Figure 1.
Experiment $ (e1) $ , $ \hat E $ (top) and $ g_1, \hat g_1 $ (bottom) for different amount of samples
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December
2020, 14(6): 1001-1024.
doi: 10.3934/ipi.2020053
Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation
| 1. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
| 2. | Department of Mathematics, Northwest Normal University, Lanzhou 730070, Gansu, China |
| 3. | School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China |
In this work, an inverse problem in the fractional diffusion equation with random source is considered. The measurements we use are the statistical moments of the realizations of single point observation $ u(x_0,t,\omega). $ We build a representation of the solution $ u $ in the integral sense, then prove some theoretical results like uniqueness and stability. After that, we establish a numerical algorithm to solve the unknowns, where a mollification method is used.
Keywords:
Inverse problem,
stochastic fractional diffusion equation,
random source,
Volterra integral equation,
mollification.
Mathematics Subject Classification:
35R11, 35R30, 65C30, 65M32.
Citation:
Chan Liu, Jin Wen, Zhidong Zhang. Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation. Inverse Problems and Imaging,
2020, 14
(6)
: 1001-1024.
doi: 10.3934/ipi.2020053
![]()
References:
| [1] |
R. A. Adams and J. J. Fournier, Sobolev Spaces, 140, Elsevier, 2003. |
| [2] |
A. Babaei and S. Banihashemi,
Reconstructing unknown nonlinear boundary conditions in a time-fractional inverse reaction-diffusion-convection problem, Numer. Methods Partial Differential Equations, 35 (2019), 976-992.
doi: 10.1002/num.22334. |
| [3] |
R. L. Bagley and P. J. Torvik,
A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201-210.
doi: 10.1122/1.549724. |
| [4] |
D. Baleanu and A. M. Lopes, Handbook of Fractional Calculus with Applications, De Gruyter, 2019. |
| [5] |
G. Bao, C. Chen and P. Li,
Inverse random source scattering problems in several dimensions, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 1263-1287.
doi: 10.1137/16M1067470. |
| [6] |
G. Bao, S.-N. Chow, P. Li and H. Zhou,
An inverse random source problem for the Helmholtz equation, Math. Comp., 83 (2014), 215-233.
doi: 10.1090/S0025-5718-2013-02730-5. |
| [7] |
G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems, 29 (2013), 015006.
doi: 10.1088/0266-5611/29/1/015006. |
| [8] |
E. Barkai, R. Metzler and J. Klafter,
From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132-138.
doi: 10.1103/PhysRevE.61.132. |
| [9] |
B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Rev. Geophys., 44.
doi: 10.1029/2005RG000178. |
| [10] |
H. Brunner, Volterra Integral Equations, Cambridge Monographs on Applied and Computational Mathematics, 30, Cambridge University Press, Cambridge, 2017
doi: 10.1017/9781316162491. |
| [11] |
X. Cao, Y.-H. Lin and H. Liu,
Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.
doi: 10.3934/ipi.2019011. |
| [12] |
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. |
| [13] |
M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 91.
doi: 10.1007/s00526-020-01740-6. |
| [14] |
X. Cheng, L. Yuan and K. Liang,
Inverse source problem for a distributed-order time fractional diffusion equation, J. Inverse Ill-Posed Probl., 28 (2020), 17-32.
doi: 10.1515/jiip-2019-0006. |
| [15] |
B. De Finetti, Theory of Probability: A Critical Introductory Treatment, 6, John Wiley & Sons, 2017.
doi: 10.1002/9781119286387. |
| [16] |
X. Feng, P. Li and X. Wang, An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion, Inverse Problems, 36 (2020), 045008.
doi: 10.1088/1361-6420/ab6503. |
| [17] |
T. Ghosh, A. Rúland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 108505.
doi: 10.1016/j.jfa.2020.108505. |
| [18] |
T. Ghosh, M. Salo and G. Uhlmann,
The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475.
