Generalized Tikhonov methods for an inverse source problem of the time-fractional diffusion equation. (March 2018)

• Record Type: Journal Article
• Authors: Ma, Yong-Ki; Prakash, P.; Deiveegan, A.
• Is Part Of: Chaos, solitons and fractals. Volume 108(2018)
• Page Start: 39
• Page End: 48
• Publication Date: 2018-03
• Subjects: Inverse problem -- Fractional diffusion equation -- Tikhonov regularization -- Convergence analysis; 35R30 -- 35R11 -- 65M32; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2018.01.003
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: This paper investigates the problem of determining a space-dependent source term in a time-fractional diffusion equation with variable coefficients in a bounded domain where additional data are consider at a fixed time. Using the generalized and revised generalized Tikhonov regularization methods, we construct regularized solutions. One important feature of our paper is that the Convergence estimates for both methods under a-priori and a-posteriori regularization parameter choice rules are given, respectively. In this work we have extended the revised generalized Tikhonov regularization method for the inverse problem of determining the source term in time-fractional diffusion equation. The obtained result shows a new contribution in the field of fractional diffusion equation. However it should be emphasized that the revised generalized Tikhonov regularization method is mainly concerned with inverse source problems for the heat equation and there have been no attempts made for studying the time-fractional diffusion problem. Abstract: In this paper, we identify the unknown space-dependent source term in a time-fractional diffusion equation with variable coefficients in a bounded domain where additional data are consider at a fixed time. Using the generalized and revised generalized Tikhonov regularization methods, we construct regularized solutions. Convergence estimates for both methods under an a-priori and a-posteriori regularization parameter choice rules are given, respectively. Numerical example shows that the proposed methods are effective and stable.