Application of Computer Algebra Software to the Derivation of Numerical Integration Rules for Singular and Hypersingular Integrals

Abstract

Computer algebra software has been used for the construction of numerical integration rules for Cauchy-type singular (principal value) integrals and the corresponding hypersingular (finite-part) integrals (like M angler-type integrals). The modern and powerful computer algebra system MATHEMATICA was employed in this task. A case of the Gauss-Chebyshev quadrature rule as well as the Gauss-Legendre quadrature rule are used for the illustration of the whole approach both for singular and for hypersingular integrals and the corresponding elementary MATHEMATICA procedures are given together with some related results. The case when the singular point coincides with a node was also considered in some detail and related sample results are presented. Finally, the question of the closed-form evaluation of the corresponding fundamental principal value integral is also studied using MATHEMATICA. The obtained results were verified with the help of the already available formulae. The present results can be used in a variety of scientific, engineering and other applications or even in the case of ordinary quadrature rules.