doi: 10.2140/apde.2020.13.455. |
| [19] |
R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, Fractional calculus and continuous-time finance. Ⅲ. The diffusion limit, in Mathematical Finance (Konstanz, 2000), Trends Math., Birkh¨auser, Basel, 2001, 171-180. |
| [20] |
B. Harrach and Y.-H. Lin,
Monotonicity-based inversion of the fractional Schrödinger equation Ⅱ. General potentials and stability, SIAM J. Math. Anal., 52 (2020), 402-436.
doi: 10.1137/19M1251576. |
| [21] |
D. Hou, M. T. Hasan and C. Xu,
Müntz spectral methods for the time-fractional diffusion equation, Comput. Methods Appl. Math., 18 (2018), 43-62.
doi: 10.1515/cmam-2017-0027. |
| [22] |
X. Huang, Z. Li and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Problems, 35 (2019), 045003.
doi: 10.1088/1361-6420/ab0138. |
| [23] |
B. Jin, R. Lazarov and Z. Zhou,
An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221.
doi: 10.1093/imanum/dru063. |
| [24] |
B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003.
doi: 10.1088/0266-5611/31/3/035003. |
| [25] |
B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction-diffusion equation, Inverse Problems, 35 (2019), 065004.
doi: 10.1088/1361-6420/ab109e. |
| [26] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006. |
| [27] |
J. Klafter and R. Silbey,
Derivation of the continuous-time random-walk equation, Phys. Rev. Lett., 44 (1980), 55-58.
doi: 10.1103/PhysRevLett.44.55. |
| [28] |
R. C. Koeller,
Applications of fractional calculus to the theory of viscoelasticity, Trans. ASME J. Appl. Mech., 51 (1984), 299-307.
doi: 10.1115/1.3167616. |
| [29] |
R.-Y. Lai and Y.-H. Lin,
Global uniqueness for the fractional semilinear Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 1189-1199.
doi: 10.1090/proc/14319. |
| [30] |
M. Li, C. Chen and P. Li, Inverse random source scattering for the Helmholtz equation in inhomogeneous media, Inverse Problems, 34 (2018), 015003.
doi: 10.1088/1361-6420/aa99d2. |
| [31] |
P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004.
doi: 10.1088/0266-5611/27/3/035004. |
| [32] |
P. Li and G. Yuan,
Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.
doi: 10.1016/j.jmaa.2017.01.074. |
| [33] |
Z. Li, X. Cheng and G. Li, An inverse problem in time-fractional diffusion equations with nonlinear boundary condition, J. Math. Phys., 60 (2019), 091502.
doi: 10.1063/1.5047074. |
| [34] |
Z. Li, Y. 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. |
| [35] |
J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. Ⅰ, Springer-Verlag, New York-Heidelberg, 1972. |
| [36] |
Y. Liu and Z. Zhang, Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation, J. Phys. A, 50 (2017), 305203.
doi: 10.1088/1751-8121/aa763a. |
| [37] |
Z. Liu, F. Liu and F. Zeng,
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations, Appl. Numer. Math., 136 (2019), 139-151.
doi: 10.1016/j.apnum.2018.10.005. |
| [38] |
Y. Luchko,
Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218-223.
doi: 10.1016/j.jmaa.2008.10.018. |
| [39] |
C. Lv, M. Azaiez and C. Xu,
Spectral deferred correction methods for fractional differential equations, Numer. Math. Theory Methods Appl., 11 (2018), 729-751.
doi: 10.4208/nmtma.2018.s03. |
| [40] |
C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput., 38 (2016), A2699–A2724.
doi: 10.1137/15M102664X. |
| [41] |
M. Magdziarz, A. Weron, K. Burnecki and J. Klafter, Fractional Brownian motion versus the continuous-time random walk: A simple test for subdiffusive dynamics, Phys. Rev. Lett., 103 (2009), 180602.
doi: 10.1103/PhysRevLett.103.180602. |
| [42] |
D. Murio, C. E. Mejía and S. Zhan,
Discrete mollification and automatic numerical differentiation, Comput. Math. Appl., 35 (1998), 1-16.
doi: 10.1016/S0898-1221(98)00001-7. |
| [43] |
R. Nigmatullin,
The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi (B), 133 (1986), 425-430.
doi: 10.1002/pssb.2221330150. |
| [44] |
P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020), 045002.
