Approximation of Cauchy-type singular integrals with high frequency Fourier kernel

Two types of splitting algorithms are proposed for approximation of Cauchy type singular integrals having high frequency Fourier kernel. To evaluate non-singular integrals, modified Levin collocation methods with multiquadric radial basis function and Chebyshev polynomials are proposed. In the scenario of interval splitting, a multi-resolution quadrature is used to tackle the singularity ridden kernel. While in the case of integrand splitting, the singular part integral is evaluated analytically. Logarithmic singular integrals with oscillatory kernels are transformed to Cauchy principal value integrals and computed with the new algorithms. Error analysis of the component algorithms, as well as the individual methods, is performed theoretically. Validation of accuracy and error estimates of the methods are performed numerically as well.