Abstract
Inverse problem to identify a space-dependent diffusivity coefficient in a generalized subdiffusion equation from final data
Highlights
Equations with time fractional derivatives are used to model subdiffusion processes in self-similar media [1,3,13]
The global uniqueness and local existence and stability of the solution to this problem is proved. Proof of these statements is based on the fixed-point principle and previously obtained results regarding an inverse source problem for a generalized subdiffusion equation
Final data are suitable for determination of space-dependent paramaters of the equations for two reasons: 1) unknown quantities and data are functions of the same type; 2) the resulting inverse problems are moderately ill-posed
Summary
Equations with time fractional derivatives (containing power-type kernels) are used to model subdiffusion processes in self-similar media [1,3,13]. Problems to reconstruct space-dependent factors of source terms of subdiffusion equations containing usual or generalized fractional derivatives from final data have been studied in several papers [8,9,11,14,18, 21]. The analysis of such problems uses the Fourier expansion or positivity principles with the Fredholm alternative. In the present paper we consider a problem to identify a space-dependent diffusivity coefficient in a one-dimensional subdiffusion equation containing a generalized time fractional derivative from the final data. A surprising result is that the inverse diffusivity problem is less ill-posed than the corresponding inverse source problem: the solution to the former one depends continuously on the 1st derivative of the final measurement uT whereas the solution to the latter one depends continuously on the 2nd derivative of uT
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