In this paper, we introduce a Matlab program method to compute Carleman estimate for the fourth order partial differential operator $\gamma\partial_t+\partial_x^4\ (\gamma\in\mathbb{R})$. We obtain two kinds of Carleman estimates with different weight functions, i.e. singular weight function and regular weight function, respectively. Based on Carleman estimate with singular weight function, one can obtain the known controllability and observability results for the 1-d fourth order parabolic-type equation, while based on Carleman estimate with regular weight function, one can deduce not only the known result on conditional stability in the inverse problem of half-order fractional diffusion equation, but also a new result on conditional stability in the inverse problem of half-order fractional Schr\"odinger equation.
翻译：在本文中, 我们引入了一种 Matlab 程序方法, 来计算 Carleman 对第四顺序部分差分操作员 $\ gamma\ part_ t ⁇ part_ x ⁇ 4\\ (\ gamma\ int\ mathb{R}) $\\ gamma\ int\ mathb{R} 的估算。 我们得到了两种不同重量函数的 Carleman 估算, 即单重量函数和定期重量函数。 根据 Carleman 的估算, 单重量函数, 一个人可以获得已知的1- 4 4 顺序抛物- 等式的可控性和可观察性结果, 而根据 Carleman 的估算, 正常重量函数, 不仅可以推断半顺序分数分数方程式反数问题中有条件稳定性的已知结果, 而且还可以推断半顺序分数 Schr\ poinger 等式反问题中有条件稳定性的新结果 。
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