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  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 & Control Theory, 2020, 9 (1) : 153-179. doi: 10.3934/eect.2020001
References:
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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.  Google Scholar

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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.  Google Scholar

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D. A. Benson D AS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412.   Google Scholar

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B. BerkowitzH. Scher and S. E. Silliman, Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resources Research, 36 (2000), 149-158.  doi: 10.1029/1999WR900295.  Google Scholar

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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.  Google Scholar

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A. V. ChechkinR. GorenfloI. M. Sokolov and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fractional Calculus and Applied Analysis, 6 (2003), 259-279.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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M. GionaS. 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.  Google Scholar

[12]

R. GorenfloY. 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.  Google Scholar

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R. GorenfloY. F. Luchko and P. P. Zabrejko, On solvability of linear fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 2 (1999), 163-176.   Google Scholar

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H. JiangF. LiuI. 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.  Google Scholar

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Y. KianL. OksanenE. 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.  Google Scholar

[22]

Y. Kian, E. Soccorsi and M. Yamamoto, A Uniqueness Result for Time-Fractional Diffusion Equations with Space-Dependent Variable Order, arXiv: 1701.04046. Google Scholar

[23]

A. A. Kilbas, H. M. Srivastave and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Hollan Math. Studies, 2006.  Google Scholar

[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.  Google Scholar

[25]

A. Kubica and K. Ryszewska, Fractional Diffusion Equation with the Distributed Order Caputo Derivative, arXiv: 1706.05591. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

Z. Y. LiY. 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.  Google Scholar

[31]

Z. Y. LiY. 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.  Google Scholar

[32]

Z. Y. LiY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999.   Google Scholar
[40]

J. Prüss, Evolutionary Integral Equations and Applications, Springer Science & Business Media, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[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.  Google Scholar

[42]

W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Education, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X. XuJ. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Courier Corporation, 1966.  Google Scholar

[2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic press, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[5]

D. A. Benson D AS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412.   Google Scholar

[6]

B. BerkowitzH. Scher and S. E. Silliman, Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resources Research, 36 (2000), 149-158.  doi: 10.1029/1999WR900295.  Google Scholar

[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.  Google Scholar

[8]

A. V. ChechkinR. GorenfloI. M. Sokolov and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fractional Calculus and Applied Analysis, 6 (2003), 259-279.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. GionaS. 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.  Google Scholar

[12]

R. GorenfloY. 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.  Google Scholar

[13]

R. GorenfloY. F. Luchko and P. P. Zabrejko, On solvability of linear fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 2 (1999), 163-176.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Y. HatanoJ. NakagawaS. Z. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, Journal of Math-for-Industry (JMI), 5 (2013), 51-57.   Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[19]

H. JiangF. LiuI. 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.  Google Scholar

[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.  Google Scholar

[21]

Y. KianL. OksanenE. 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.  Google Scholar

[22]

Y. Kian, E. Soccorsi and M. Yamamoto, A Uniqueness Result for Time-Fractional Diffusion Equations with Space-Dependent Variable Order, arXiv: 1701.04046. Google Scholar

[23]

A. A. Kilbas, H. M. Srivastave and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Hollan Math. Studies, 2006.  Google Scholar

[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.  Google Scholar

[25]

A. Kubica and K. Ryszewska, Fractional Diffusion Equation with the Distributed Order Caputo Derivative, arXiv: 1706.05591. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

Z. Y. LiY. 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.  Google Scholar

[31]

Z. Y. LiY. 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.  Google Scholar

[32]

Z. Y. LiY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999.   Google Scholar
[40]

J. Prüss, Evolutionary Integral Equations and Applications, Springer Science & Business Media, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[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.  Google Scholar

[42]

W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Education, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X. XuJ. 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.  Google Scholar

[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.  Google Scholar

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