International Journal of Analysis and Applications | 2026
Authors: Dad K.; Shaikh M.M.; Memon K.; Shaikh A.W.
DOI: 10.28924/2291-8639-24-2026-41
Journal: International Journal of Analysis and Applications
Year: 2026
Publisher: Etamaths Publishing
Document Type: Article
Open Access: All Open Access; Gold Open Access
Cited by: 0
Most of the difficulties in control theory and probability distributions are described in terms of the Riemann-Stieltjes (RS) integral rather than the standard Riemann integral (RI). Numerical approximations for the approximation of the RS integral are required due to the nonlinearity of the integrand and the complexity of the analytical process. The numerical techniques, besides convergence features, should also be computationally effective and time-efficient. In this study, some time-efficient and cost effective numerical approaches for approximating the RS integral are proposed. The proposed approximations are based on Newton-Cotes' standard open-type schemes. We derive derivative-based open Newton-Cotes quadrature schemes in both basic and composite forms, as well as the error terms for the Riemann-Stieltjes integral's numerical evaluation. For the suggested schemes, theorems associated with the degree of precision and order of accuracy are studied with proofs. For all suggested and current rules on the test integrals, the absolute error distributions, computational costs, execution times, and computational orders of accuracy have been calculated. To demonstrate the efficacy of the proposed approaches, a numerical verification method will be used. MATLAB R2022a software was used to achieve the results. The proposed method's quick convergence and high efficacy over the current methods have been demonstrated by the results and theoretical properties. © 2026 the author(s).
and phrases. quadrature; derivative-based scheme; Riemann-Stieltjes integral