一类对偶积分方程组正则化为第一类含对数核的Fredholm奇异积分方程组解法

A Method of Solving a Kind of Dual Integral Equations by Decoupling Reduced to Regularized Fredholm Singular Integral Equations with Logarithmic Kernel of First Kind

引入辅助未知函数及辅助未知函数的积分关系式,表示原未知函数,将对偶积分方程组退耦.应用Sonine第一有限积分公式,实现化为Abel型积分方程组,应用Abel反演变换并化简,正则化为含对数核的第一类Fredholm奇异积分方程组.由此给出奇异积分方程组的一般性解.进而获得对偶积分方程组的解析解,同时严格地证明了,对偶积分方程组和由它化成的含对数核的奇异积分方程组的等价性,以及对偶积分方程组解的存在性和唯一性.

Original unknown functions are expressed by introducing auxiliary unknown functions and integral relations of auxiliary unknown functions.The dual integral equations are decoupled and reduced to Abel integral equations by using Sonine first finite integral formula and the it is further reduced to regularized Fredholm singular integral equations with logarithmic kernels of first kind by Abel anti-transformation.Thus general solutions of singular integral equations are given.And then analytic solutions of dual integral equations are obtained.Simultaneously the equivalence between dual integral equations and corresponding Fredholm singular integral equations with logarithmic kernel of first kind,the existence and uniqueness of solutions are proved exactly.