On the Spectrum Computation of Non-oscillatory and Highly Oscillatory Kernel with Weak Singularity

We compute the spectra of integral compact operators with weak singularity. Jacobi-spectral collocation methods are applied for problems without high oscillation. A convergence rate is obtained for general non-oscillatory operators. Furthermore, if the bilinear form associated with the kernel is positive definite, the convergence rate is doubled. A spectral Galerkin method with modified Fourier expansion is developed to compute the spectra of highly oscillatory kernel. Numerical results are presented to demonstrate the effectiveness and accuracy of our algorithms and theorems.

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