Wavelet regularization strategy for the fractional inverse diffusion problem
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Wavelet regularization strategy for the fractional inverse diffusion problem Milad Karimi1 · Fatemeh Zallani2 · Khosro Sayevand2 Received: 20 June 2019 / Accepted: 30 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This manuscript deals with an inverse fractional-diffusing problem, the timefractional 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 H¨older-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. Keywords Fractional diffusion equation · Ill-posed problem · Meyer wavelet · Multiresolution analysis Mathematics Subject Classification (2010) 35K05 · 35K99 · 65F22 · 42C40 Milad Karimi
m [email protected]; [email protected] 1
Department of Mathematics, Sahand University of Technology, P.O. Box: 51335-1996, Tabriz, Iran
2
Department of Mathematics, Malayer University, P.O. Box: 16846-13114, Malayer, Iran
Numerical Algorithms
1 Introduction In the past few decades, fractional calculus and derivative have been major impact on science and technology for instance, heat conduction [6, 11, 18], electron transportation [17], dissipation [19], high frequency financial data [12], semiconductors [1, 16] and viscoelasticity [22]. To describe modeling of some real-world applications such as Brownian motion and sea pollution, many experiments have evinced that the fractional calculus is more accurate than the classical integer order calculus. Also, flexibility of the fractional calculus, especially fractional derivatives, in description of viscoelastic behaviors is more than the classical calculus. In particular, a trenchant analys
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