Yin Chuntao, Li Changpin, Bi Qinsheng
Department of Mathematics, Shanghai University, Shanghai 200444, China.
Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 050043, China.
Entropy (Basel). 2018 Dec 18;20(12):983. doi: 10.3390/e20120983.
In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green's function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique.
1923年，哈达玛在使用特定格林函数求解柱面波动方程时遇到了一类具有强奇点的积分。在分部积分后，他忽略了此类积分的无穷部分。这种想法在许多物理模型中非常实用且有用，例如平面和三维弹性力学的裂纹问题。在本文中，我们提出了矩形公式和梯形公式，通过有限部分积分的思想来近似哈达玛导数。然后，我们将所提出的数值方法应用于含哈达玛导数的微分方程。最后，展示了几个数值例子以说明基本思想和技术的有效性。
