# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2022019
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## Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation

 1 School of Sciences, Chang'an University, Xi'an, 710064, China 2 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan 3 Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti, Romania, Correspondence Member of Accademia Peloritana dei Pericolanti, Palazzo Università, Piazza S. Pugliatti 1 98122 Messina, Italy

*Corresponding author: Xiaohua Jing

Received  December 2021 Revised  February 2022 Early access April 2022

We consider two kinds of inverse problems on determining multiple parameters simultaneously for one-dimensional time-fractional diffusion-wave equations with derivative order $\alpha \in (0, 2)$. Based on the analysis of the poles of Laplace transformed data and a transformation formula, we first prove the uniqueness in identifying multiple parameters, including the order of the derivative in time, a spatially varying potential, initial values, and Robin coefficients simultaneously from boundary measurement data, provided that no eigenmodes are zero. Our main results show that the uniqueness of four kinds of parameters holds simultaneously by such observation for the time-fractional diffusion-wave model where unknown orders $\alpha$ vary order (0, 2) including 1, restricted to neither $\alpha \in (0, 1]$ nor $\alpha \in (1, 2)$. Furthermore, for another formulation of the fractional diffusion-wave equation with input source term in place of the initial value, we can also prove the simultaneous uniqueness of multiple parameters, including a spatially varying potential and Robin coefficients by means of the uniqueness result in the case of non-zero initial value and Duhamel's principle.

Citation: Xiaohua Jing, Masahiro Yamamoto. Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation. Inverse Problems and Imaging, doi: 10.3934/ipi.2022019
##### References:
 [1] X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schr$\ddot{o}$dinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011. [2] X. Cao and H. Liu, Determining a fractional Helmholtz equation with unknown source and scattering potential, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5. [3] J. Cheng, Y. Ke and T. Wei, The backward problem of parabolic equations with the measurements on a discrete set, Journal of Inverse and Ill-posed Problems, 28 (2020), 137-144.  doi: 10.1515/jiip-2019-0079. [4] 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. [5] M. Ginoa, S. Cerbelli and H. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A, 191 (1992), 449-453. [6] 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. [7] J. Jia, J. Peng and J. Yang, Harnack's inequality for a space-time fractional diffusion equation and applications to an inverse source problem, J. Differential Equations, 262 (2017), 4415-4450.  doi: 10.1016/j.jde.2017.01.002. [8] D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013, 22 pp. doi: 10.1088/1361-6420/aa58d1. [9] D. Jiang, Z. Li, M. Pauron and M. Yamamoto, Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem, preprint, arXiv: 2103.01692. [10] B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28 (2012), 075010, 19 pp. doi: 10.1088/0266-5611/28/7/075010. [11] 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. [12] B. Jin and Z. Zhou, Recovering the potential in one-dimensional time-fractional diffusion with unknown initial condition and source, Inverse Problems, 37 (2021), 105009, 28 pp. doi: 10.1088/1361-6420/ac1f6d. [13] 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. [14] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Translations of Mathematical Monographs, Vol. 39. American Mathematical Society, Providence, R.I., 1975. [15] B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), 59. Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3748-5. [16] G. Li, D. Zhang, X. Jia and M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems, 29 (2013), 065014, 36 pp. doi: 10.1088/0266-5611/29/6/065014. [17] 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. [18] Z. Li, Y. Liu and M. Yamamoto, Inverse problems of determining parameters of the fractional partial differential equations, Handbook of Fractional Calculus with Applications, De Gruyter, Berlin, 2 (2019), 431-442. [19] 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. [20] 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. [21] Y. Liu, G. Hu and M. Yamamoto, Inverse moving source problem for time-fractional evolution equations: Determination of profiles, Inverse Problems, 37 (2021), 084001, 24 pp. doi: 10.1088/1361-6420/ac0c20. [22] Y. Liu, W. Rundell and M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem, Fractional Calculus Appl. Anal., 19 (2016), 888-906.  doi: 10.1515/fca-2016-0048. [23] Y. Liu and Z. Zhang, Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation, J. Phys. A: Math. Theor., 50 (2017), 305203, 27 pp. doi: 10.1088/1751-8121/aa763a. [24] Y. Luchko, W. Rundell, M. Yamamoto and L. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Problems, 29 (2013), 065019, 16 pp. doi: 10.1088/0266-5611/29/6/065019. [25] F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoust., 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826. [26] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77 pp. doi: 10.1016/S0370-1573(00)00070-3. [27] L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems, 29 (2013), 075013, 8 pp. doi: 10.1088/0266-5611/29/7/075013. [28] R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status Solidi B-Basic Solid State Phys, 133 (1986), 425-430. [29] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. [30] Z. Ruan, W. Zhang and W. Wang, Simultaneous inversion of the fractional order and the space-dependent source term for the time-fractional diffusion equation, Appl. Math. Comput., 328 (2018), 365-379.  doi: 10.1016/j.amc.2018.01.025. [31] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058. [32] L. L. Sun, Y. S. Li and Y. Zhang, Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation, Inverse Problems, 37 (2021), 055007, 26 pp. doi: 10.1088/1361-6420/abf162. [33] L. Sun and T. Wei, Identification of the zeroth-order coefficient in a time fractional diffusion equation, Appl. Numer. Math., 111 (2017), 160-180.  doi: 10.1016/j.apnum.2016.09.005. [34] T. Suzuki, Uniqueness and nonuniqueness in an inverse problem for the parabolic equation, J. Differential Equations, 47 (1983), 296-316.  doi: 10.1016/0022-0396(83)90038-4. [35] T. Suzuki, Gel'fand-Levitan's theory, deformation formulas and inverse problems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 32 (1985), 223-271. [36] T. Suzuki and R. Murayama, A uniqueness theorem in an identification problem for coefficients of parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 259-263. [37] E. C. Titchmarsh, The zeros of certain integral functions, Proc. London Math. Soc., 2 (1926), 283-302.  doi: 10.1112/plms/s2-25.1.283. [38] T. Wei, X. Li and Y. Li, An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Problems, 32 (2016), 085003, 24 pp. doi: 10.1088/0266-5611/32/8/085003. [39] T. Wei and K. Liao, Identifying a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by using the measured data at a boundary point, Appl. Anal., (2021), 1–26. [40] T. Wei and X. B. Yan, Uniqueness for identifying a space-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation from a single boundary point measurement, Appl. Math. Lett., 112 (2021), 106814, 7 pp. doi: 10.1016/j.aml.2020.106814. [41] M. Yamamoto, Weak solutions to non-homogeneous boundary value problems for time-fractional diffusion equations, J. Math. Anal. Appl., 460 (2018), 365-381.  doi: 10.1016/j.jmaa.2017.11.048. [42] M. Yamamoto, Uniqueness in determining fractional orders of derivatives and initial values, Inverse Problems, 37 (2021), 095006, 34 pp. doi: 10.1088/1361-6420/abf9e9. [43] Y. Zhang, T. Wei and Y.-X. Zhang, Simultaneous inversion of two initial values for a time-fractional diffusion-wave equation, Num. Meth. Partial Differential Equations, 37 (2021), 24-43.  doi: 10.1002/num.22517. [44] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2021), 035010, 12 pp. doi: 10.1088/0266-5611/27/3/035010. [45] L. Zhou and H. Selim, Application of the fractional advection-dispersion equation in porous media, Soil Sci. Soc. Am. J., 67 (2003), 1079-1084. [46] T. Zhu, Seismic Modeling, Inversion, and Imaging in Attenuating, Ph.D thesis, Stanford University, 2014. [47] T. Zhu and J. Harris, Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians, Geophysics, 79 (2014), T105–T116.

