A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equation. (English) Zbl 07489711
Summary: This paper is devoted to identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. By the series expression of the solution of the direct problem, the inverse source problem can be formulated by a first kind of Fredholm integral equation. The existence and uniqueness, ill-posedness and a conditional stability in Hilbert scale for the considered inverse problem are provided. We propose a generalized quasi-boundary value regularization method to solve the inverse source problem and also prove that the regularized problem is well-posed. Further, two kinds of convergence rates in Hilbert scale for the regularized solution can be obtained by using an a priori and an a posteriori regularization parameter choice rule, respectively. The numerical examples in one-dimensional case and two-dimensional case are given to confirm our theoretical results for the constant coefficients problem. We also propose a finite difference method based on a variant of L1 scheme to solve the regularized problem for the variable coefficients problem and give its convergence rate. One finite difference method based on a convolution quadrature is provided to solve the regularized problem for comparison. The numerical results for three examples by two algorithms are provided to show the effectiveness and stability of the proposed algorithms.
Keywords:
time-fractional diffusion-wave equation; inverse source problem; generalized quasi-boundary value regularization method; finite difference algorithms; convergence rateReferences:
| [1] | Berkowitz, B.; Scher, H.; Silliman, S. E., Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resour. Res., 36, 149-158 (2000) · doi:10.1029/1999wr900295 |
| [2] | Chen, A.; Li, C., Numerical solution of fractional diffusion-wave equation, Numer. Funct. Anal. Optim., 37, 19-39 (2016) · Zbl 1382.65236 · doi:10.1080/01630563.2015.1078815 |
| [3] | Cheng, J.; Nakagawa, J.; Yamamoto, M.; Yamazaki, T., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009) · Zbl 1181.35322 · doi:10.1088/0266-5611/25/11/115002 |
| [4] | Denche, M.; Bessila, K., A modified quasi-boundary value method for ill-posed problems, J. Math. Anal. Appl., 301, 419-426 (2005) · Zbl 1084.34536 · doi:10.1016/j.jmaa.2004.08.001 |
| [5] | Du, R.; Cao, W. R.; Sun, Z. Z., A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Modelling, 34, 2998-3007 (2010) · Zbl 1201.65154 · doi:10.1016/j.apm.2010.01.008 |
| [6] | Hào, D. N.; Liu, J.; Van Duc, N.; Van Thang, N., Stability results for backward time-fractional parabolic equations, Inverse Problems, 35 (2019) · Zbl 1425.35237 · doi:10.1088/1361-6420/ab45d3 |
| [7] | Hào, D. N.; Van Duc, N.; Lesnic, D., Regularization of parabolic equations backward in time by a non-local boundary value problem method, IMA J. Appl. Math., 75, 291-315 (2010) · Zbl 1194.35501 · doi:10.1093/imamat/hxp026 |
| [8] | Hào, D. N.; Van Duc, N.; Sahli, H., A non-local boundary value problem method for parabolic equations backward in time, J. Math. Anal. Appl., 345, 805-815 (2008) · Zbl 1160.35070 · doi:10.1016/j.jmaa.2008.04.064 |
| [9] | Huang, J.; Tang, Y.; Vázquez, L.; Yang, J., Two finite difference schemes for time fractional diffusion-wave equation, Numer. Algorithms, 64, 707-720 (2013) · Zbl 1284.65103 · doi:10.1007/s11075-012-9689-0 |
| [10] | Jiang, Y.; Ma, J., High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235, 3285-3290 (2011) · Zbl 1216.65130 · doi:10.1016/j.cam.2011.01.011 |
| [11] | Jin, B.; Lazarov, R.; Zhou, Z., Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput., 38, A146-A170 (2016) · Zbl 1381.65082 · doi:10.1137/140979563 |
| [12] | Jin, B. T.; Rundell, W., An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28 (2012) · Zbl 1247.35203 · doi:10.1088/0266-5611/28/7/075010 |
| [13] | Kian, Y.; Yamamoto, M., On existence and uniqueness of solutions for semilinear fractional wave equations, Fractional Calculus Appl. Anal., 20, 117-138 (2015) · Zbl 1362.35323 · doi:10.1515/fca-2017-0006 |
| [14] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, vol 204, 2453-2461 (2006), Amsterdam: Elsevier, Amsterdam · Zbl 1092.45003 |
| [15] | Liao, K.; Wei, T., Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously, Inverse Problems, 35 (2019) · Zbl 1442.65230 · doi:10.1088/1361-6420/ab383f |
| [16] | Liu, J. J.; Yamamoto, M., A backward problem for the time-fractional diffusion equation, Appl. Anal., 89, 1769-1788 (2010) · Zbl 1204.35177 · doi:10.1080/00036810903479731 |
| [17] | Liu, Y. K.; Hu, G. H.; Yamamoto, M., Inverse moving source problem for time-fractional evolution equations: determination of profiles, Inverse Problems, 37 (2021) · Zbl 1469.35257 · doi:10.1088/1361-6420/ac0c20 |
