Abstract
In this work we introduce new rational transformations which are available for numerical evaluation of weakly singular integrals and Cauchy principal value integrals. The proposed rational transformations include parameters playing an important role in accelerating the accuracy of the Gauss quadrature rule used for the singular integrals. Results of some selected numerical examples show the efficiency of the proposed transformation method compared with some existing transformation methods.
Highlights
We consider numerical evaluation of weakly singular integrals and Cauchy principal value(CPV)integrals which frequently appear, for instance, in the dominant coefficients of the boundary element method [1,2,3,4,5]
Among the methods one can notice that the nonlinear coordinate transformation techniques [6,7,8,9,10,11,12,13,14,15,16,17,18] are prominent as they are easy to use in adaptive approaches
We explore the availability of the proposed rational transformation method for each type of singular integrals
Summary
We consider numerical evaluation of weakly singular integrals and Cauchy principal value(CPV). Integrals which frequently appear, for instance, in the dominant coefficients of the boundary element method [1,2,3,4,5] Accurate evaluation of these singular integrals is a very practical and important problem in many areas of engineering including elastostatics and electromagnetics. Aiming enhancement of the accuracy of the numerical integration method via the coordinate transformation, in this paper we present simple rational functions including parameters. The proposed rational functions, satisfying some appropriate properties, weaken the singularity of the integrand These rational functions associated with a standard quadrature rule are expected to improve approximation errors further by controlling the parameters η and k. Numerical results of the presented method associated with the Gauss-Legendre quadrature rule show the competent errors for the end-point weakly singular integrals. For the CPV integral the presented transformation method with the parameter η selected within an appropriate region results in much better errors than the existing transformation methods
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