doi: 10.1088/1361-6420/ab532c. |
| [45] |
B. Øksendal, Stochastic Differential Equations, 6$^th$ edition, Universitext, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-642-14394-6. |
| [46] |
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
|
| [47] |
S. Qasemi, D. Rostamy and N. Abdollahi,
The time-fractional diffusion inverse problem subject to an extra measurement by a local discontinuous Galerkin method, BIT, 59 (2019), 183-212.
doi: 10.1007/s10543-018-0731-z. |
| [48] |
Z. Ruan, S. Zhang and S. Xiong,
Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method, Evol. Equ. Control Theory, 7 (2018), 669-682.
doi: 10.3934/eect.2018032. |
| [49] |
A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003.
doi: 10.1088/1361-6420/aaac5a. |
| [50] |
W. Rundell and Z. Zhang, Fractional diffusion: Recovering the distributed fractional derivative from overposed data, Inverse Problems, 33 (2017), 035008.
doi: 10.1088/1361-6420/aa573e. |
| [51] |
W. Rundell and Z. Zhang,
Recovering an unknown source in a fractional diffusion problem, J. Comput. Phys., 368 (2018), 299-314.
doi: 10.1016/j.jcp.2018.04.046. |
| [52] |
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. |
| [53] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. |
| [54] |
S. Shen, F. Liu and V. V. Anh,
The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation, J. Comput. Appl. Math., 345 (2019), 515-534.
doi: 10.1016/j.cam.2018.05.020. |
| [55] |
T. N. Thach, T. N. Huy, P. T. M. Tam, M. N. Minh and N. H. Can,
Identification of an inverse source problem for time-fractional diffusion equation with random noise, Math. Methods Appl. Sci., 42 (2019), 204-218.
doi: 10.1002/mma.5334. |
| [56] |
F. G. Tricomi, Integral Equations, Dover Publications, Inc., New York, 1985. |
| [57] |
N. H. Tuan, V. C. H. Luu and S. Tatar, An inverse problem for an inhomogeneous time-fractional diffusion equation: A regularization method and error estimate, Comput. Appl. Math., 38 (2019), Art. 32.
doi: 10.1007/s40314-019-0776-x. |
| [58] |
X. B. Yan and T. Wei,
Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach, J. Inverse Ill-Posed Probl., 27 (2019), 1-16.
doi: 10.1515/jiip-2017-0091. |
| [59] |
F. Zeng, C. Li, F. Liu and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35 (2013), A2976–A3000.
doi: 10.1137/130910865. |
| [60] |
Z. Zhang,
An undetermined time-dependent coefficient in a fractional diffusion equation, Inverse Probl. Imaging, 11 (2017), 875-900.
doi: 10.3934/ipi.2017041. |
| [61] |
Z. Zhang and Z. Zhou,
Recovering the potential term in a fractional diffusion equation, IMA J. Appl. Math., 82 (2017), 579-600.
doi: 10.1093/imamat/hxx004. |
show all references
References:
| [1] |
R. A. Adams and J. J. Fournier, Sobolev Spaces, 140, Elsevier, 2003. |
| [2] |
A. Babaei and S. Banihashemi,
Reconstructing unknown nonlinear boundary conditions in a time-fractional inverse reaction-diffusion-convection problem, Numer. Methods Partial Differential Equations, 35 (2019), 976-992.
doi: 10.1002/num.22334. |
| [3] |
R. L. Bagley and P. J. Torvik,
A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201-210.
doi: 10.1122/1.549724. |
| [4] |
D. Baleanu and A. M. Lopes, Handbook of Fractional Calculus with Applications, De Gruyter, 2019. |
| [5] |
G. Bao, C. Chen and P. Li,
Inverse random source scattering problems in several dimensions, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 1263-1287.
doi: 10.1137/16M1067470. |
| [6] |
G. Bao, S.-N. Chow, P. Li and H. Zhou,
An inverse random source problem for the Helmholtz equation, Math. Comp., 83 (2014), 215-233.
doi: 10.1090/S0025-5718-2013-02730-5. |
| [7] |
G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems, 29 (2013), 015006.
