一类多节对偶积分方程组正则化为超(强)奇异积分方程组求解法

Solving Method of Regularized Integral Equations with Hyperstrongly Singular Kernels on a Kind of General Dual Integral Equations with Multiple Intermittent Case on Path of Integration

基于Mellin变换法,首先方程组进行Mellin变换,然后,通过引入新的未知函数的Mellin变换代换原来未知函数的Mellin变换,使对偶积分方程组退耦正则化为超(强)奇异积分方程组．将未知函数分解并表示成未知函数和已知幂函数的乘积,幂指数(a_i,v_i)需使超(强)奇异积分方程组中的超(强)奇异积分,在端点(a_i,b_i)有界或可积奇异,求解超(强)奇异积分方程组可以使用有限部分积分式．将未知函数展成任意完备函数系(?)_n*(u)的级数,将超(强)奇异积分方程组,化成线性代数方程组,通过求解级数中的各项系数,由此给出对偶积分方程组的一般性解．并严格证明了对偶积分方程组和由它化成的超(强)奇异积分方程组的等价性,解的存在性和解的表示形式不唯一性．本文给出的理论解和解法,可供求解数学,物理,力学中的混合边值问题应用．

Based on the method of Mellin transform of equations, firstly, make Mellin transform, and then decouple and regularize the dual integral equations into integral equations with strong singularity through substitution of Mellin transform of a new introduced unknown function for Mellin transform of original unknown function. The unknown function is factorized as a product of an unknown function and a known power function. The power exponent （ai, vi） enables the strongly singular integral of integral equations with strong singularity to be bounded or integrally singular at endpoints （ai, bi）. For solving the integral equations with strong singularity, the ifitegral formula of finite part is applied. The unknown function is expanded into a series of arbitrary complete functional set φn＊ （u）. The integral equations with strong singularity are changed into linear algebraic equations. Thus general solutions of dual integral equations are given by finding coefficient of every term in the series. The equivalence between dual integral equations and corresponding integral equations with strong singularity, the existence of solutions and non-uniqueness of expressive form on solutions are proved exactly. The theoretical solutions and solving method given here are applicable to solving mixed boundary value problems in mathematics, physics and mechanics.