# American Institute of Mathematical Sciences

March  2020, 9(1): 153-179. doi: 10.3934/eect.2020001

## Initial-boundary value problems for multi-term time-fractional diffusion equations with $x$-dependent coefficients

 1 School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255049, China 2 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan 3 Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, no 54, 050094 Bucharest Romania 4 Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

* Corresponding author: Zhiyuan Li

Received  October 2018 Published  March 2020 Early access  August 2019

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with $x$-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

Citation: Zhiyuan Li, Xinchi Huang, Masahiro Yamamoto. Initial-boundary value problems for multi-term time-fractional diffusion equations with $x$-dependent coefficients. Evolution Equations and Control Theory, 2020, 9 (1) : 153-179. doi: 10.3934/eect.2020001
##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Courier Corporation, 1966. [2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic press, 2003. [3] 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. [4] S. Beckers and M. Yamamoto, Regularity and unique existence of solution to linear diffusion equation with multiple time-fractional derivatives, in Control and Optimization with PDE Constraints, Springer Basel, 164 (2013), 45-55. [5] D. A. Benson D A, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412. [6] B. Berkowitz, H. Scher and S. E. Silliman, Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resources Research, 36 (2000), 149-158.  doi: 10.1029/1999WR900295. [7] A. V. Chechkin, R. Gorenflo and I. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Physical Review E, 66 (2002), 046129. doi: 10.1103/PhysRevE.66.046129. [8] A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fractional Calculus and Applied Analysis, 6 (2003), 259-279. [9] 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. [10] V. Daftardar-Gejji and S. Bhalekar, Boundary value problems for multi-term fractional differential equations, Journal of Mathematical Analysis and Applications, 345 (2008), 754-765.  doi: 10.1016/j.jmaa.2008.04.065. [11] M. Giona, S. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A: Statistical Mechanics and its Applications, 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9. [12] R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fractional Calculus and Applied Analysis, 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048. [13] R. Gorenflo, Y. F. Luchko and P. P. Zabrejko, On solvability of linear fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 2 (1999), 163-176. [14] R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics, (eds. A. Carpinteri and F. Mainardi), Springer-Verlag, New York, 378 (1997), 223-276. [15] 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. [16] Y. Hatano, J. Nakagawa, S. Z. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, Journal of Math-for-Industry (JMI), 5 (2013), 51-57. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [18] D. J. Jiang, Z. Y. Li, Y. K. 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. [19] H. Jiang, F. Liu, I. Turner and K. Burrage, Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain, Journal of Mathematical Analysis and Applications, 389 (2012), 1117-1127.  doi: 10.1016/j.jmaa.2011.12.055. [20] 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. [21] Y. Kian, L. Oksanen, E. Soccorsi and M. Yamamoto, Global uniqueness in an inverse problem for time fractional diffusion equations, Journal of Differential Equations, 264 (2018), 1146-1170.  doi: 10.1016/j.jde.2017.09.032. [22] Y. Kian, E. Soccorsi and M. Yamamoto, A Uniqueness Result for Time-Fractional Diffusion Equations with Space-Dependent Variable Order, arXiv: 1701.04046. [23] A. A. Kilbas, H. M. Srivastave and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Hollan Math. Studies, 2006. [24] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, Journal of Mathematical Analysis and Applications, 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024. [25] A. Kubica and K. Ryszewska, Fractional Diffusion Equation with the Distributed Order Caputo Derivative, arXiv: 1706.05591. [26] A. Kubica and M. Yamamoto, Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients, Fract. Calc. Appl. Anal., 21 (2018), 276-311.  doi: 10.1515/fca-2018-0018. [27] M. Levy and B. Berkowitz, Measurement and analysis of non-Fickian dispersion in heterogeneous porous media, Journal of Contaminant Hydrology, 64 (2003), 203-226.  doi: 10.1016/S0169-7722(02)00204-8. [28] G. S. Li, D. L. Zhang, X. Z. 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, 36pp. doi: 10.1088/0266-5611/29/6/065014. [29] Z. Y. 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. [30] Z. Y. Li, Y. K. Liu and M. Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Applied Mathematics and Computation, 257 (2015), 381-397.  doi: 10.1016/j.amc.2014.11.073. [31] Z. Y. Li, Y. R. Luchko and M. Yamamoto, Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations, Fractional Calculus and Applied Analysis, 17 (2014), 1114-1136.  doi: 10.2478/s13540-014-0217-x. [32] Z. Y. Li, Y. R. Luchko and M. Yamamoto, Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem, Comput. Math. Appl., 73 (2017), 1041-1052.  doi: 10.1016/j.camwa.2016.06.030. [33] Z. Y. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Applicable Analysis, 94 (2015), 570-579.  doi: 10.1080/00036811.2014.926335. [34] Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59 (2010), 1766-1772.  doi: 10.1016/j.camwa.2009.08.015. [35] Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 374 (2011), 538-548.  doi: 10.1016/j.jmaa.2010.08.048. [36] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Mathematica Vietnamica, 24 (1999), 207-233. [37] R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physica A: Statistical Mechanics and its Applications, 278 (2000), 107-125.  doi: 10.1016/S0378-4371(99)00503-8. [38] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [39] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999. [40] J. Prüss, Evolutionary Integral Equations and Applications, Springer Science & Business Media, 1993. doi: 10.1007/978-3-0348-8570-6. [41] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, Journal of Physics A: Mathematical and General, 27 (1994), 3407-3410.  doi: 10.1088/0305-4470/27/10/017. [42] W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Education, 1987. [43] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058. [44] J. L. Schiff, The Laplace Transform: Theory and Applications, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-22757-3. [45] X. Xu, J. Cheng and M. Yamamoto, Carleman estimate for a fractional diffusion equation with half order and application, Applicable Analysis, 90 (2011), 1355-1371.  doi: 10.1080/00036811.2010.507199. [46] R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcialaj Ekvacioj, 52 (2009), 1-18.  doi: 10.1619/fesi.52.1.