doi: 10.1088/0266-5611/29/1/015006. |
| [8] |
E. Barkai, R. Metzler and J. Klafter,
From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132-138.
doi: 10.1103/PhysRevE.61.132. |
| [9] |
B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Rev. Geophys., 44.
doi: 10.1029/2005RG000178. |
| [10] |
H. Brunner, Volterra Integral Equations, Cambridge Monographs on Applied and Computational Mathematics, 30, Cambridge University Press, Cambridge, 2017
doi: 10.1017/9781316162491. |
| [11] |
X. Cao, Y.-H. Lin and H. Liu,
Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.
doi: 10.3934/ipi.2019011. |
| [12] |
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. |
| [13] |
M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 91.
doi: 10.1007/s00526-020-01740-6. |
| [14] |
X. Cheng, L. Yuan and K. Liang,
Inverse source problem for a distributed-order time fractional diffusion equation, J. Inverse Ill-Posed Probl., 28 (2020), 17-32.
doi: 10.1515/jiip-2019-0006. |
| [15] |
B. De Finetti, Theory of Probability: A Critical Introductory Treatment, 6, John Wiley & Sons, 2017.
doi: 10.1002/9781119286387. |
| [16] |
X. Feng, P. Li and X. Wang, An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion, Inverse Problems, 36 (2020), 045008.
doi: 10.1088/1361-6420/ab6503. |
| [17] |
T. Ghosh, A. Rúland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 108505.
doi: 10.1016/j.jfa.2020.108505. |
| [18] |
T. Ghosh, M. Salo and G. Uhlmann,
The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475.
doi: 10.2140/apde.2020.13.455. |
| [19] |
R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, Fractional calculus and continuous-time finance. Ⅲ. The diffusion limit, in Mathematical Finance (Konstanz, 2000), Trends Math., Birkh¨auser, Basel, 2001, 171-180. |
| [20] |
B. Harrach and Y.-H. Lin,
Monotonicity-based inversion of the fractional Schrödinger equation Ⅱ. General potentials and stability, SIAM J. Math. Anal., 52 (2020), 402-436.
doi: 10.1137/19M1251576. |
| [21] |
D. Hou, M. T. Hasan and C. Xu,
Müntz spectral methods for the time-fractional diffusion equation, Comput. Methods Appl. Math., 18 (2018), 43-62.
doi: 10.1515/cmam-2017-0027. |
| [22] |
X. Huang, Z. Li and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Problems, 35 (2019), 045003.
doi: 10.1088/1361-6420/ab0138. |
| [23] |
B. Jin, R. Lazarov and Z. Zhou,
An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221.
doi: 10.1093/imanum/dru063. |
| [24] |
B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003.
doi: 10.1088/0266-5611/31/3/035003. |
| [25] |
B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction-diffusion equation, Inverse Problems, 35 (2019), 065004.
doi: 10.1088/1361-6420/ab109e. |
| [26] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006. |
| [27] |
J. Klafter and R. Silbey,
Derivation of the continuous-time random-walk equation, Phys. Rev. Lett., 44 (1980), 55-58.
doi: 10.1103/PhysRevLett.44.55. |
| [28] |
R. C. Koeller,
Applications of fractional calculus to the theory of viscoelasticity, Trans. ASME J. Appl. Mech., 51 (1984), 299-307.
doi: 10.1115/1.3167616. |
| [29] |
R.-Y. Lai and Y.-H. Lin,
Global uniqueness for the fractional semilinear Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 1189-1199.
doi: 10.1090/proc/14319. |
| [30] |
M. Li, C. Chen and P. Li, Inverse random source scattering for the Helmholtz equation in inhomogeneous media, Inverse Problems, 34 (2018), 015003.
doi: 10.1088/1361-6420/aa99d2. |
| [31] |
P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004.
doi: 10.1088/0266-5611/27/3/035004. |
| [32] |
P. Li and G. Yuan,
Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.
doi: 10.1016/j.jmaa.2017.01.074. |
| [33] |
Z. Li, X. Cheng and G. Li, An inverse problem in time-fractional diffusion equations with nonlinear boundary condition, J. Math. Phys., 60 (2019), 091502.
