The work in this paper is concerned with application of a very simple numerical technique called line element method to solve singular integral equations with weakly singular kernel and hypersingular kernel. Of the integral equations considered here are first kind Abel integral equation and integral equation with log kernel and hypersingular integral equations of first and second kind. In this method, the range of integration as well as the interval of definition of the integral equation are discretised into finite number of small subintervals and the unknown function satisfying the integral equation is assumed to be constant in each small subinterval. This reduces the integral equations to a system of algebraic equations which is then solved to obtain the unknown function in each subinterval. The method is illustrated with examples. It is observed that a very accurate result is obtained by applying this method. The error analysis for this method is also given.

本文的工作涉及一种称为线元法的非常简单的数值技术的应用。求解具有弱奇异核和超奇异核的奇异积分方程。这里考虑的积分方程是第一类阿贝尔积分方程和具有对数核的积分方程以及第一和第二类超奇异积分方程。该方法将积分方程的积分范围和定义区间离散为有限个小子区间，假设满足积分方程的未知函数在每个小子区间内为常数。这将积分方程简化为代数方程组，然后求解该方程组以获得每个子区间中的未知函数。举例说明了该方法。据观察，通过应用该方法获得了非常准确的结果。还给出了该方法的误差分析。