# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2022025
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## Identifying a space-dependent source term in distributed order time-fractional diffusion equations

 Department of Economic Mathematics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

Received  February 2022 Revised  April 2022 Early access May 2022

The aim of this paper is to investigate an inverse problem of recovering a space-dependent source term governed by distributed order time-fractional diffusion equations in Hilbert scales. Such a problem is ill-posed and has important practical applications. For this problem, we propose a general regularization method based on the idea of the filter method. With a suitable source condition, we prove that the method is of optimal order under various choices of regularization parameter. One is based on the a priori regularization parameter choice rule and another one is the discrepancy principle. Finally, the capabilities of our method are illustrated by both the Tikhonov and the Landweber method.

Citation: Dinh Nguyen Duy Hai. Identifying a space-dependent source term in distributed order time-fractional diffusion equations. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022025
##### References:
 [1] W. Bu, A. Xiao and W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 72 (2017), 422-441.  doi: 10.1007/s10915-017-0360-8. [2] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, Springer, 2016. doi: 10.1007/978-3-319-28739-3. [3] J. R. Cannon, Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. Anal., 5 (1968), 275-286.  doi: 10.1137/0705024. [4] 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. [5] R. Courant and D. Hilbert, Methods of Mathematical Physics, Translated from the English by Ch'ien Min and Kuo Tun-jen Science Press, Peking 1965. [6] N. M. Dien, D. N. D. Hai, T. Q. Viet and D. D. Trong, On Tikhonov's method and optimal error bound for inverse source problem for a time-fractional diffusion equation, Comput. Math. Appl., 80 (2020), 61-81.  doi: 10.1016/j.camwa.2020.02.024. [7] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers Group, Dordrecht, 1996. [8] N. J. Ford, M. L. Morgado and M. Rebelo, An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time, Electron. Trans. Numer. Anal., 44 (2015), 289-305. [9] D. N. D. Hai and D. D. Trong, Optimal error bound and truncation regularization method for a backward time-fractional diffusion problem in Hilbert scales, Appl. Math. Lett., 107 (2020), 106448, 9 pp. doi: 10.1016/j.aml.2020.106448. [10] A. Hasanov and B. Pektaș, Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method, Comput. Math. Appl., 65 (2013), 42-57.  doi: 10.1016/j.camwa.2012.10.009. [11] A. Hazanee, D. Lesnic, M. I. Ismailov and N. B. Kerimov, Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions, Appl. Math. Comput., 346 (2019), 800-815.  doi: 10.1016/j.amc.2018.10.059. [12] V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034. [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] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [15] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024. [16] A. Kubica and K. Ryszewska, Decay of solutions to parabolic-type problem with distributed order Caputo derivative, J. Math. Anal. Appl., 465 (2018), 75-99.  doi: 10.1016/j.jmaa.2018.04.067. [17] Z. Li, K. Fujishiro and G. Li, Uniqueness in the inversion of distributed orders in ultraslow diffusion equations, J. Comput. Appl. Math., 369 (2020), 112564, 13 pp. doi: 10.1016/j.cam.2019.112564. [18] Z. Li, Y. Kian and E. Soccorsi, Initial-boundary value problem for distributed order time-fractional diffusion equations, Asymptot. Anal., 115 (2019), 95-126.  doi: 10.3233/ASY-191532. [19] Z. Li, Y. Luchko and M. Yamamoto, Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations, Fract. Calc. Appl. Anal., 17 (2014), 1114-1136.  doi: 10.2478/s13540-014-0217-x. [20] 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. [21] Y. S. Li, L. L. Sun, Z. Q. Zhang and T. Wei, Identification of the time-dependent source term in a multi-term time-fractional diffusion equation, Numer. Algor., 82 (2019), 1279-1301.  doi: 10.1007/s11075-019-00654-5. [22] J. J. Liu, C. L. Sun and M. Yamamoto, Recovering the weight function in distributed order fractional equation from interior measurement, Appl. Numer. Math., 168 (2021), 84-103.  doi: 10.1016/j.apnum.2021.05.026. [23] S. Lu, S. V. Pereverzev, Y. Shao and U. Tautenhahn, On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales, J. Integral Equations Appl., 22 (2010), 483-517. [24] Y. Luchko, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12 (2009), 409-422. [25] Z. Ruan and Z. Wang, Identification of a time-dependent source term for a time fractional diffusion problem, Appl. Anal., 96 (2017), 1638-1655.  doi: 10.1080/00036811.2016.1232400. [26] W. Rundell and Z. Zhang, Fractional diffusion: Recovering the distributed fractional derivative from overposed data, Inverse Probl., 33 (2017), 035008, 27 pp. doi: 10.1088/1361-6420/aa573e. [27] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [28] C. Sun and J. Liu, An inverse source problem for distributed order time-fractional diffusion equation, Inverse Probl., 36 (2020), 055008.  doi: 10.1088/1361-6420/ab762c. [29] S. Tatar and S. Ulusoy, An inverse source problem for a one-dimensional space-time fractional diffusion equation, Appl. Anal., 94 (2015), 2233-2244.  doi: 10.1080/00036811.2014.979808. [30] U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Num. Funct. Anal. Optim., 19 (1998), 377-398.  doi: 10.1080/01630569808816834. [31] D. D. Trong, D. N. D. Hai and N. D. Minh, Optimal regularization for an unknown source of space-fractional diffusion equation, Appl. Math. Comput., 349 (2019), 184-206.  doi: 10.1016/j.amc.2018.12.030. [32] H. T. Nguyen, D. L. Le and V. T. Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Appl. Math. Model., 40 (2016), 8244-8264.  doi: 10.1016/j.apm.2016.04.009. [33] J. G. Wang, Y. B. Zhou and T. Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl. Numer. Math., 68 (2013), 39-57.  doi: 10.1016/j.apnum.2013.01.001. [34] H. Ye, F. Liu, V. Anh and I. Turner, Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains, IMA J. Appl. Math., 80 (2015), 825-838.  doi: 10.1093/imamat/hxu015. [35] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Probl., 27 (2011), 035010, 12 pp. doi: 10.1088/0266-5611/27/3/035010.

