On Tikhonov's method and optimal error bound for inverse source problem for a time-fractional diffusion equation

Abstract 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 H r ( 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 H r ( R n ) . Locally optimal choices of parameters for the family of regularization operator in the Hilbert scales H r ( R n ) are analyzed by a-priori and a-posteriori methods. Numerical implementations are presented to illustrate our theoretical findings.

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