第二类端点奇异Fredholm积分方程的分数阶退化核方法

本文针对第二类端点奇异Fredholm积分方程构造基于分数阶Taylor展开的退化核方法,设计了两种计算格式,一是在全区间上使用分数阶Taylor展开式近似核函数,二是在包含奇点的小区间上采用分数阶插值,在剩余区间上采用分段二次多项式插值逼近核函数.讨论了两种退化核方法收敛的条件,并给出了混合插值法的收敛阶估计.数值算例表明对于非光滑核函数分数阶退化核方法有着良好的计算效果,且混合二次插值法比全区间上的分数阶退化核方法有着更广泛的适用范围.

FRACTIONAL ORDER DEGENERATE KERNEL METHODS FOR FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND WITH ENDPOINT SINGULARITIES

This paper constructs fractional order degenerate kernel methods based on the fractional Taylor's expansion for Fredholm integral equations of the second kind with endpoint singularities. Two computational schemes are designed, one is approximating the kernel function by its fractional Taylor's expansion in the whole interval, and the other is constructing a fractional order interpolation in a small interval containing the singularity and using piecewise quadratic interpolation to approximate the kernel function in the remaining part of the interval. The conditions that the two degenerate kernel methods converge are discussed, and the convergence order is estimated for hybrid interpolation scheme. Numerical examples demonstrate that the fractional order degenerate kernel methods have good computational results for the kernel functions with endpoint singularities, and the hybrid quadratic interpolation scheme has broader scope of application than the fractional order degenerate kernel method in the whole interval.