| [18] | Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal., 17, 704-719 (1986) · Zbl 0624.65015 · doi:10.1137/0517050 |
| [19] | Luchko, Y., Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59, 1766-1772 (2010) · Zbl 1189.35360 · doi:10.1016/j.camwa.2009.08.015 |
| [20] | Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/s0370-1573(00)00070-3 |
| [21] | Metzler, R.; Klafter, J., Subdiffusive transport close to thermal equilibrium: from the Langevin equation to fractional diffusion, Phys. Rev. E, 61, 6308-6311 (2000) · doi:10.1103/physreve.61.6308 |
| [22] | Murio, D. A., Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56, 1138-1145 (2008) · Zbl 1155.65372 · doi:10.1016/j.camwa.2008.02.015 |
| [23] | Podlubny, I., Fractional Differential Equations (1999), New York: Academic, New York · Zbl 0918.34010 |
| [24] | Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A, 314, 749-755 (2002) · Zbl 1001.91033 · doi:10.1016/s0378-4371(02)01048-8 |
| [25] | Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 426-447 (2011) · Zbl 1219.35367 · doi:10.1016/j.jmaa.2011.04.058 |
| [26] | Showalter, R. E., Trends in The Theory and Practice of Non-Linear Analysis, Cauchy problem for hyper-parabolic partial differential equations, 421-425 (1985), Amsterdam: North-Holland, Amsterdam · Zbl 0587.35044 · doi:10.1016/S0304-0208(08)72739-7 |
| [27] | Šišková, K.; Slodička, M., Recognition of a time-dependent source in a time-fractional wave equation, Appl. Numer. Math., 121, 1-17 (2017) · Zbl 1372.65264 · doi:10.1016/j.apnum.2017.06.005 |
| [28] | Sokolov, I. M.; Klafter, J., From diffusion to anomalous diffusion: a century after Einsteins Brownian motion, Chaos, 15 (2005) · Zbl 1080.82022 · doi:10.1063/1.1860472 |
| [29] | Sun, Z-z; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 193-209 (2006) · Zbl 1094.65083 · doi:10.1016/j.apnum.2005.03.003 |
| [30] | Tang, T., A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11, 309-319 (1993) · Zbl 0768.65093 · doi:10.1016/0168-9274(93)90012-g |
| [31] | Tatar, S.; Ulusoy, S., An inverse source problem for a one-dimensional space-time fractional diffusion equation, Appl. Anal., 94, 2233-2244 (2015) · Zbl 1327.35416 · doi:10.1080/00036811.2014.979808 |
| [32] | Wang, J-G; Zhou, Y-B; Wei, T., A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem, Appl. Math. Lett., 26, 741-747 (2013) · Zbl 1311.65123 · doi:10.1016/j.aml.2013.02.006 |
| [33] | Wang, W.; Yamamoto, M.; Han, B., Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation, Inverse Problems, 29 (2013) · Zbl 1296.65126 · doi:10.1088/0266-5611/29/9/095009 |
| [34] | Wei, T.; Li, X. L.; Li, Y. S., An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Problems, 32 (2016) · Zbl 1351.65072 · doi:10.1088/0266-5611/32/8/085003 |
| [35] | Wei, T.; Wang, J., A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math., 78, 95-111 (2014) · Zbl 1282.65141 · doi:10.1016/j.apnum.2013.12.002 |
| [36] | Wei, T.; Wang, J-G, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM: Math. Modelling Numer. Anal., 48, 603-621 (2014) · Zbl 1295.35378 · doi:10.1051/m2an/2013107 |
| [37] | Wei, T.; Zhang, Y., The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Comput. Math. Appl. B, 75, 3632-3648 (2018) · Zbl 1417.35224 · doi:10.1016/j.camwa.2018.02.022 |
| [38] | Xian, J.; Wei, T., Determination of the initial data in a time-fractional diffusion-wave problem by a final time data, Comput. Math. Appl., 78, 2525-2540 (2019) · Zbl 1443.35179 · doi:10.1016/j.camwa.2019.03.056 |
| [39] | Yamamoto, M., Uniqueness in determining fractional orders of derivatives and initial values, Inverse Problems, 37 (2021) · Zbl 1478.35227 · doi:10.1088/1361-6420/abf9e9 |
| [40] | Yan, X. B.; Wei, T., Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach, J. Inverse Ill-Posed Probl., 27, 1-16 (2019) · Zbl 1503.65279 · doi:10.1515/jiip-2017-0091 |
| [41] | Yan, X. B.; Wei, T., Determine a space-dependent source term in a time fractional diffusion-wave equation, Acta Appl. Math., 165, 163-181 (2020) · Zbl 1432.35232 · doi:10.1007/s10440-019-00248-2 |
| [42] | Yan, X. B.; Zhang, Y. X.; Wei, T., Identify the fractional order and diffusion coefficient in a fractional diffusion wave equation, J. Comput. Appl. Math., 393 (2021) · Zbl 1468.65130 · doi:10.1016/j.cam.2021.113497 |
| [43] | Yang, M.; Liu, J., Solving a final value fractional diffusion problem by boundary condition regularization, Appl. Numer. Math., 66, 45-58 (2013) · Zbl 1269.65093 · doi:10.1016/j.apnum.2012.11.009 |
| [44] | Zhang, Y.; Xu, X., Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2011) · Zbl 1211.35280 · doi:10.1088/0266-5611/27/3/035010 |
| [45] | Zhang, Y-n; Sun, Z-z; Zhao, X., Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal., 50, 1535-1555 (2012) · Zbl 1251.65126 · doi:10.1137/110840959 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
