The Higher-Order Modified Method Of Backward Heat Conduction Problem

The inverse problem of mathematical physics is one of the fastest growing and the most valuable research of applied mathematics in the 21st century, and its prospects are very extensive. The difficulty of the inverse problem is that its ill-posed characteristic. Backward heat equation is one of the typical inverse problem. In this paper, we consider the following forms of backward heat equation: We want to find the distribution of temperature u(·, t) at the time 0≤ t< T by the original data φT(x).For the backward heat equation problem, many scholars have made signif-icant contributions. The article which written by Zhi Qian, Chu-Li Fu and Rui Shi solved the stability problems by adding a third-order partial differential term, however, its drawback is that the approximate solution did not reach convergent optimal order. In order to solve this problem, this paper improved the modified method to higher-order modified method, that is, considering the higher-order modified equation as follows: By selecting the appropriate regularization parameter λ and the positive inte-ger κ, we have reached the Holder and logarithmic type stability estimation respectively under different prior information. Finally, numerical tests also verify our conclusions.