第二类两端奇异Fredholm积分方程的分数阶线性插值方法

考虑第二类两端奇异的Fredholm积分方程,假设核函数在区间的两个端点非光滑,存在分数阶的Taylor展开式.对于这种类型的核函数,在包含端点的小区间上采用分数阶插值,在剩余区间上采用分段线性插值逼近,由此得到一种分数阶线性插值退化核方法.本文讨论该方法收敛的条件,给出收敛阶估计.数值算例表明这种分数阶混合线性插值方法对于两端奇异核函数有着较好的计算效果.

Fractional Linear Interpolation Method for Fredholm Integral Equations of the Second Kind with Two-Endpoint Singularities

The Fredholm integral equations of the second kind with two-endpoint singularities are considered, and it’s supposed that the kernel function possesses fractional Taylor’s expansions at both endpoints of the interval. For this type of kernel, the approximation is done by fractional order interpolation in a small interval involving the singularity and piecewise linear interpolation in the remaining part of the interval, which leads to a kind of fractional degenerate kernel method based on linear interpolation. We discuss the condition that the method can converge and give the convergence order estimation. Numerical examples demonstrate that the fractional order hybrid linear interpolation algorithm has good computational results for the kernel functions with two-endpoint singularities.