THE REVISED GENERALIZED TIKHONOV METHOD FOR THE BACKWARD TIME-FRACTIONAL DIFFUSION EQUATION

Arumugam Deiveegan; Juan J. Nieto; Periasamy Prakash

doi:10.11948/2019.45

2019 Volume 9 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2019 > 9(1): 45-56

Next Article Previous Article

Arumugam Deiveegan, Juan J. Nieto, Periasamy Prakash. THE REVISED GENERALIZED TIKHONOV METHOD FOR THE BACKWARD TIME-FRACTIONAL DIFFUSION EQUATION[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 45-56. doi: 10.11948/2019.45

Citation:

Arumugam Deiveegan, Juan J. Nieto, Periasamy Prakash. THE REVISED GENERALIZED TIKHONOV METHOD FOR THE BACKWARD TIME-FRACTIONAL DIFFUSION EQUATION[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 45-56. doi: 10.11948/2019.45

THE REVISED GENERALIZED TIKHONOV METHOD FOR THE BACKWARD TIME-FRACTIONAL DIFFUSION EQUATION

1.
Department of Mathematics, Periyar University, Salem - 636 011, India
2.
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela 15782, Spain

Corresponding author: Email address: pprakashmaths@gmail.com(P. Prakash)
Fund Project: This work was supported by University Grants Commission, New Delhi, India, under Major Research Project (41-798/2012(SR)). The second author was partially supported by the Ministerio de Economia y Competitividad of Spain under grant (MTM2013-43014-P) and Xunta de Galicia under grant (GRC 2015/004). The third author was supported by the Department of Science and Technology, New Delhi, India under FIST Programme (SR/FST/MSI-115/2016)

Abstract

Abstract

In this paper, we solve the backward problem for a time-fractional diffusion equation with variable coefficients in a bounded domain by using the revised generalized Tikhonov regularization method. Convergence estimates under an a-priori and a-posteriori regularization parameter choice rules are given. Numerical example shows that the proposed method is effective and stable.
- Inverse problem /
- fractional diffusion equation /
- Tikhonov regularization /
- convergence analysis
MSC: 35R30, 35R11, 65M32

FullText(HTML)

References

[1]	J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 2009, 25, 115 002. doi: 10.1088/0266-5611/25/11/115002 CrossRef Google Scholar
[2]	G. H. Gao and Z. Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 2011, 230, 586-595. doi: 10.1016/j.jcp.2010.10.007 CrossRef Google Scholar
[3]	A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, New York, 2006. Google Scholar
[4]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problem, Springer, New York, 1999. Google Scholar
[5]	J. J. Liu and M. M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal., 2010, 89, 1769-1788. doi: 10.1080/00036810903479731 CrossRef Google Scholar
[6]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. Google Scholar
[7]	A. Qian and Y. Li, Optimal error bound and generalized Tikhonov regularization for identifying an unknown source in the heat equation, J. Math. Chem., 2011, 49, 765-775. doi: 10.1007/s10910-010-9774-3 CrossRef Google Scholar
[8]	C. X. Ren, X. Xu and S. Lu, Regularization by projection for a backward problem of the time-fractional diffusion equation, J. Inverse Ill-posed Probl., 2014, 22, 121-139. Google Scholar
[9]	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., 2011, 382, 426-447. doi: 10.1016/j.jmaa.2011.04.058 CrossRef Google Scholar
[10]	G. Sudha Priya, P. Prakash, J. J. Nieto and Z. Kayar, Higher order numerical scheme for fractional heat equation with Dirichlet and Neumann boundary conditions, Numer. Heat Tr. B-Fund, 2013, 63, 540-559. doi: 10.1080/10407790.2013.778719 CrossRef Google Scholar
[11]	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., 2013, 37, 8518-8532. doi: 10.1016/j.apm.2013.03.071 CrossRef Google Scholar
[12]	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-fractional diffusion equation, Appl. Math. Lett., 2013, 26, 741-747. doi: 10.1016/j.aml.2013.02.006 CrossRef Google Scholar
[13]	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., 2013, 68, 39-57. doi: 10.1016/j.apnum.2013.01.001 CrossRef Google Scholar
[14]	T. Wei and J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM Math. Model. Numer. Anal., 2014, 48, 603-621. doi: 10.1051/m2an/2013107 CrossRef Google Scholar
[15]	F. Yang and C. L. Fu, The revised generalized Tikhonov regularization for the inverse time-dependent heat source problem, J. Appl. Math. Comput., 2013, 41, 81-98. doi: 10.1007/s12190-012-0596-2 CrossRef Google Scholar
[16]	H. Zhang and X. Zhang, Generalized Tikhonov method for the final value problem of time-fractional diffusion equation, Int. J. Comput. Math., 2017, 94, 66-78. doi: 10.1080/00207160.2015.1089354 CrossRef Google Scholar