Wavelet regularization strategy for the fractional inverse diffusion problem

Abstract

This manuscript deals with an inverse fractional-diffusing problem, the time-fractional heat conduction equation, which is a physical model of a problem, where one needs to identify the temperature distribution of a semi-conductor, but one transient temperature data is unreachable to measurement. Mathematically, it is designed as a time-fractional diffusion problem in a semi-infinite region, with polluted data measured at x = 1, where the solution is wanted for 0 ≤ x < 1. In view of Hadamard, the problem extremely suffers from an intrinsic ill-posedness, i.e., the true solution of the problem is computationally impossible to measure since any measurement or numerical computation is polluted by inevitable errors. In order to capture the solution, a regularization scheme based on the Meyer wavelet is therefore applied to treat the underlying problem in the presence of polluted data. The regularized solution is restored by the Meyer wavelet projection on elements of the Meyer multiresolution analysis (MRA). Furthermore, the concepts of convergence rate and stability of the proposed scheme are investigated and some new order-optimal stable estimates of the so-called Holder-Logarithmic type are rigorously derived by carrying out both an a priori and a posteriori choice approaches in Sobolev scales. It turns out that both approaches yield the same convergence rate, except for some different constants. Finally, the computational performance of the proposed method effectively verifies the applicability and validity of our strategy. Meanwhile, the thrust of the present paper is compared with other sophisticated methods in the literature.