 477 (2021), Paper No. 20210468, 21 pp. doi: 10.1098/rspa.2021.0468. [24] B. Jin and Y. Kian, Recovery of the order of derivation for fractional diffusion equations in an unknown medium, to appear in SIAM J. Appl. Math.. [25] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse problems, 31 (2015), 035003, 40 pp. doi: 10.1088/0266-5611/31/3/035003. [26] Y. Kian, Simultaneous determination of coefficients and internal source of a diffusion equation from a single measurement, to appear in Inverse Problems. [27] Y. Kian, Equivalence of definitions of solutions for some class of fractional diffusion equations, preprint, arXiv: 2111.06168. [28] Y. Kian, E. Soccorsi and M. Yamamoto, On time-fractional diffusion equations with space-dependent variable order, Ann. Henri Poincaré, 19 (2018), 3855-3881.  doi: 10.1007/s00023-018-0734-y. [29] Y. Kian and M. Yamamoto, Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations, Fract. Calc. Appl. Anal., 24 (2021), 168–201. doi: 10.1515/fca-2021-0008. [30] Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems, 32 (2016), 015004, 16 pp. doi: 10.1088/0266-5611/32/1/015004. [31] Z. Li, Y. Liu and M. Yamamoto, Inverse problems of determining parameters of the fractional partial differential equations, Handbook of Fractional Calculus with Applications., Vol. 2, pages 431–442. De Gruyter, Berlin, 2019. [32] Z. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Appl. Anal., 94 (2015), 570-579.  doi: 10.1080/00036811.2014.926335. [33] Z. Li and Z. Zhang, Unique determination of fractional order and source term in a fractional diffusion equation from sparse boundary data, Inverse Problems, 36 (2020), 115013, 20 pp. doi: 10.1088/1361-6420/abbc5d. [34] K. Liao and T. Wei, Identifying a fractional order and a space source term in a time-fractional diffusion wave equation simultaneously, Inverse Problems, 35 (2019), 115002, 23 pp. doi: 10.1088/1361-6420/ab383f. [35] F. W. Olver, Asymptotics and Special Functions, Academic Press, New York and London, 1974. [36] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [37] W. Rudin, Real and Complex Analysis, McGraw Hill, New York, 1974. [38] W. Smit and H. De Vries, Rheological models containing fractional derivatives, Rheol. Acta, 9 (1970), 525-534. [39] B. A. Stickler and E. Schachinger, Continuous time anomalous diffusion in a composite medium, J. Phys. E., 84 (2011), 021116. [40] H. Sun, W. Chen and Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388 (2009), 4586-4592. [41] M. Yamamoto, Uniqueness in determining fractional orders of derivatives and initial values, Inverse Probl., 37 (2021), 095006, 34 pp. doi: 10.1088/1361-6420/abf9e9. [42] H. Zhang, G.-H. Li and M.-K. Luo, Fractional Feynman-Kac equation with space-dependent anomalous exponent, J. Stat. Phys., 152 (2013), 1194-1206.  doi: 10.1007/s10955-013-0810-0.
The sets $\Omega$, $D$ and $\mbox{dist}\,(K_{\star},D)$
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