Approximating Riemann-Stieltjes Integral Using New Time and Cost Efficient Trapezoid-type Quadrature

Abstract

Engineers and scientists adopt numerical integration to achieve an approximate solution for definite integrals that have no analytic solution. This research focuses on developing some new derivative based quadrature schemes for numerically integrating the integral of Riemann-Stieltjes (Rs-integral) by proposing new schemes which are based on trapezoid-type quadrature. The process of undetermined coefficients has been used for the derivation of proposed schemes. The theoretical derivation and numerical verification of orders of accuracy have been addressed in line with the degrees of precision, and a sufficient improvement has been demonstrated over the existing schemes. The theorems regarding single and mul-
tiple use of the suggested schemes in a finite interval have been proved along with the theoretical results on residual terms, both locally and globally. All suggested schemes have been verified to reduce to corresponding variants for the Riemann integral in the case when integrator g(t) = t. Numerical experiments have been performed on all the discussed schemes with the help of MATLAB coding. The experimental results assure the smaller numerical errors by the proposed schemes in the comparison of existing schemes. The obtained results show the efficiency of proposed schemes in light of computational burden and CPU time (in seconds) with reference to the existing quadrature. The proposed work substantially advances the existing knowledge with an addition of derivative-based correction terms in the usual quadrature in a way that the consequent computational costs and execution times are also minimized.