Fractional Hermite degenerate kernel method for linear Fredholm integral equations involving endpoint weak singularities

In this article, the Fredholm integral equation of the second kind with endpoint weakly singular kernel is considered and suppose that the kernel possesses fractional Taylor''s expansions about the endpoints of the interval. For this type kernel, the fractional order interpolation is adopted in a small interval involving the singularity and piecewise cubic Hermite interpolation is used in the remaining part of the interval, which leads to a kind of fractional degenerate kernel method. We discuss the condition that the method can converge and give the convergence order. Furthermore, we design an adaptive mesh adjusting algorithm to improve the computational accuracy of the degenerate kernel method. Numerical examples confirm that the fractional order hybrid interpolation method has good computational results for the kernels involving endpoint weak singularities.