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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. ChinaSchool 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. ChinaComputational 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. ChinaComputational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, P.R. China
*
*Corresponding author. Email addresses: zhshruan@whu.edu.cn (Z. Ruan), zjyang.math@whu.edu.cn (J. Z. Yang), xllv.math@whu.edu.cn (X. Lu)

Abstract

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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

Information

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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