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Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation. (English) Zbl 1266.65161
Authors’ abstract: The inverse problem of identifying a space-dependent source for the time-fractional diffusion equation is investigated. Such a problem is obtained from the classical diffusion equation in which the time derivative is replaced with a Caputo derivative of order $\alpha \in \left(0,1\right]$. We show that such a problem is ill-posed and apply the Tikhonov regularization method and a simplified Tikhonov regularization method to solve it based on the solution given by the separation of variables. Convergence estimates are presented under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. Finally, numerical examples are given to show that the regularization methods are effective and stable.
##### MSC:
 65M32 Inverse problems (IVP of PDE, numerical methods) 35R11 Fractional partial differential equations 35R30 Inverse problems for PDE 35K05 Heat equation 65M12 Stability and convergence of numerical methods (IVP of PDE) 35R25 Improperly posed problems for PDE 65M30 Improperly posed problems (IVP of PDE, numerical methods)