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On Tikhonov’s method and optimal error bound for inverse source problem for a time-fractional diffusion equation. (English) Zbl 1445.35328
Summary: We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale \(\mathbb{H}^r(\mathbb{R}^n)\) and establish global order optimal lower bound for the worst case error. Next, we use the Tikhonov regularization method to deal with this problem in the Hilbert scale \(\mathbb{H}^r(\mathbb{R}^n)\). Locally optimal choices of parameters for the family of regularization operator in the Hilbert scales \(\mathbb{H}^r(\mathbb{R}^n)\) are analyzed by a-priori and a-posteriori methods. Numerical implementations are presented to illustrate our theoretical findings.
MSC:
35R30 Inverse problems for PDEs
35R25 Ill-posed problems for PDEs
35R11 Fractional partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
35K15 Initial value problems for second-order parabolic equations
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