Two new regularization methods for solving sideways heat equation

Abstract

We consider a non-standard inverse heat conduction problem in a bounded domain which appears in some applied subjects. We want to know the surface temperature in a body from a measured temperature history at a fixed location inside the body. This is an exponentially ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we introduce the two new classes of quasi-type methods and iteration methods to solve the problem and prove that our methods are stable under both a priori and a posteriori parameter choice rules. An appropriate selection of a parameter in the scheme will get a satisfactory approximate solution. Furthermore, if we use the discrepancy principle we can avoid the selection of the a priori bound. MSC: 35K05; 35K99; 47J06; 47H10

On the Regularization-Homotopy Analysis Method for Linear and Nonlinear Fredholm Integral Equations of the First Kind

“…This method was introduced in [25,26,27] by Tikhonov and Phillips respectively. Both linear and nonlinear Fredholm integral equations of the first kind is transformed by a regularization method [25,26,27,28,20,23,24,29,30,31,32] to the Fredholm integral equation of the second kind. However, ߙ is introduced with the unknown function to approximate Fredholm integral equation respectively as follows…”

Regularization Error Analysis for a Sideways Problem of the 2D Nonhomogeneous Time-Fractional Diffusion Equation

“…For the homogeneous fractional diffusion equation, i.e., f (x, t) = 0 in the first equation in (1), we refer to [14][15][16][17][18][19][20][21][22] and the references therein. As to the nonhomogeneous system, there are also a few articles, e.g., Tuan [23] proposes a truncation regularization method to obtain a regularized solution, error estimates are established under some a priori assumptions for the exact solution.…”