American Institute of Mathematical Sciences

April  2018, 12(2): 315-330. doi: 10.3934/ipi.2018014

Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation

 Department of Mathematical Sciences, The University of Tokyo, Komaba Meguro Tokyo 153-8914, Japan

Received  November 2016 Revised  November 2017 Published  February 2018

We consider the first and half order time fractional equation with the zero initial condition. We investigate an inverse source problem of determining the time-independent source factor by the spatial data at an arbitrarily fixed time and we establish the conditional stability estimate of Hölder type in our inverse problem. Our method is based on the Bukhgeim-Klibanov method by means of the Carleman estimate. We also derive the Carleman estimate for the first and half order time fractional diffusion equation.

Citation: Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
References:
 [1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2.Spatial moments analysis, Water Resources Research, 28 (1992), 3293-3307.  doi: 10.1029/92WR01757. [2] B. Amaziane, L. Pankratov and A. Piatnitski, Homogenization of a single-phase flow through a porous medium in a thin layer, Math. Models Methods Appl. Sci., 17 (2007), 1317-1349.  doi: 10.1142/S0218202507002339. [3] A. Ashyralyev, Well-posedness of the Basset problem in spaces of smooth functions, Appl. Math. Lett., 24 (2011), 1176-1180.  doi: 10.1016/j.aml.2011.02.002. [4] E. Bazhlekova and I. Dimovski, Exact solution of two-term time-fractional Thornley's problem by operational method, Integral Transforms Spec. Funct., 25 (2014), 61-74.  doi: 10.1080/10652469.2013.815184. [5] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272 (in Russian). [6] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independantes, Ark. Mat. Astr. Fys., 26B (1939), 1-9. [7] J. Cheng, C.-L. Lin and G. Nakamura, Unique continuation property for the anomalous diffusion and its application, J. Differ. Equ., 254 (2013), 3715-3728.  doi: 10.1016/j.jde.2013.01.039. [8] M. Eller and V. Isakov, Carleman estimates with two large parameters and applications, Proc. Conf. Differential-Geometric Methods in the Control of Partial Differential Equations (Boulder, CO, July 1999), Contemp. Math., 268 (2000), 117-136. [9] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, (Lecture Notes Series vol. 34) Seoul National University, Seoul (Korea), 1996. [10] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Research, 34 (1998), 1027-1033.  doi: 10.1029/98WR00214. [11] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963. [12] O. Y. Imanuvilov, Controllability of parabolic equations, Sbornik Math., 186 (1995), 879-900. [13] O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009. [14] V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2006. [15] V. Isakov and N. Kim, Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress, Appl. Math., 35 (2008), 447-465.  doi: 10.4064/am35-4-4. [16] V. Isakov and N. Kim, Carleman estimates with second large parameter for second-order operators, Some Applications of Sobolev Spaces to PDEs, International Math. Ser., SpringerVerlag, 10 (2009), 135-159. [17] 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 (22p). [18] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003 (40pp). [19] A. Kawamoto, Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates, UTMS Preprint Series, UTMS 2016-3. [20] A. Kawamoto and M. Machida, Global Lipschitz stability for a fractional inverse transport problem by Carleman estimates, arXiv preprint, arXiv: 1608.07914 (2016). [21] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009. [22] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21 (2013), 477-560. [23] M. V. Klibanov and A. A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. [24] Z. Li, Y. Liu and M. Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Appl. Math. Comput., 257 (2015), 381-397.  doi: 10.1016/j.amc.2014.11.073. [25] Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations Zhiyuan, Inverse Problems, 32 (2016), 015004 (16pp). [26] 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. [27] C.-L. Lin and G. Nakamura, Unique continuation property for anomalous slow diffusion equation, Commun. Partial Differ. Equations, 41 (2016), 749-758. [28] Y. Liu, Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem, Comput. Math. Appl., 73 (2017), 96-108. [29] Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548. [30] M. Machida, The time-fractional radiative transport equation: Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion, J. Math. Phys. , 58 (2017), 12pp. [31] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. [32] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [33] X. Xu, J. Cheng and M. Yamamoto, Carleman estimate for fractional diffusion equation with half order and applicatioin, Appl. Anal., 90 (2011), 1355-1371. [34] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013 (75pp). [35] M. Yamamoto and Y. Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Problems, 28 (2012), 105010 (10pp).

