Fourier regularization for a backward heat equation

Abstract

In this paper a simple and convenient new regularization method for solving backward heat equationFourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.

On an initial inverse problem in nonlinear heat equation associated with time-dependent coefficient

“…This error estimate is much better than the logarithmic order estimates obtained in some previous results [8], [4].…”

“…Liu in [6] introduced a group preserving scheme. Some papers [1], [4] have approximated (1.1) by truncated methods. A modified quasi-reversibility for problem (1.2) is investigeted by Denche et al [3], Fu et al [8].…”

A simple regularization method for the ill-posed evolution equation

“…So, the right-hand side of (16) is logarithmic stability estimate. This logarithmic order is also given in [2], [9], [14], [11], [21], [22].…”

“…Recently, the truncated regularization method has been effectively applied to solve the sideways heat equation [6], [7], a more general sideways parabolic equation [8] and backward heat [9]. This regularization method is rather simple and convenient for dealing with some ill-posed problems.…”

Determination temperature of a backward heat equation with time-dependent coefficients

“…The problem is called the backward heat problem with time-dependent coefficient. In the simple case a(t) = 1, the problem (1.1) is investigated in many papers, such as Clark and Oppenheimer [3], Denche and Bessila [5], Tautenhahn et al [24] Melnikova et al [15,16], ChuLiFu [4,10,9], Tautenhahn [24], Trong et al [21,22], B. Yildiz et al [25,26]. Although there are many papers on the backward heat equation with the constant coefficient, there are rarely works considered the backward heat with the time-dependent coefficient, such as (1.1).…”