show all references

##### References:
 [1] X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schr$\ddot{o}$dinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011. [2] X. Cao and H. Liu, Determining a fractional Helmholtz equation with unknown source and scattering potential, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5. [3] J. Cheng, Y. Ke and T. Wei, The backward problem of parabolic equations with the measurements on a discrete set, Journal of Inverse and Ill-posed Problems, 28 (2020), 137-144.  doi: 10.1515/jiip-2019-0079. [4] 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. [5] M. Ginoa, S. Cerbelli and H. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A, 191 (1992), 449-453. [6] 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. [7] J. Jia, J. Peng and J. Yang, Harnack's inequality for a space-time fractional diffusion equation and applications to an inverse source problem, J. Differential Equations, 262 (2017), 4415-4450.  doi: 10.1016/j.jde.2017.01.002. [8] D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013, 22 pp. doi: 10.1088/1361-6420/aa58d1. [9] D. Jiang, Z. Li, M. Pauron and M. Yamamoto, Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem, preprint, arXiv: 2103.01692. [10] B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28 (2012), 075010, 19 pp. doi: 10.1088/0266-5611/28/7/075010. [11] 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. [12] B. Jin and Z. Zhou, Recovering the potential in one-dimensional time-fractional diffusion with unknown initial condition and source, Inverse Problems, 37 (2021), 105009, 28 pp. doi: 10.1088/1361-6420/ac1f6d. [13] 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. [14] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Translations of Mathematical Monographs, Vol. 39. American Mathematical Society, Providence, R.I., 1975. [15] B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), 59. Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3748-5. [16] G. Li, D. Zhang, X. Jia and M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems, 29 (2013), 065014, 36 pp. doi: 10.1088/0266-5611/29/6/065014. [17] 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. [18] Z. Li, Y. Liu and M. Yamamoto, Inverse problems of determining parameters of the fractional partial differential equations, Handbook of Fractional Calculus with Applications, De Gruyter, Berlin, 2 (2019), 431-442. [19] 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. [20] 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. [21] Y. Liu, G. Hu and M. Yamamoto, Inverse moving source problem for time-fractional evolution equations: Determination of profiles, Inverse Problems, 37 (2021), 084001, 24 pp. doi: 10.1088/1361-6420/ac0c20. [22] Y. Liu, W. Rundell and M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem, Fractional Calculus Appl. Anal., 19 (2016), 888-906.  doi: 10.1515/fca-2016-0048. [23] Y. Liu and Z. Zhang, Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation, J. Phys. A: Math. Theor., 50 (2017), 305203, 27 pp. doi: 10.1088/1751-8121/aa763a. [24] Y. Luchko, W. Rundell, M. Yamamoto and L. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Problems, 29 (2013), 065019, 16 pp. doi: 10.1088/0266-5611/29/6/065019. [25] F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoust., 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826. [26] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77 pp. doi: 10.1016/S0370-1573(00)00070-3. [27] L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems, 29 (2013), 075013, 8 pp. doi: 10.1088/0266-5611/29/7/075013. [28] R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status Solidi B-Basic Solid State Phys, 133 (1986), 425-430. [29] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. [30] Z. Ruan, W. Zhang and W. Wang, Simultaneous inversion of the fractional order and the space-dependent source term for the time-fractional diffusion equation, Appl. Math. Comput., 328 (2018), 365-379.  doi: 10.1016/j.amc.2018.01.025. [31] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058. [32] L. L. Sun, Y. S. Li and Y. Zhang, Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation, Inverse Problems, 37 (2021), 055007, 26 pp. doi: 10.1088/1361-6420/abf162. [33] L. Sun and T. Wei, Identification of the zeroth-order coefficient in a time fractional diffusion equation, Appl. Numer. Math., 111 (2017), 160-180.  doi: 10.1016/j.apnum.2016.09.005. [34] T. Suzuki, Uniqueness and nonuniqueness in an inverse problem for the parabolic equation, J. Differential Equations, 47 (1983), 296-316.  doi: 10.1016/0022-0396(83)90038-4. [35] T. Suzuki, Gel'fand-Levitan's theory, deformation formulas and inverse problems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 32 (1985), 223-271. [36] T. Suzuki and R. Murayama, A uniqueness theorem in an identification problem for coefficients of parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 259-263. [37] E. C. Titchmarsh, The zeros of certain integral functions, Proc. London Math. Soc., 2 (1926), 283-302.  doi: 10.1112/plms/s2-25.1.283. [38] T. Wei, X. Li and Y. Li, An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Problems, 32 (2016), 085003, 24 pp. doi: 10.1088/0266-5611/32/8/085003. [39] T. Wei and K. Liao, Identifying a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by using the measured data at a boundary point, Appl. Anal., (2021), 1–26. [40] T. Wei and X. B. Yan, Uniqueness for identifying a space-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation from a single boundary point measurement, Appl. Math. Lett., 112 (2021), 106814, 7 pp. doi: 10.1016/j.aml.2020.106814. [41] M. Yamamoto, Weak solutions to non-homogeneous boundary value problems for time-fractional diffusion equations, J. Math. Anal. Appl., 460 (2018), 365-381.  doi: 10.1016/j.jmaa.2017.11.048. [42] M. Yamamoto, Uniqueness in determining fractional orders of derivatives and initial values, Inverse Problems, 37 (2021), 095006, 34 pp. doi: 10.1088/1361-6420/abf9e9. [43] Y. Zhang, T. Wei and Y.-X. Zhang, Simultaneous inversion of two initial values for a time-fractional diffusion-wave equation, Num. Meth. Partial Differential Equations, 37 (2021), 24-43.  doi: 10.1002/num.22517. [44] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2021), 035010, 12 pp. doi: 10.1088/0266-5611/27/3/035010. [45] L. Zhou and H. Selim, Application of the fractional advection-dispersion equation in porous media, Soil Sci. Soc. Am. J., 67 (2003), 1079-1084. [46] T. Zhu, Seismic Modeling, Inversion, and Imaging in Attenuating, Ph.D thesis, Stanford University, 2014. [47] T. Zhu and J. Harris, Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians, Geophysics, 79 (2014), T105–T116.
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