超奇异积分及其积分方程的数值算法和应用

Numerical algorithms of the hypersingular integrals and hypersingular integral equations in mathematical physics and engineering calculation are of great significance. Professor Huang Jin and his research team have presented a series of research results in recent years. The results contain error asymptotic expansions of one-dimensional hypersingular integral, point type and product type of two-dimensional hypersingular integrals, point type of multi-dimensional hypersingular integral, and the kinds with parameters. On the basis of the conclusion of Professor Huang and his research team, the project discuss the following questions: (1) study the modified numerica methods of the quadrature formulas for solving hypersingular integrals, and the high accuracy algorithms and their applications for solving one-dimensional hypersingular integral equations. (2) study the multi-parameter asymptotic expansions with errors for two-dimensional hypersingular integrals mixed with other singular integrals, and the numerical methods for solving the corresponding integrals; (3) study the numerical algorithms of two-dimensional hypersingular integral equations and mixed hypersingular integral equations, and the applications of numerical algorithms in fracture mechanics, aerodynamics, electrodynamics, quantum field theory, etc.. The research of the above problems provides a new method for solving the mathematical modeling based on physical background, and provides the basis for understanding and analysis of physical phenomena accurately.

超奇异积分及超奇异积分方程的数值计算在数学物理及工程计算中具有重要的意义. 黄晋教授及其研究团队近几年给出了关于一维超奇异积分、二维点型及乘积型超奇异积分、多维点型超奇异积分以及与上述各种情况相应的含参数类型超奇异积分的误差渐近展式等一系列结论. 本项目拟在此已有结论的基础上, 研究以下问题: (1) 研究一维超奇异积分求积公式的修正算法, 以及一维超奇异积分方程的高精度数值算法及应用; (2) 研究含其它奇异类型的二维混合超奇异积分的误差多参数渐近展式及相应类型积分的求值算法; (3) 研究二维超奇异积分方程及混合超奇异积分方程的数值算法, 以及相应结论在断裂力学、空气动力学、电动力学、量子场理论等方面的应用. . 以上问题的研究, 给基于物理背景的数理模型的数值求解方法提供了新的算法支持, 也为准确的了解和分析物理现象提供了依据.

超奇异积分及超奇异积分方程数值求解方法在数学物理及科学与工程计算中具有重要的应用价值和意义。常用的配置法、Galerkin方法会受到超奇异性和维数效应的影响。在黄晋教授及其研究团队研究的基础上，讨论了一维超奇异积分求积公式中含有负指数幂项的处理方法， 应用负指数幂的外推方法可以达到将其消除的目的，并将之应用到超奇异积分方程的数值求解中；在含参数的Love’s 积分方程及积分的求解中应用求积公式进行数值求解，并讨论了其积分项中所带有参数得临界值问题；讨论了泛函修正平均法，并将之应用到第二类Volterra积分方程及具有特殊形式核函数的Fredholm积分方程的近似求解中；与合作者应用格林公式将二维带有Robin边界条件的Helmholtz 方程转化为积分方程，应用高阶的求积公式及外推法得到方程的数值解。

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