show all references

##### References:
 [1] W. Bu, A. Xiao and W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 72 (2017), 422-441.  doi: 10.1007/s10915-017-0360-8. [2] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, Springer, 2016. doi: 10.1007/978-3-319-28739-3. [3] J. R. Cannon, Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. Anal., 5 (1968), 275-286.  doi: 10.1137/0705024. [4] 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. [5] R. Courant and D. Hilbert, Methods of Mathematical Physics, Translated from the English by Ch'ien Min and Kuo Tun-jen Science Press, Peking 1965. [6] N. M. Dien, D. N. D. Hai, T. Q. Viet and D. D. Trong, On Tikhonov's method and optimal error bound for inverse source problem for a time-fractional diffusion equation, Comput. Math. Appl., 80 (2020), 61-81.  doi: 10.1016/j.camwa.2020.02.024. [7] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers Group, Dordrecht, 1996. [8] N. J. Ford, M. L. Morgado and M. Rebelo, An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time, Electron. Trans. Numer. Anal., 44 (2015), 289-305. [9] D. N. D. Hai and D. D. Trong, Optimal error bound and truncation regularization method for a backward time-fractional diffusion problem in Hilbert scales, Appl. Math. Lett., 107 (2020), 106448, 9 pp. doi: 10.1016/j.aml.2020.106448. [10] A. Hasanov and B. Pektaș, Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method, Comput. Math. Appl., 65 (2013), 42-57.  doi: 10.1016/j.camwa.2012.10.009. [11] A. Hazanee, D. Lesnic, M. I. Ismailov and N. B. Kerimov, Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions, Appl. Math. Comput., 346 (2019), 800-815.  doi: 10.1016/j.amc.2018.10.059. [12] V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034. [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] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [15] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024. [16] A. Kubica and K. Ryszewska, Decay of solutions to parabolic-type problem with distributed order Caputo derivative, J. Math. Anal. Appl., 465 (2018), 75-99.  doi: 10.1016/j.jmaa.2018.04.067. [17] Z. Li, K. Fujishiro and G. Li, Uniqueness in the inversion of distributed orders in ultraslow diffusion equations, J. Comput. Appl. Math., 369 (2020), 112564, 13 pp. doi: 10.1016/j.cam.2019.112564. [18] Z. Li, Y. Kian and E. Soccorsi, Initial-boundary value problem for distributed order time-fractional diffusion equations, Asymptot. Anal., 115 (2019), 95-126.  doi: 10.3233/ASY-191532. [19] Z. Li, Y. Luchko and M. Yamamoto, Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations, Fract. Calc. Appl. Anal., 17 (2014), 1114-1136.  doi: 10.2478/s13540-014-0217-x. [20] 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. [21] Y. S. Li, L. L. Sun, Z. Q. Zhang and T. Wei, Identification of the time-dependent source term in a multi-term time-fractional diffusion equation, Numer. Algor., 82 (2019), 1279-1301.  doi: 10.1007/s11075-019-00654-5. [22] J. J. Liu, C. L. Sun and M. Yamamoto, Recovering the weight function in distributed order fractional equation from interior measurement, Appl. Numer. Math., 168 (2021), 84-103.  doi: 10.1016/j.apnum.2021.05.026. [23] S. Lu, S. V. Pereverzev, Y. Shao and U. Tautenhahn, On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales, J. Integral Equations Appl., 22 (2010), 483-517. [24] Y. Luchko, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12 (2009), 409-422. [25] Z. Ruan and Z. Wang, Identification of a time-dependent source term for a time fractional diffusion problem, Appl. Anal., 96 (2017), 1638-1655.  doi: 10.1080/00036811.2016.1232400. [26] W. Rundell and Z. Zhang, Fractional diffusion: Recovering the distributed fractional derivative from overposed data, Inverse Probl., 33 (2017), 035008, 27 pp. doi: 10.1088/1361-6420/aa573e. [27] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [28] C. Sun and J. Liu, An inverse source problem for distributed order time-fractional diffusion equation, Inverse Probl., 36 (2020), 055008.  doi: 10.1088/1361-6420/ab762c. [29] S. Tatar and S. Ulusoy, An inverse source problem for a one-dimensional space-time fractional diffusion equation, Appl. Anal., 94 (2015), 2233-2244.  doi: 10.1080/00036811.2014.979808. [30] U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Num. Funct. Anal. Optim., 19 (1998), 377-398.  doi: 10.1080/01630569808816834. [31] D. D. Trong, D. N. D. Hai and N. D. Minh, Optimal regularization for an unknown source of space-fractional diffusion equation, Appl. Math. Comput., 349 (2019), 184-206.  doi: 10.1016/j.amc.2018.12.030. [32] H. T. Nguyen, D. L. Le and V. T. Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Appl. Math. Model., 40 (2016), 8244-8264.  doi: 10.1016/j.apm.2016.04.009. [33] J. G. Wang, Y. B. Zhou and T. Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl. Numer. Math., 68 (2013), 39-57.  doi: 10.1016/j.apnum.2013.01.001. [34] H. Ye, F. Liu, V. Anh and I. Turner, Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains, IMA J. Appl. Math., 80 (2015), 825-838.  doi: 10.1093/imamat/hxu015. [35] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Probl., 27 (2011), 035010, 12 pp. doi: 10.1088/0266-5611/27/3/035010.
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