Fan, Bin; Azaïez, Mejdi; Xu, Chuanju An extension of the Landweber regularization for a backward time fractional wave problem. (English) Zbl 1476.65218 Discrete Contin. Dyn. Syst., Ser. S 14, No. 8, 2893-2916 (2021). Summary: In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity. MSC: 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs 35R11 Fractional partial differential equations 47A52 Linear operators and ill-posed problems, regularization 35L05 Wave equation Keywords:backward problem; time-fractional wave equation; fractional filter regularization; convergence Software:Mittag-Leffler; mlf PDFBibTeX XMLCite \textit{B. Fan} et al., Discrete Contin. Dyn. Syst., Ser. S 14, No. 8, 2893--2916 (2021; Zbl 1476.65218) Full Text: DOI References: [1] O. P. Agrawal, Formulation of euler-lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272, 368-379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4 [2] O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29, 145-155 (2002) · Zbl 1009.65085 · doi:10.1023/A:1016539022492 [3] D. A. Benson; S. W. Wheatcraft; M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36, 1403-1412 (2000) · doi:10.1029/2000WR900031 [4] D. Bianchi, A. Buccini, M. Donatelli and S. Serra-Capizzano, Iterated fractional tikhonov regularization, Inverse Problems, 31 (2015), 055005, 34pp. · Zbl 1434.65081 [5] H. Cheng; C. L. Fu, An iteration regularization for a time-fractional inverse diffusion problem, Applied Mathematical Modelling, 36, 5642-5649 (2012) · Zbl 1254.65100 · doi:10.1016/j.apm.2012.01.016 [6] E. Cuesta; M. Kirane; S. A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Processing, 92, 553-563 (2012) · doi:10.1016/j.sigpro.2011.09.001 [7] Y. Deng and Z. Liu, Iteration methods on sideways parabolic equations, Inverse Problems, 25 (2009), 095004, 14pp. · Zbl 1173.35724 [8] Y. Deng; Z. Liu, New fast iteration for determining surface temperature and heat flux of general sideways parabolic equation, Nonlinear Analysis: Real World Applications, 12, 156-166 (2011) · Zbl 1205.35337 · doi:10.1016/j.nonrwa.2010.06.005 [9] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Science & Business Media, 2010. · Zbl 1215.34001 [10] R. Du; W. R. Cao; Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Applied Mathematical Modelling, 34, 2998-3007 (2010) · Zbl 1201.65154 · doi:10.1016/j.apm.2010.01.008 [11] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publisher, Dordrecht, Boston, London, 1996. · Zbl 0859.65054 [12] G. H. Gao; Z. Z. Sun, The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain, Journal of Computational Physics, 236, 443-460 (2013) · Zbl 1286.35251 · doi:10.1016/j.jcp.2012.11.011 [13] D. Gerth; E. Klann; R. Ramlau; L. Reichel, On fractional tikhonov regularization, Journal of Inverse and Ill-posed Problems, 23, 611-625 (2015) · Zbl 1327.65075 · doi:10.1515/jiip-2014-0050 [14] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations, 104p, Boston Pitman Publication, 1984. [15] Y. Han; X. Xiong; X. Xue, A fractional landweber method for solving backward time-fractional diffusion problem, Computers & Mathematics with Applications, 78, 81-91 (2019) · Zbl 1442.65224 · doi:10.1016/j.camwa.2019.02.017 [16] M. E. Hochstenbach; L. Reichel, Fractional tikhonov regularization for linear discrete ill-posed problems, BIT Numerical Mathematics, 51, 197-215 (2011) · Zbl 1215.65075 · doi:10.1007/s10543-011-0313-9 [17] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Science & Business Media, 2011. · Zbl 1213.35004 [18] E. Klann; P. Maass; R. Ramlau, Two-step regularization methods for linear inverse problems, Journal of Inverse and Ill-posed Problems, 14, 583-607 (2006) · Zbl 1107.65045 · doi:10.1515/156939406778474523 [19] X. J. Li; C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47, 2108-2131 (2009) · Zbl 1193.35243 · doi:10.1137/080718942 [20] Y. M. Lin; C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of computational physics, 225, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 [21] J. J. Liu; M. Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89, 1769-1788 (2010) · Zbl 1204.35177 · doi:10.1080/00036810903479731 [22] R. L. Magin, Fractional Calculus in Bioengineering, volume 2(6)., Begell House Redding, 2006. [23] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. · Zbl 1210.26004 [24] R. Metzler; J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 [25] I. Podlubny and M. Kacenak, Mittag-leffler Function, the matlab routine, 2006. [26] M. Richter, Inverse Problems: Basics, Theory and Applications in Geophysics, Birkhäuser, 2016. · Zbl 1365.65157 [27] K. Sakamoto; 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, 426-447 (2011) · Zbl 1219.35367 · doi:10.1016/j.jmaa.2011.04.058 [28] F. Y. Song; C. J. Xu, Spectral direction splitting methods for two-dimensional space fractional diffusion equations, Journal of Computational Physics, 299, 196-214 (2015) · Zbl 1352.65400 · doi:10.1016/j.jcp.2015.07.011 [29] Z. Z. Sun; X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56, 193-209 (2006) · Zbl 1094.65083 · doi:10.1016/j.apnum.2005.03.003 [30] J. G. Wang; T. Wei, An iterative method for backward time-fractional diffusion problem, Numerical Methods for Partial Differential Equations, 30, 2029-2041 (2014) · Zbl 1314.65120 · doi:10.1002/num.21887 [31] J. G. Wang; T. Wei; Y. B. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Applied Mathematical Modelling, 37, 8518-8532 (2013) · Zbl 1427.65229 · doi:10.1016/j.apm.2013.03.071 [32] L. Wang; J. J. Liu, Data regularization for a backward time-fractional diffusion problem, Computers & Mathematics with Applications, 64, 3613-3626 (2012) · Zbl 1268.65128 · doi:10.1016/j.camwa.2012.10.001 [33] T. Wei; J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM: Mathematical Modelling and Numerical Analysis, 48, 603-621 (2014) · Zbl 1295.35378 · doi:10.1051/m2an/2013107 [34] T. Wei; Y. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Computers & Mathematics with Applications, 75, 3632-3648 (2018) · Zbl 1417.35224 · doi:10.1016/j.camwa.2018.02.022 [35] X. Xiong; X. Xue; Z. Qian, A modified iterative regularization method for ill-posed problems, Applied Numerical Mathematics, 122, 108-128 (2017) · Zbl 1375.65077 · doi:10.1016/j.apnum.2017.08.004 [36] F. Yang; Y. Zhang; X. X. Li, Landweber iterative method for identifying the initial value problem of the time-space fractional diffusion-wave equation, Numerical Algorithms, 83, 1509-1530 (2020) · Zbl 07191904 · doi:10.1007/s11075-019-00734-6 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.