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Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation. (English) Zbl 1467.65090

The authors analyse an algorithm for recovering the distribution \(u(r,0)\) for an inhomogeneous time-fractional diffusion problem in a spherical symmetric domain \((0,R) \times (0,T)\), where the right-hand side of the differential equation and the function \(g\) in the condition \(u(r,T) = g(r)\), \(r \in [0,R]\) are noisy data. Hereby, the deterministic case and the random noise case are considered. An iterated version of the fractional Tikhonov regularization method is proposed. At first the deterministic situation is studied. Convergence estimates are proved in the case of an a priori choice of the regularization parameter and in the case of an a posteriori choice rule for the regularization parameter, where a modified discrepancy principle is used. In the random noise case a convergence estimate is given when an a priori choice rule is used for the regularization parameter. Finally, a numerical example is given to show the effectiveness of the proposed method.

MSC:

65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
60H40 White noise theory
35R11 Fractional partial differential equations
35R30 Inverse problems for PDEs
35R25 Ill-posed problems for PDEs

Software:

Mittag-Leffler; mlf
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Full Text: DOI

References:

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