On regularization and error estimates for the backward heat conduction problem with time-dependent thermal diffusivity factor

Abstract

In this work, an accurate regularization technique based on the Meyer wavelet method is developed to solve the ill-posed backward heat conduction problem with time-dependent thermal diffusivity factor in an infinite "strip". In principle, the extremely ill-posedness of the considered problem is caused by the amplified infinitely growth in the frequency components which lead to a blow-up in the representation of the solution. Using the Meyer wavelet technique, some new stable estimates are proposed in the Hölder and Logarithmic types which are optimal in the sense of given by Tautenhahn. The stability and convergence rate of the proposed regularization technique are proved. The good performance and the high-accuracy of this technique is demonstrated through various one and two dimensional examples. Numerical simulations and some comparative results are presented.

Regularization of ill-posed problems involving constant-coefficient pseudo-differential operators

“…where k(t) ∈ C([0, T]) and g T (•) ∈ L 2 (R N ) respectively denote the positive thermal conductivity and the terminal distribution [21]. By the technique of Fourier transform we obtain in the frequency space the following operator equation B t û(ξ, t) = û(ξ, T) or equivalently the pseudo-differential operator equation…”

“…It turns out that the process of computing the ΨDOs leads to an ill-posed problem and thereby provides a general framework for studying a much wider class of inverse and ill-posed problems. For instance, many classical and non-classical ill-posed problems can be classified by the language of ΨDOs, including numerical differentiation [13], the Cauchy problems associated with the Laplace and Helmholtz equations [20,32], ill-posed analytic continuation problem [10,24], inverse and backward problems [7,[21][22][23][33][34][35] and so on. During the last forty years, much progress has been made on numerical ΨDOs including wavelet approximation methods, Fourier method and Galerkin methods.…”

“…In contrast to Daubechies wavelets, the Meyer wavelets are of special importance, mainly because they have the so-called oversampling property for which the coefficients can be calculated by sampling procedures [41,42]. Meyer wavelets are frequently used for studying a wide range of inverse and ill-posed problems including numerical analytic continuation problem [10,24], Cauchy problem for the Helmholtz and Laplace equations [20,32], backward problem in heat propagation [7,[21][22][23][33][34][35], some inverse problems involving parabolic PDEs [26] and so on.…”

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

“…In the same manner, a wavelet approach based on Meyer wavelet theory can be developed to regularize the source inverse problem under consideration. We refer to [12,13,15,14] for more details. Finally, it should be noted that the main purpose of this contribution is to show theoretically (based on energy estimates) and numerically (based on a novel formulated nonuniform Rothe scheme) how to reconstruct a time-dependent source from the knowledge of an integral measurement for a non-autonomous time Caputo fractional diffusion equation of order 0 < β < 1.…”

Regularized solution for backward heat equation in Sobolev space

“…Later, Clack and Oppenheimer used this idea to regularize homogeneous problems by adding the small perturbation to the final data. Recently, Karimi et al and Vo regularized homogenous heat equation with time‐dependent thermal conductivity. In contrast with homogenous case, we have only very few papers for nonhomogeneous case.…”