Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 3, 2020

Regularization of backward time-fractional parabolic equations by Sobolev-type equations

  • Dinh Nho Hào EMAIL logo , Nguyen Van Duc , Nguyen Van Thang and Nguyen Trung Thành ORCID logo

Abstract

The problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

MSC 2010: 35R30; 35R11

Dedicated to Professor Anatoly Yagola in the occasion of his 75th birthday


Award Identifier / Grant number: 101.01-2017.319

Funding statement: The work of N. V. Thang was partly supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.319.

References

[1] M. F. Al-Jamal, A backward problem for the time-fractional diffusion equation, Math. Methods Appl. Sci. 40 (2017), 2466–2474. 10.1002/mma.4151Search in Google Scholar

[2] A. Chadha, D. Bahuguna and D. N. Pandeym, Faedo–Galerkin approximate solutions for nonlocal fractional differential equation of Sobolev type, Fract. Differ. Calc. 8 (2018), 205–222. 10.7153/fdc-2018-08-13Search in Google Scholar

[3] R. E. Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math.Anal. 6 (1975), 283–294. 10.1137/0506029Search in Google Scholar

[4] M. A. Fury, Nonautonomous ill-posed evolution problems with strongly elliptic differential operators, Electron. J. Differential Equations 92 (2013), 1–25. Search in Google Scholar

[5] H. Gajewski and K. Zacharias, Zur Ruguliarisierung einer nichtkorrekter Probleme bei Evolutionsgleichungen, J. Math. Anal. Appl. 38 (1972), 784–789. 10.1016/0022-247X(72)90083-2Search in Google Scholar

[6] D. N. Hào, J. Liu, N. V. Duc and N. V. Thang, Stability results for backward time-fractional parabolic equations, Inverse Problems 35 (2019), Article ID 125006. 10.1088/1361-6420/ab45d3Search in Google Scholar

[7] Y. Huang and Z. Quan, Regularization for a class of illposed Cauchy problem, Proc. Amer. Math. Soc. 133 (2005), 3005–3012. 10.1090/S0002-9939-05-07822-6Search in Google Scholar

[8] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems 31 (2015), Article ID 035003. 10.1088/0266-5611/31/3/035003Search in Google Scholar

[9] N. T. Long and A. P. N. Dinh, Approximation of a parabolic non-linear evolution equation backward in time, Inverse Problems 10 (1994), 905–914. 10.1088/0266-5611/10/4/010Search in Google Scholar

[10] N. T. Long and A. P. N. Dinh, Note on a regularization of a parabolic nonlinear evolution equation backwards in time, Inverse Problems 4 (1996), 455–462. 10.1088/0266-5611/12/4/008Search in Google Scholar

[11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Search in Google Scholar

[12] J. J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal. 89 (2010), 1769–1788. 10.1080/00036810903479731Search in Google Scholar

[13] V. Padrón, Sobolev regularization of some nonlinear illposed problems, PhD thesis, University of Minnensota, Minneapolis, 1990. Search in Google Scholar

[14] V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Differential Equations 23 (1998), 457–486. 10.1080/03605309808821353Search in Google Scholar

[15] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, New York, 1999. Search in Google Scholar

[16] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd ed., Springer, New York, 2004. Search in Google Scholar

[17] 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. 10.1016/j.jmaa.2011.04.058Search in Google Scholar

[18] R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl. 47 (1974), 563–572. 10.1016/0022-247X(74)90008-0Search in Google Scholar

[19] J. G. Wang, T. Wei and Y. B. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Appl. Math. Model. 37 (2013), 8518–8532. 10.1016/j.apm.2013.03.071Search in Google Scholar

[20] J. G. Wang, T. Wei and Y. B. Zhou, Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, J. Comput. Appl. Math. 279 (2015), 277–292. 10.1016/j.cam.2014.11.026Search in Google Scholar

[21] J. G. Wang, Y. B. Zhou and T. Wei, A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fraction diffusion problem, Appl. Math. Lett. 26 (2013), 741–747. 10.1016/j.aml.2013.02.006Search in Google Scholar

[22] L. Wang and J. Liu, Data regularization for a backward time-fractional diffusion problem, Comput. Math. Appl. 64 (2012), 3613–3626. 10.1016/j.camwa.2012.10.001Search in Google Scholar

[23] L. Wang and J. Liu, Total variation regularization for a backward time-fractional diffusion problem, Inverse Problems 29 (2013), Article ID 115013. 10.1088/0266-5611/29/11/115013Search in Google Scholar

[24] T. Wei and J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM Math. Model. Numer. Anal. 48 (2014), 603–621. 10.1051/m2an/2013107Search in Google Scholar

[25] M. Yang and J. Liu, Solving a final value fractional diffusion problem by boundary condition regularization, Appl. Numer. Math. 66 (2013), 45–58. 10.1016/j.apnum.2012.11.009Search in Google Scholar

[26] M. Yang and J. Liu, Fourier regularization for a final value time-fractional diffusion problem, Appl. Anal. 94 (2015), 1508–1526. 10.1080/00036811.2014.936402Search in Google Scholar

Received: 2020-05-27
Accepted: 2020-08-14
Published Online: 2020-10-03
Published in Print: 2020-11-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 5.3.2024 from https://www.degruyter.com/document/doi/10.1515/jiip-2020-0062/html
Scroll to top button