In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.

This paper is devoted to an inverse space-dependent source problem for spacefractional diffusion equation. Furthermore, we show that this problem is ill-posed in the sense of Hadamard, i.e., the solution (if it exists) does not depend continuously on the data. In addition, we propose a simplified generalized Tikhonov regularization method and prove the corresponding convergence estimates by using a priori regularization parameter choice rule and a posteriori parameter choice rule, respectively. Finally, numerical examples are carried to support the theoretical results and illustrate the effectiveness of the proposed method.

In this article, we first transform the problems of the bounded stability and asymptotical stability of impulsive neutral delay differential equations(INDDEs) into the problems of neutral delay differential equations (NDDEs) without impulsive perturbations, then we further transform them into the problems of ordinary different equations with a forcing term. On this basis, some new sufficient conditions for bounded stability and asymptotical stability of the exact solutions of INDDEs are obtained and the numerical methods for INDDEs are constructed. In these sufficient conditions, the numerical methods which can preserve the bounded stability and asymptotical stability of the exact solutions are provided. And two numerical examples are given to demonstrate the theoretical results.