用正交函数求超奇异积分的近似值及其误差估计

基于 Hadamard有限部分积分定义, 当密度函数是多项式、正弦函数和余弦函数时, 本文推导出了计算超奇异积分准确值的公式, 进而利用这些公式给出了密度函数为一般连续函数的超奇异积分近似值的计算方法. 本文还对近似值进行了误差分析, 据此可以在事先给定的误差下来计算超奇异积分的近似值. 最后将前面的理论应用到超奇异积分方程求近似解的问题. 数值算例表明该方法的可行性和有效性.

USING THE ORTHOGONAL FUNCTIONS TO SOLVE THE APPROXIMATE SOLUTIONS AND THE ERROR OF THE SUPERSINGULAR

Based on the definition of Hadamard finite-part integral, we deduce the formula computing the exact value of the supersingular integral when the density function is a polynomial, sine or cosine. Then we gain the calculation method solving the approximate value of the supersingular integral for a general density function. In this paper, we analyse the error so that we can compute the approximate value of supersingular integral in the limit of the error which has been given. Finally, the theory is applied to resolve the approximate solution of the supersingular integral equation. The feasibility and validity of the method can be proved by the examples shown in the work.