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In this paper, we present an inverse problem of identifying the reaction coefficient for time fractional diffusion equations in two dimensional spaces by using boundary Neumann data. It is proved that the forward operator is continuous with respect to the unknown parameter. Because the inverse problem is often ill-posed, regularization strategies are imposed on the least fit-to-data functional to overcome the stability issue. There may exist various kinds of functions to reconstruct. It is crucial to choose a suitable regularization method. We present a multi-parameter regularization $L^{2}+BV$ method for the inverse problem. This can extend the applicability for reconstructing the unknown functions. Rigorous analysis is carried out for the inverse problem. In particular, we analyze the existence and stability of regularized variational problem and the convergence. To reduce the dimension in the inversion for numerical simulation, the unknown coefficient is represented by a suitable set of basis functions based on a priori information. A few numerical examples are presented for the inverse problem in time fractional diffusion equations to confirm the theoretic analysis and the efficacy of the different regularization methods.
In this paper, the time fractional reaction-diffusion equations with the Caputo fractional derivative are solved by using the classical $L1$-formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the
In this paper, a second-order backward difference formula (abbr. BDF2) is used to approximate first-order time partial derivative, the Riesz fractional derivatives are approximated by fourth-order compact operators, a class of new alternating-directi
We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these
We present and analyze a novel wavelet-Fourier technique for the numerical treatment of multidimensional advection-diffusion-reaction equations based on the CORSING (COmpRessed SolvING) paradigm. Combining the Petrov-Galerkin technique with the compr
In this paper, we develop a robust fast method for mobile-immobile variable-order (VO) time-fractional diffusion equations (tFDEs), superiorly handling the cases of small or vanishing lower bound of the VO function. The valid fast approximation of th