Recovery of the temperature and the heat flux by a novel meshless method from the measured noisy data

In this paper, we give an invariant method of fundamental solutions (MFS) for recovering the temperature and the heat flux. The invariant MFS is to keep a very basic natural property, which is called the invariance property under trivial coordinate changes in the problem description. The optimal regularization parameter is chosen by Morozov discrepancy principle. Then the reason for introducing the regularization is explained clearly by using the potential function. Three kinds of boundary value problems are investigated to show the effectiveness of this method with some examples. In especial, when the classical MFS does not give accurate results for some problems, it is shown that the proposed method is effective and stable. For each example, the numerical convergence, accuracy, and stability with respect to the number of source points, the distance between the pseudo and real boundary, and decreasing the amount of noise added into the input data, respectively, are also analyzed.