第二类代数和对数奇异Fredholm积分方程的退化核方法

Degenerate kernel method for the Fredholm integral equations of the second kind involving algebraic and logarithmic singularities

考虑核函数在端点奇异的第二类Fredholm积分方程,设核函数在区间端点代数和对数奇异,且存在Puiseux级数展开式.针对该类方程,在包含奇点的小区间采用Puiseux基函数插值,在其他区间采用线性插值,构造了一种混合型的退化核方法,对奇异积分采用修正的复合Gauss-Legendre求积公式计算.对所得格式进行数值分析,证明了格式的收敛性.数值算例表明,该方法对核函数在区间端点奇异的情形有良好的计算效果,且计算精度较高.

Fredholm integral equation of the second kind involving algebraic and logarithmic endpoint singularities is considered. It is supposed that the kernel function possesses the Puiseux expansion at the endpoint of the interval. For this equation,the Puiseux interpolation is adopted in a small interval involving the singularity and piecewise linear interpolation is used in the remaining part of the interval,and then a hybrid degenerate kernel method is established. The deduced singular integrals are evaluated by the modified composite Gauss-Legendre algorithm. On the basis of numerical analysis,the con-vergence of the algorithm is proved. A numerical example shows that the method has good computational results for the ker-nel with algebraic and logarithmic singularity,and the precision is high.