AUTOMATIC QUADRATURE SCHEME FOR CAUCHY TYPE SINGULAR INTEGRAL ON THE VARIABLE INTERVAL

Abstract

In this note, we consider the product indefinite integral of the form
 
 An automatic quadrature scheme (AQS) is constructed for evaluating Cauchy principal singular integrals in two cases. In the first case c∈ [y,z] ⊂ [-1,1] where -1 < y < z < 1, density function h(t) is approximated by the truncated sum of Chebyshev polynomials of the first kind. Direct substitution does not give solutions so we have used the AQS and reduced problems into algebraic equation with unknown parameters bk which can be found in terms of the singular point with some front conditions. In the second case c ∈ [-1,1], the application of the AQS reduced the number of calculations twice and accuracy is increased. As a theoretical result, the convergence theorem of the proposed method is proven in a Hilbert space. Numerical examples with exact solutions and comparisons with other methods are also given, and they are in the line with theoretical findings.