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References:
 [1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2.Spatial moments analysis, Water Resources Research, 28 (1992), 3293-3307.  doi: 10.1029/92WR01757. [2] B. Amaziane, L. Pankratov and A. Piatnitski, Homogenization of a single-phase flow through a porous medium in a thin layer, Math. Models Methods Appl. Sci., 17 (2007), 1317-1349.  doi: 10.1142/S0218202507002339. [3] A. Ashyralyev, Well-posedness of the Basset problem in spaces of smooth functions, Appl. Math. Lett., 24 (2011), 1176-1180.  doi: 10.1016/j.aml.2011.02.002. [4] E. Bazhlekova and I. Dimovski, Exact solution of two-term time-fractional Thornley's problem by operational method, Integral Transforms Spec. Funct., 25 (2014), 61-74.  doi: 10.1080/10652469.2013.815184. [5] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272 (in Russian). [6] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independantes, Ark. Mat. Astr. Fys., 26B (1939), 1-9. [7] J. Cheng, C.-L. Lin and G. Nakamura, Unique continuation property for the anomalous diffusion and its application, J. Differ. Equ., 254 (2013), 3715-3728.  doi: 10.1016/j.jde.2013.01.039. [8] M. Eller and V. Isakov, Carleman estimates with two large parameters and applications, Proc. Conf. Differential-Geometric Methods in the Control of Partial Differential Equations (Boulder, CO, July 1999), Contemp. Math., 268 (2000), 117-136. [9] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, (Lecture Notes Series vol. 34) Seoul National University, Seoul (Korea), 1996. [10] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Research, 34 (1998), 1027-1033.  doi: 10.1029/98WR00214. [11] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963. [12] O. Y. Imanuvilov, Controllability of parabolic equations, Sbornik Math., 186 (1995), 879-900. [13] O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009. [14] V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2006. [15] V. Isakov and N. Kim, Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress, Appl. Math., 35 (2008), 447-465.  doi: 10.4064/am35-4-4. [16] V. Isakov and N. Kim, Carleman estimates with second large parameter for second-order operators, Some Applications of Sobolev Spaces to PDEs, International Math. Ser., SpringerVerlag, 10 (2009), 135-159. [17] 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 (22p). [18] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003 (40pp). [19] A. Kawamoto, Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates, UTMS Preprint Series, UTMS 2016-3. [20] A. Kawamoto and M. Machida, Global Lipschitz stability for a fractional inverse transport problem by Carleman estimates, arXiv preprint, arXiv: 1608.07914 (2016). [21] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009. [22] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21 (2013), 477-560. [23] M. V. Klibanov and A. A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. [24] Z. Li, Y. Liu and M. Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Appl. Math. Comput., 257 (2015), 381-397.  doi: 10.1016/j.amc.2014.11.073. [25] Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations Zhiyuan, Inverse Problems, 32 (2016), 015004 (16pp). [26] 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. [27] C.-L. Lin and G. Nakamura, Unique continuation property for anomalous slow diffusion equation, Commun. Partial Differ. Equations, 41 (2016), 749-758. [28] Y. Liu, Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem, Comput. Math. Appl., 73 (2017), 96-108. [29] Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548. [30] M. Machida, The time-fractional radiative transport equation: Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion, J. Math. Phys. , 58 (2017), 12pp. [31] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. [32] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [33] X. Xu, J. Cheng and M. Yamamoto, Carleman estimate for fractional diffusion equation with half order and applicatioin, Appl. Anal., 90 (2011), 1355-1371. [34] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013 (75pp). [35] M. Yamamoto and Y. Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Problems, 28 (2012), 105010 (10pp).
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