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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Courier Corporation, 1966. [2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic press, 2003. [3] 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. [4] S. Beckers and M. Yamamoto, Regularity and unique existence of solution to linear diffusion equation with multiple time-fractional derivatives, in Control and Optimization with PDE Constraints, Springer Basel, 164 (2013), 45-55. [5] D. A. Benson D A, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412. [6] B. Berkowitz, H. Scher and S. E. Silliman, Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resources Research, 36 (2000), 149-158.  doi: 10.1029/1999WR900295. [7] A. V. Chechkin, R. Gorenflo and I. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Physical Review E, 66 (2002), 046129. doi: 10.1103/PhysRevE.66.046129. [8] A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fractional Calculus and Applied Analysis, 6 (2003), 259-279. [9] 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. [10] V. Daftardar-Gejji and S. Bhalekar, Boundary value problems for multi-term fractional differential equations, Journal of Mathematical Analysis and Applications, 345 (2008), 754-765.  doi: 10.1016/j.jmaa.2008.04.065. [11] M. Giona, S. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A: Statistical Mechanics and its Applications, 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9. [12] R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fractional Calculus and Applied Analysis, 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048. [13] R. Gorenflo, Y. F. Luchko and P. P. Zabrejko, On solvability of linear fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 2 (1999), 163-176. [14] R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics, (eds. A. Carpinteri and F. Mainardi), Springer-Verlag, New York, 378 (1997), 223-276. [15] 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. [16] Y. Hatano, J. Nakagawa, S. Z. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, Journal of Math-for-Industry (JMI), 5 (2013), 51-57. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [18] D. J. Jiang, Z. Y. Li, Y. K. 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. [19] H. Jiang, F. Liu, I. Turner and K. Burrage, Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain, Journal of Mathematical Analysis and Applications, 389 (2012), 1117-1127.  doi: 10.1016/j.jmaa.2011.12.055. [20] 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. [21] Y. Kian, L. Oksanen, E. Soccorsi and M. Yamamoto, Global uniqueness in an inverse problem for time fractional diffusion equations, Journal of Differential Equations, 264 (2018), 1146-1170.  doi: 10.1016/j.jde.2017.09.032. [22] Y. Kian, E. Soccorsi and M. Yamamoto, A Uniqueness Result for Time-Fractional Diffusion Equations with Space-Dependent Variable Order, arXiv: 1701.04046. [23] A. A. Kilbas, H. M. Srivastave and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Hollan Math. Studies, 2006. [24] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, Journal of Mathematical Analysis and Applications, 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024. [25] A. Kubica and K. Ryszewska, Fractional Diffusion Equation with the Distributed Order Caputo Derivative, arXiv: 1706.05591. [26] A. Kubica and M. Yamamoto, Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients, Fract. Calc. Appl. Anal., 21 (2018), 276-311.  doi: 10.1515/fca-2018-0018. [27] M. Levy and B. Berkowitz, Measurement and analysis of non-Fickian dispersion in heterogeneous porous media, Journal of Contaminant Hydrology, 64 (2003), 203-226.  doi: 10.1016/S0169-7722(02)00204-8. [28] G. S. Li, D. L. Zhang, X. Z. 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, 36pp. doi: 10.1088/0266-5611/29/6/065014. [29] Z. Y. 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. [30] Z. Y. Li, Y. K. Liu and M. Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Applied Mathematics and Computation, 257 (2015), 381-397.  doi: 10.1016/j.amc.2014.11.073. [31] Z. Y. Li, Y. R. Luchko and M. Yamamoto, Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations, Fractional Calculus and Applied Analysis, 17 (2014), 1114-1136.  doi: 10.2478/s13540-014-0217-x. [32] Z. Y. Li, Y. R. Luchko and M. Yamamoto, Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem, Comput. Math. Appl., 73 (2017), 1041-1052.  doi: 10.1016/j.camwa.2016.06.030. [33] Z. Y. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Applicable Analysis, 94 (2015), 570-579.  doi: 10.1080/00036811.2014.926335. [34] Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59 (2010), 1766-1772.  doi: 10.1016/j.camwa.2009.08.015. [35] Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 374 (2011), 538-548.  doi: 10.1016/j.jmaa.2010.08.048. [36] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Mathematica Vietnamica, 24 (1999), 207-233. [37] R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physica A: Statistical Mechanics and its Applications, 278 (2000), 107-125.  doi: 10.1016/S0378-4371(99)00503-8. [38] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [39] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999. [40] J. Prüss, Evolutionary Integral Equations and Applications, Springer Science & Business Media, 1993. doi: 10.1007/978-3-0348-8570-6. [41] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, Journal of Physics A: Mathematical and General, 27 (1994), 3407-3410.  doi: 10.1088/0305-4470/27/10/017. [42] W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Education, 1987. [43] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058. [44] J. L. Schiff, The Laplace Transform: Theory and Applications, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-22757-3. [45] X. Xu, J. Cheng and M. Yamamoto, Carleman estimate for a fractional diffusion equation with half order and application, Applicable Analysis, 90 (2011), 1355-1371.  doi: 10.1080/00036811.2010.507199. [46] R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcialaj Ekvacioj, 52 (2009), 1-18.  doi: 10.1619/fesi.52.1.
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