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Tikhonov Regularisation Method for Simultaneous Inversion of the Source Term and Initial Data in a Time-Fractional Diffusion Equation

Published online by Cambridge University Press:  07 September 2015

Zhousheng Ruan ,
Jerry Zhijian Yang  and
Xiliang Lu
Zhousheng Ruan
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China School of Science, East China Institute of Technology, Nanchang, Jiangxi, 330013, P.R. China
Jerry Zhijian Yang*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, P.R. China
Xiliang Lu
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, P.R. China
*
*Corresponding author. Email addresses: [email protected] (Z. Ruan), [email protected] (J. Z. Yang), [email protected] (X. Lu)
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Abstract

The inverse problem of identifying the time-independent source term and initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated into an operator equation based on the Fourier method. Under a certain smoothness assumption, conditional stability is established. A standard Tikhonov regularisation method is proposed to solve the inverse problem. Furthermore, the convergence rate is given for an a priori and a posteriori regularisation parameter choice rule, respectively. Several numerical examples, including one-dimensional and two-dimensional cases, show the efficiency of our proposed method.

Keywords

65N2149M15Time-fractional diffusion equationconditional stabilityTikhonov regularisationMorozov discrepancy principleconvergence rate
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 3 , August 2015 , pp. 273 - 300
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Kenichi, S. and Masahiro, Y., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382, 426–447 (2011).Google Scholar
[2]Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problems, Applied Mathematical Sciences, vol. 120, Springer-Verlag (2011).Google Scholar
[3]Fan, Q., Jiao, Y. and Lu, X., A primal dual active set algorithm with continuation for compressed sensing, IEEE Trans. Signal Processing 62, 6276–6285 (2014).Google Scholar
[4]Jiao, Y., Jin, B. and Lu, X., A primal dual active set with continuation algorithm for the l 0regularized optimization problem, to appear in Appl. Comp. Harmonic Anal., DOI: http://dx.doi.org/10.1016/j.acha.2014.10.001.Google Scholar
[5]Jin, B., Lazarov, R. and Zhou, Z., Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Num. Anal. 55, 445–466 (2013).Google Scholar
[6]Jin, B. and William, R., An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems 28, 075010 (2012).CrossRefGoogle Scholar
[7]Johansson, B. and Lesnic, D., A procedure for determining a spacewise dependent heat source and the initial temperature, Appl. Anal. 87, 265–276 (2008).CrossRefGoogle Scholar
[8]Li, G., Zhang, D., Jia, X. and Masahiro, Y., Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems 29, 065014 (2013).Google Scholar
[9]Li, X. and Xu, C., A space-time spectral method for the time-fractional diffusion equation, SIAM J. Num. Anal. 47, 2108–2131 (2009).CrossRefGoogle Scholar
[10]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comp. Phys, 225, 1533–1552 (2007).CrossRefGoogle Scholar
[11]Liu, J. and Yamamoto, M., A backward problem for the time-fractional diffusion equation, Appl. Anal. 89, 1769–1788 (2010).Google Scholar
[12]Luc, M. and Masahiro, Y., Coefficient inverse problem for a fractional diffusion equation, Inverse Problems 29, 075013 (2013).Google Scholar
[13]Metzler, R. and Klafter, J., Boundary value problems for fractional diffusion equations, Phys. A 278, 107–125 (2000).Google Scholar
[14]Podlubny, L., Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Mathematics in Science and Engineering 198, Academic Press, San Diego (1999).Google Scholar
[15]Roman, E. and Alemany, A., Continuous-time random walks and the fractional diffusion equation, J. Phys. A: Math. Gen 27, 3407–3410 (1994).Google Scholar
[16]Groetsch, C.W., The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman-Boston (1984).Google Scholar
[17]Ren, C., Xu, X. and Lu, S., Regularization by projection for a backward problem of the time-fractional diffusion equation, J. Inverse Ill-Posed Problems 22, 121–139 (2014).Google Scholar
[18]Ruan, Z., Yang, Z. and Lu, X., An inverse source problem with sparsity constraint for the time-fractional diffusion equation, to appear in Adv. Appl. Math. Mech., (2015).Google Scholar
[19]Sun, Z., Jiao, Y., Jin, B. and Lu, X., Numerical Identification of a sparse Robin coefficient, Adv. Comp. Math. 41, 131–148 (2015).CrossRefGoogle Scholar
[20]Wang, J., Wei, T. and Zhou, Y., Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Appl. Math. Mod. 37, 8518–8532 (2013).Google Scholar
[21]Wang, J., Zhou, Y. and Wei, T., Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl. Num. Math. 68, 39–57 (2013).Google Scholar
[22]Wang, L. and Liu, J., Data regularization for a backward time-fractional diffusion problem, Comp. Math. Appl. 64, 3613–3626 (2012).CrossRefGoogle Scholar
[23]Wang, Z. and Liu, J., New model function methods for determining regularization parameters in linear inverse problems, Appl. Num, Math. 59, 2489–2506 (2009).Google Scholar
[24]Wang, Z., Qiu, S. and Ruan, Z., A regularized optimization method for identifying the space-dependent source and the initial value simultaneously in a parabolic equation, Comp. Math. Appl. 67, 1345–1357 (2014).Google Scholar
[25]Wei, T. and Wang, J., Simultaneous determination for a space-dependent heat source and the initial data by the MFS, Eng. Anal. Boundary Elements 36, 1848–1855 (2012).Google Scholar
[26]Wei, T. and Zhang, Z., Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Eng. Anal. Boundary Elements 37, 23–31 (2013).Google Scholar
[27]William, R., Xu, X. and Zuo, L.H., The determination of an unknown boundary condition in a fractional diffusion equation, Appl. Analysis 92, 1511–1526 (2013).Google Scholar
[28]Xiong, X., Wang, J. and Li, M., An optimal method for fractional heat conduction problem backward in time, Appl. Analysis 91, 823–840 (2012).Google Scholar
[29]Zhang, Y. and Xu, X., Inverse source problem for a fractional diffusion equation, Inverse Problems 27, 035010 (2011).Google Scholar
[30]Zheng, G. and Wei, T., Recovering the source and initial value simultaneously in a parabolic equation, Inverse Problems 30, 065013 (2014).CrossRefGoogle Scholar