A New Family of Efficient Open-Type Quadrature for the Approximation of Riemann-Stieltjes Integrals Using Derivatives

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DOI: https://doi.org/10.28924/2291-8639-24-2026-41
Published: Feb 19, 2026
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Khuda Dad, Muhammad Mujtaba Shaikh, Kashif Memon, Abdul Wasim Shaikh, A New Family of Efficient Open-Type Quadrature for the Approximation of Riemann-Stieltjes Integrals Using Derivatives, Int. J. Anal. Appl., 24 (2026), 41.

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Khuda Dad, Muhammad Mujtaba Shaikh, Kashif Memon, Abdul Wasim Shaikh

Abstract

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.

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References

  1. S.D. Conte, C. de Boor, Elementary Numerical Analysis, SIAM, Philadelphia, 2017. https://doi.org/10.1137/1.9781611975208.
  2. R. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, American Mathematical Society, Providence, Rhode Island, 1994. https://doi.org/10.1090/gsm/004.
  3. P. Billingsley, Probability and Measure, John Wiley & Sons, (2017).
  4. F. Zafar, S. Saleem, C.O.E. Burg, New Derivative Based Open Newton-Cotes Quadrature Rules, Abstr. Appl. Anal. 2014 (2014), 109138. https://doi.org/10.1155/2014/109138.
  5. F.B. Hildebrand, Introduction to Numerical Analysis, Courier Corporation, (1987).
  6. W. Zhao, H. Li, Midpoint Derivative-Based Closed Newton-Cotes Quadrature, Abstr. Appl. Anal. 2013 (2013), 492507. https://doi.org/10.1155/2013/492507.
  7. M.M. Shaikh, M.S. Chandio, A.S. Soomro, A Modified Four-Point Closed Mid-Point Derivative Based Quadrature Rule for Numerical Integration, Sindh Univ. Res. J. (Sci. Ser.), 48 (2016), 389-392.
  8. W.H. Press, Numerical Recipes 3rd Edition, Cambridge University Press, (2007).
  9. W. Zhao, Z. Zhang, Z. Ye, Composite Trapezoid Rule for the Riemann-Stieltjes Integral and Its Richardson Extrapolation Formula, Italian J. Pure Appl. Math. 35 (2015), 311-318.
  10. W. Zhao, Z. Zhang, Derivative‐Based Trapezoid Rule for the Riemann‐Stieltjes Integral, Math. Probl. Eng. 2014 (2014), 874651. https://doi.org/10.1155/2014/874651.
  11. W. Zhao, Z. Zhang, Simpson’s Rule for the Riemann-Stieltjes Integral, J. Interdiscip. Math. 24 (2021), 1305-1314. https://doi.org/10.1080/09720502.2020.1848317.
  12. W. Zhao, Z. Zhang, Derivative-Based Trapezoid Rule for a Special Kind of Riemann-Stieltjes Integral, Italian J. Pure Appl. Math. 50 (2023), 672–681.
  13. W. Zhao, Z. Zhang, Z. Ye, Midpoint Derivative-Based Trapezoid Rule for the Riemann-Stieltjes Integral, Italian J. Pure Appl. Math. 33 (2014), 369-376.
  14. K. Memon, A New and Efficient Simpson’s 1/3-Type Quadrature Rule for Riemann-Stieltjes Integral, J. Mech. Contin. Math. Sci. 15 (2020), 132-148. https://doi.org/10.26782/jmcms.2020.11.00012.
  15. K. Memon, A New Harmonic Mean Derivative-Based Simpson’s 1/3-Type Scheme for Riemann-Stieltjes Integral, J. Mech. Contin. Math. Sci. 16 (2021), 22-46. https://doi.org/10.26782/jmcms.2021.04.00003.
  16. M. Dehghan, M. Masjed-Jamei, M. Eslahchi, On Numerical Improvement of Open Newton–Cotes Quadrature Rules, Appl. Math. Comput. 175 (2006), 618-627. https://doi.org/10.1016/j.amc.2005.07.030.
  17. M. Dehghan, M. Masjed-Jamei, M. Eslahchi, The Semi-Open Newton–Cotes Quadrature Rule and Its Numerical Improvement, Appl. Math. Comput. 171 (2005), 1129-1140. https://doi.org/10.1016/j.amc.2005.01.137.
  18. M. Dehghan, M. Masjed-Jamei, M. Eslahchi, On Numerical Improvement of Closed Newton–Cotes Quadrature Rules, Appl. Math. Comput. 165 (2005), 251-260. https://doi.org/10.1016/j.amc.2004.07.009.
  19. E. Babolian, M. MasjedJamei, M. Eslahchi, On Numerical Improvement of Gauss–Legendre Quadrature Rules, Appl. Math. Comput. 160 (2005), 779-789. https://doi.org/10.1016/j.amc.2003.11.031.
  20. M.M. Shaikh, K. Memon, S. Mahesar, O.A. Rajput, A Theoretical Analysis of Simpson’s 1/3-Type Scheme for the Riemann-Stieltjes Integral Based on Geometric Mean Derivative, Sindh Univ. Res. J. Sci. Ser. 57 (2025), 1-13. https://doi.org/10.26692/surjss.v57i1.7711.
  21. K. Memon, M.M. Shaikh, S. Mahesar, Numerical Investigation of a Four-Point Simpson’s Quadrature for Computing Riemann-Stieltjes Integrals, NED Univ. J. Res. XXII (2025), 27-44. https://doi.org/10.35453/nedjr-ascn-2024-0017.r3.
  22. R. Marjulisa, M. Imran, Aturan Titik Tengah Double Untuk Mengaproksimasi Integral Riemann-Stieltjes, Jurnal Matematika Integratif, 19 (2023), 29–39.
  23. S. Mahesar, M.M. Shaikh, M.S. Chandio, A.W. Shaikh, Some New Time and Cost Efficient Quadrature Formulas to Compute Integrals Using Derivatives with Error Analysis, Symmetry 14 (2022), 2611. https://doi.org/10.3390/sym14122611.
  24. S. Mahesar, M.M. Shaikh, M.S. Chandio, A.W. Shaikh, Centroidal Mean Derivative-Based Open Newton-Cotes Quadrature Rules, VFAST Trans. Math. 11 (2023), 138-154. https://doi.org/10.21015/vtm.v11i2.1601.
  25. P.R. Mercer, Hadamard's Inequality and Trapezoid Rules for the Riemann–Stieltjes Integral, J. Math. Anal. Appl. 344 (2008), 921-926. https://doi.org/10.1016/j.jmaa.2008.03.026.
  26. K. Memon, Heronian Mean Derivative-Based Simpson’s-Type Scheme for Riemann-Stieltjes Integral, J. Mech. Contin. Math. Sci. 16 (2021), 53-68. https://doi.org/10.26782/jmcms.2021.03.00005.
  27. R.L. Burden, J.D. Faires, Numerical Analysis, Brooks/Cole, Boston, (2011).
  28. M.M. Shaikh, Analysis of Polynomial Collocation and Uniformly Spaced Quadrature Methods for Second Kind Linear Fredholm Integral Equations–A Comparison, Turk. J. Anal. Number Theory 7 (2019), 91-97.