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Fractional Hermite Degenerate Kernel Method for Linear Fredholm Integral Equations involving Endpoint Singularities
Jiawei Guo,Tongke Wang
Keywords:Linear Fredholm integral equation of the second kind; Kernel with two-endpoint singularities; Fractional Taylor"s expansion; Piecewise hybrid Hermite interpolation; Degenerate kernel method; Adaptive mesh
      In this article, the Fredholm integral equation of the second kind with endpoint singularities 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 which 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 demonstrate that the fractional order hybrid interpolation method has good computational results for the kernels involving endpoint singularities.