doi: 10.1063/1.5047074. |
| [34] |
Z. Li, Y. 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. |
| [35] |
J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. Ⅰ, Springer-Verlag, New York-Heidelberg, 1972. |
| [36] |
Y. Liu and Z. Zhang, Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation, J. Phys. A, 50 (2017), 305203.
doi: 10.1088/1751-8121/aa763a. |
| [37] |
Z. Liu, F. Liu and F. Zeng,
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations, Appl. Numer. Math., 136 (2019), 139-151.
doi: 10.1016/j.apnum.2018.10.005. |
| [38] |
Y. Luchko,
Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218-223.
doi: 10.1016/j.jmaa.2008.10.018. |
| [39] |
C. Lv, M. Azaiez and C. Xu,
Spectral deferred correction methods for fractional differential equations, Numer. Math. Theory Methods Appl., 11 (2018), 729-751.
doi: 10.4208/nmtma.2018.s03. |
| [40] |
C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput., 38 (2016), A2699–A2724.
doi: 10.1137/15M102664X. |
| [41] |
M. Magdziarz, A. Weron, K. Burnecki and J. Klafter, Fractional Brownian motion versus the continuous-time random walk: A simple test for subdiffusive dynamics, Phys. Rev. Lett., 103 (2009), 180602.
doi: 10.1103/PhysRevLett.103.180602. |
| [42] |
D. Murio, C. E. Mejía and S. Zhan,
Discrete mollification and automatic numerical differentiation, Comput. Math. Appl., 35 (1998), 1-16.
doi: 10.1016/S0898-1221(98)00001-7. |
| [43] |
R. Nigmatullin,
The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi (B), 133 (1986), 425-430.
doi: 10.1002/pssb.2221330150. |
| [44] |
P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020), 045002.
doi: 10.1088/1361-6420/ab532c. |
| [45] |
B. Øksendal, Stochastic Differential Equations, 6$^th$ edition, Universitext, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-642-14394-6. |
| [46] |
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
|
| [47] |
S. Qasemi, D. Rostamy and N. Abdollahi,
The time-fractional diffusion inverse problem subject to an extra measurement by a local discontinuous Galerkin method, BIT, 59 (2019), 183-212.
doi: 10.1007/s10543-018-0731-z. |
| [48] |
Z. Ruan, S. Zhang and S. Xiong,
Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method, Evol. Equ. Control Theory, 7 (2018), 669-682.
doi: 10.3934/eect.2018032. |
| [49] |
A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003.
doi: 10.1088/1361-6420/aaac5a. |
| [50] |
W. Rundell and Z. Zhang, Fractional diffusion: Recovering the distributed fractional derivative from overposed data, Inverse Problems, 33 (2017), 035008.
doi: 10.1088/1361-6420/aa573e. |
| [51] |
W. Rundell and Z. Zhang,
Recovering an unknown source in a fractional diffusion problem, J. Comput. Phys., 368 (2018), 299-314.
doi: 10.1016/j.jcp.2018.04.046. |
| [52] |
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. |
| [53] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. |
| [54] |
S. Shen, F. Liu and V. V. Anh,
The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation, J. Comput. Appl. Math., 345 (2019), 515-534.
doi: 10.1016/j.cam.2018.05.020. |
| [55] |
T. N. Thach, T. N. Huy, P. T. M. Tam, M. N. Minh and N. H. Can,
Identification of an inverse source problem for time-fractional diffusion equation with random noise, Math. Methods Appl. Sci., 42 (2019), 204-218.
doi: 10.1002/mma.5334. |
| [56] |
F. G. Tricomi, Integral Equations, Dover Publications, Inc., New York, 1985. |
| [57] |
N. H. Tuan, V. C. H. Luu and S. Tatar, An inverse problem for an inhomogeneous time-fractional diffusion equation: A regularization method and error estimate, Comput. Appl. Math., 38 (2019), Art. 32.
doi: 10.1007/s40314-019-0776-x. |
| [58] |
X. B. Yan and T. Wei,
Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach, J. Inverse Ill-Posed Probl., 27 (2019), 1-16.
