An iterative regularization method for an abstract ill-posed biparabolic problem

Abstract

In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a regularizing strategy based on the Kozlov-Maz'ya iteration method. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.MSC: Primary 47A52; secondary 65J22

Stability and Regularization Method for Inverse Initial Value Problem of Biparabolic Equation

“…However, it is well-known that the classical parabolic equation can not accurately describe the procedure of heat conduction [5] [6], so many models have been proposed to describe this procedure; among them the biparabolic model proposed in [7] can give a more adequate mathematical description for the process of heat conduction than the classical case. Meanwhile we note that, for the biparabolic model, up to now the literatures devoted to it are relatively scarce, except for [7]- [9]. On other models, we can see [10]- [13], etc.…”

“…Problem 2is ill-posed and the regularization techniques are required to stabilize numerical computations [14] [15]. In 2015, [9] considered this problem and proved a condition stability result of Hölder type, and then applied the Kozlov-Maz'ya iteration method to deal with it; the corresponding convergence results have been given, but unfortunately the condition stability result in [9] is not useful for the case of 0 t = . In this paper, we firstly establish a conditional stability of Hölder type, which is valid at the point 0 t = , then use a modified regularization method to overcome its ill-posedness and give the convergence estimate under an a-priori assumption for the exact solution.…”

Recovering the initial distribution for a logarithmic nonlinear biparabolic equation

“…NBPE describes the conduction of heat and has many applications for heat processes, [8][9][10] and NBPEs are also used to describe specials phenomena on the dynamics of filtration consolidation. 11,12 Although the inverse problem of NBPE (as of the form BP) has been studied in the case of homogeneous and nonlinear problems, [13][14][15] the forms of source terms are logarithmic nonlinearities (such as Problem BP); as far as we know, there have not been any research findings related to it. The qualitative properties of the solutions of the initial and inverse problems are very different.…”

Modified Quasi Boundary Value method for inverse source biparabolic

“…In [14], Tuan and his group provided an impressive result of the final value problem for a biparabolic problem with statistical discrete data. In [16], by applying the iteration method, Abdelghani Lakhdari and Nadjib Boussetila give some other convergent rates under a-priori bound assumptions on the sought solution.…”