Authors:
Kashif Memon,Muhammad Mujtaba Shaikh,Kamran Malik,Muhammad Saleem Chandio,Abdul Wasim Shaikh,DOI NO:
https://doi.org/10.26782/jmcms.2021.04.00003Keywords:
Quadrature rule,Riemann-Stieltjes,Harmonic Mean,Simpson’s 1/3 rule,Local error,Global error,Cost-effectiveness,Time-efficiency,Abstract
In this research paper, a new harmonic mean derivative-based Simpson’s 1/3 scheme has been presented for the Riemann-Stieltjes integral (RS-integral). The basic and composite forms of the proposed scheme with local and global error terms have been derived for the RS-integral. The proposed scheme has been reduced using g(t) = t for Riemann integral. Experimental work has been discussed to verify the theoretical results of the new proposed scheme against existing schemes using MATLAB. The order of accuracy, computational cost and average CPU time (in seconds) of the new proposed scheme have been computed. Finally, it is observed from computational results that the proposed scheme is better than existing schemes.Refference:
I. Bartle, R.G. and Bartle, R.G., The elements of real analysis, (Vol. 2). John Wiley & Sons, 1964.
II. Burden, R.L., Faires, J.D., Numerical Analysis, Brooks/Cole, Boston, Mass, USA, 9th edition, 2011.
III. Dragomir, S.S., and Abelman S., Approximating the Riemann-Stieltjes integral of smooth integrands and of bounded variation integrators, Journal of Inequalities and Applications, 2013.1 (2013), 154.
IV. Malik K., Shaikh, M. M., Chandio, M. S. and Shaikh, A. W. : Some new and efficient derivative-based schemes for numerical cubature. J. Mech. Cont. & Math. Sci., Vol.-15, No.10 October (2020): pp 67-78. DOI : 10.26782/jmcms.2020.10.00005
V. Malik K., Shaikh, M. M., Chandio, M. S. and Shaikh, A. W. : Error analysis of closed Newton-Cotes cubature schemes for double integrals. J. Mech. Cont. & Math. Sci., Vol.-15, No.-11, November (2020) pp 95-107, 2020. DOI: 10.26782/jmcms.2020.11.00009
VI. Memon K, Shaikh MM, Chandio MS and Shaikh AW, A Modified Derivative-Based Scheme for the Riemann-Stieltjes Integral, Sindh University Research Journal-SURJ (Science Series) 52.1, (2020): 37-40.
VII. Memon K., Shaikh, M. M., Chandio, M. S. and Shaikh, A. W. : A new and efficient Simpson’s 1/3-type quadrature rule for Riemann-Stieltjes integral. J. Mech. Cont. & Math. Sci., Vol.-15, No.-11, November (2020) pp 132-148. DOI : 10.26782/jmcms.2020.11.00012
VIII. Memon K., Shaikh, M. M., Malik, K., M., Chandio, M. S. and Shaikh, A. W. : Heronian Mean Derivative-Based Simpson’s-type scheme for Riemann-Stieltjes integral, J. Mech. Cont. & Math. Sci., Vol.-16, No.-3, March (2021) pp 53-68. DOI : 10.26782/jmcms.2021.03.00005
IX. Memon K., Shaikh, M. M., Malik, K., M., Chandio, M. S. and Shaikh, A. W. : Efficient Derivative-Based Simpson’s 1/3-type scheme using Centroidal Mean for Riemann-Stieltjes integral, J. Mech. Cont. & Math. Sci., Vol.-16, No.-3, March (2021) pp 69-85. DOI : 10.26782/jmcms.2021.03.00006
X. Mercer, P.R., Hadamard’s inequality and Trapezoid rules for the Riemann-Stieltjes integral, Journal of Mathematica Analysis and Applications, 344 (2008), 921-926.
XI. Mercer, P.R., Relative convexity and quadrature rules for the Riemann-Stieltjes integral, Journal of Mathematica inequality, 6 (2012), 65-68.
XII. Protter, M.H. and Morrey, C.B., A First Course in Real Analysis . Springer, New York, NY, 1977.
XIII. Ramachandran Thiagarajan, Udayakumar.D, Parimala .R, Geometric mean derivative-based closed Newton-Cotes quadrature, International Journal of Pure & Engineering Mathematics, 4, 107-116, April 2016.
XIV. Ramachandran Thiagarajan, Udayakumar.D, Parimala .R, Harmonic mean derivative-based closed Newton-Cotes quadrature, IOSR-Journal of Mathematics, 12, 36-41, May-June 2016.
XV. Ramachandran Thiagarajan, Udayakumar.D, Parimala .R, Heronian mean derivative- based closed Newton cotes quadrature, International Journal of Mathematical Archive, 7, 53-58, July 2016.
XVI. Ramachandran Thiagarajan, Parimala .R, Centroidal mean derivative–based
closed Newton cotes quadrature, International Journal of Science and Research, 5, 338-343, August 2016.
XVII. Zhao, W., and H. Li, Midpoint Derivative-Based Closed Newton-Cotes
Quadrature, Abstract And Applied Analysis, Article ID 492507, (2013).
XVIII. Zhao, W., Z. Zhang, and Z. Ye, Midpoint Derivative-Based Trapezoid Rule
for the Riemann-Stieltjes Integral, Italian Journal of Pure and Applied
Mathematics, 33, (2014), 369-376.
XIX. Zhao, W., Z. Zhang, and Z. Ye, Composite Trapezoid rule for the Riemann
-Stieltjes Integral and its Richardson Extrapolation Formula, Italian Journal of
Pure and Applied Mathematics, 35 (2015), 311-318.