doi: 10.1515/jiip-2017-0091. |
| [59] |
F. Zeng, C. Li, F. Liu and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35 (2013), A2976–A3000.
doi: 10.1137/130910865. |
| [60] |
Z. Zhang,
An undetermined time-dependent coefficient in a fractional diffusion equation, Inverse Probl. Imaging, 11 (2017), 875-900.
doi: 10.3934/ipi.2017041. |
| [61] |
Z. Zhang and Z. Zhou,
Recovering the potential term in a fractional diffusion equation, IMA J. Appl. Math., 82 (2017), 579-600.
doi: 10.1093/imamat/hxx004. |
Figure 2.
Experiment $ (e1), $ $ \hat E,J_\epsilon * \hat E $ (top) and $ g_1, \hat g_1 $ (bottom) for different $ \epsilon $ , $ 10^3 $ realizations
Figure 3.
Experiment $ (e1) $ , $ \hat V $ (top) and $ |g_2|,|\hat g_2| $ (bottom) for different amount of samples
Figure 4.
Experiment $ (e1) $ , $ \hat V, J_\epsilon *\hat V $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \epsilon $ , $ 10^3 $ realizations
Figure 5.
Experiment $ (e1) $ , $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \epsilon $
Figure 6.
Experiment $ (e1) $ , $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 7.
Experiment $ (e2) $ , $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 8.
Experiment $ (e3) $ , $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 9.
Experiment $ (e4) $ , $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 10.
Experiment $ (e5) $ , $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 12.
Experiment $ (e1) $ , $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Figure 13.
Experiment $ (e2) $ , $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Figure 14.
Experiment $ (e3) $ , $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Figure 15.
Experiment $ (e4) $ , $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Figure 16.
Experiment $ (e5) $ , $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Table 1.
Errors without mollification
| 0.299794 | 0.056932 | 0.091171 | 0.018218 | 0.028705 | 0.009218 | |
| 0.449422 | 0.084018 | 0.142783 | 0.029317 | 0.046238 | 0.012576 | |
| 0.447672 | 0.083309 | 0.231421 | 0.043802 | 0.081429 | 0.020119 | |
| 1.570322 | 0.288016 | 0.503009 | 0.104327 | 0.151976 | 0.055286 | |
| 1.542568 | 0.310195 | 0.509957 | 0.115601 | 0.171542 | 0.057249 | |
| 0.299794 | 0.056932 | 0.091171 | 0.018218 | 0.028705 | 0.009218 | |
| 0.449422 | 0.084018 | 0.142783 | 0.029317 | 0.046238 | 0.012576 | |
| 0.447672 | 0.083309 | 0.231421 | 0.043802 | 0.081429 | 0.020119 | |
| 1.570322 | 0.288016 | 0.503009 | 0.104327 | 0.151976 | 0.055286 | |
| 1.542568 | 0.310195 | 0.509957 | 0.115601 | 0.171542 | 0.057249 | |
| Without mollification | ||||||
| 0.299794 | 0.056932 | 0.124498 | 0.030254 | 0.050135 | 0.055231 | |
| 0.449422 | 0.084018 | 0.203604 | 0.039741 | 0.097088 | 0.034309 | |
| 0.447672 | 0.083309 | 0.216069 | 0.041439 | 0.096336 | 0.033850 | |
| 1.570322 | 0.288016 | 0.827782 | 0.172930 | 0.266984 | 0.262352 | |
| 1.542568 | 0.310195 | 0.624018 | 0.195870 | 0.165958 | 0.281415 | |
| Without mollification | ||||||
| 0.299794 | 0.056932 | 0.124498 | 0.030254 | 0.050135 | 0.055231 | |
| 0.449422 | 0.084018 | 0.203604 | 0.039741 | 0.097088 | 0.034309 | |
| 0.447672 | 0.083309 | 0.216069 | 0.041439 | 0.096336 | 0.033850 | |
| 1.570322 | 0.288016 | 0.827782 | 0.172930 | 0.266984 | 0.262352 | |
| 1.542568 | 0.310195 | 0.624018 | 0.195870 | 0.165958 | 0.281415 | |
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