An Automatic Quadrature Schemes and Error Estimates for Semibounded Weighted Hadamard Type Hypersingular Integrals

Abstract

The approximate solutions for the semibounded Hadamard type hypersingular integrals (HSIs) for smooth density function are investigated. The automatic quadrature schemes (AQSs) are constructed by approximating the density function using the third and fourth kinds of Chebyshev polynomials. Error estimates for the semibounded solutions are obtained in the class of $h(t)\in C^{N,\alpha }[-\mathrm{1,1}]$. Numerical results for the obtained quadrature schemes revealed that the proposed methods are highly accurate when the density function $h\hspace{0.17em}(t)$ is any polynomial or rational functions. The results are in line with the theoretical findings.