A high-precision hypersingular quadrature formula of order seven for analytic functions

In our study, we undertake the evaluation of a hyper-singular integral of the form  using quadrature rules with a precision degree of ۵. It's recognized that ordinary quadrature rules may not be suitable for hyper-singular integrals, as they typically depend on smooth integrands. The presence of poles of order ۲ can lead to challenges, causing the integrand to become unbounded, divergent, and numerically unstable. As such, there is a clear need for specialized quadrature rules that include explicit singularity treatment for more accurate evaluations. In this work, we focus on analyzing the hyper-singular integral through quadrature rules, each initially demonstrating a precision of ۴. Following extensive evaluation, we were pleased to develop two new quadrature rules, both achieving a commendable precision of ۵. By creating a convex combination of these two rules, we successfully derived a new hyper-singular quadrature rule with an enhanced precision of ۷. We have illustrated several problems to demonstrate the efficiency of the newly developed hyper-singular quadrature rules. Our numerical computations indicate that the new quadrature rule yields lower error rates compared to the existing counterparts. Furthermore, its degree of precision surpasses that of the previous rules. Consequently, we believe that the new quadrature rule represents a more efficient option for such evaluations.

کلیدواژه ها:

Analytic function ، Quadrature rule ، Degree of precision ، Absolute Error ، Hyper Singular Integral