A collocation scheme for a certain Cauchy singular integral equation based on the superconvergence analysis

In this paper, we investigate the composite midpoint rule for the evaluation of Cauchy principal value integral in an interval and place the key point on its pointwise superconver-gence phenomenon. The error expansion of the rule is obtained, which shows that the superconvergence phenomenon occurs at the points of each subinterval whose local coordinate is the zeros of some function. Then, by applying the midpoint rule to approximate the Cauchy principal value integral and choosing the superconvergence points as the col-location points, we obtain a collocation scheme for solving a certain Cauchy singular integral equation. The more interesting thing is that the coefficient matrix of the resulting linear system possesses some good properties, from which we obtain an optimal error estimate. Finally, some numerical examples are provided to validate the theoretical analysis. Consider the Cauchy singular integral equation Z-b a uðxÞ x À s dx ¼ f ðsÞ; ð1Þ where f ðsÞ is a given function and uðxÞ the density function to be determined. Throughout the paper, R-denotes the Cauchy principal value integral or Hilbert transform. This integral has many different but mathematically equivalent definitions, among which we adopt the following definition Z-b a uðxÞ x À s dx :¼ lim !0 Z sÀ a uðxÞ x À s dx þ Z b sþ uðxÞ x À s dx () s 2 ða; bÞ: ð2Þ The density function uðxÞ is said to be Cauchy principal value integrable with respect to the weight ðx À sÞ À1 if the limit on the right-hand side of (2) exists. A sufficient condition for uðxÞ to be Cauchy principal value integrable is that uðxÞ is Hölder continuous. Integrals of this kind are widely used in many areas of mathematical physics in terms of boundary integral equations, such as potential theory, elasticity problems as well as electromagnetic scattering [6,29]. Numerous work has been devoted to developing efficient quadrature formulas, such as the Gaussian method [10,14,19], the (composite) Newton–Cotes method 0096-3003/$-see front matter Ó 2012 Elsevier Inc. All rights reserved.