Authors:
Kashif Memon ,Muhammad Mujtaba Shaikh,Kamran Malik,Muhammad Saleem Chandio,Abdul Wasim Shaikh,DOI NO:
https://doi.org/10.26782/jmcms.2021.03.00006Keywords:
Quadrature rule,Riemann-Stieltjes integral,Centroidal Mean,Simpson’s 1/3 rule,Composite form,Local error,Global error,Cost-effectiveness,Time-efficiency,Abstract
In this paper, a new efficient derivative-based quadrature scheme of Simpson’s 1/3-type is proposed using the centroidal mean for the approximation of Riemann-Stieltjes integral (RS-integral). Theorems are proved related to the basic form, composite form, local and global errors of the new scheme for the RS-integral. The reduction of the new proposed scheme is verified using g(t) = t for Riemann integral. The theoretical results of new proposed scheme have been proved by experimental work using programming in MATLAB against existing schemes. The order of accuracy, computational cost and average CPU time (in seconds) of the new proposed scheme are determined. The results obtained show the effectiveness of the proposed scheme compared to the existing schemes.Refference:
I. Bartle, R.G. and Bartle, R.G., The elements of real analysis, (Vol. 2). John Wiley and Sons, 1964.
II. Bhatti AA, Chandio MS, Memon RA and Shaikh MM, A Modified Algorithm for Reduction of Error in Combined Numerical Integration, Sindh University Research Journal-SURJ (Science Series) 51.4, (2019): 745-750.
III. Burden, R.L., Faires, J.D., Numerical Analysis, Brooks/Cole, Boston, Mass, USA, 9th edition, 2011.
IV. 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.
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., 15 (11): 95-107, 2020.
VI. 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., 15 (10): 67-78, 2020.
VII. Mastoi, Adnan Ali, Muhammad Mujtaba Shaikh, and Abdul Wasim Shaikh. A new third-order derivative-based iterative method for nonlinear equations. : J. Mech. Cont. & Math. Sci., 15 (10): 110-123, 2020.
VIII. 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.
IX. 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., 15 (11): 132-148, 2020.
X. Memon, A. A., Shaikh, M. M., Bukhari, S. S. H., & Ro, J. S. Look-up Data Tables-Based Modeling of Switched Reluctance Machine and Experimental Validation of the Static Torque with Statistical Analysis. Journal of Magnetics, 25(2): 233-244, 2020.
XI. Mercer, P.R., Hadamard’s inequality and Trapezoid rules for the Riemann-Stieltjes integral, Journal of Mathematica Analysis and Applications, 344 (2008), 921-926.
XII. Mercer, P.R., Relative convexity and quadrature rules for the Riemann-Stieltjes integral, Journal of Mathematica inequality, 6 (2012), 65-68.
XIII. Protter, M.H. and Morrey, C.B., A First Course in Real Analysis . Springer, New York, NY, 1977.
XIV. 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.
XV. Ramachandran Thiagarajan, Udayakumar.D, Parimala .R, Harmonic mean derivative-based closed Newton-Cotes quadrature, IOSR-Journal of Mathematics, 12, 36-41, May-June 2016.
XVI. Ramachandran Thiagarajan, Udayakumar.D, Parimala .R, Heronian mean derivative- based closed Newton cotes quadrature, International Journal of Mathematical Archive, 7, 53-58, July 2016.
XVII. Ramachandran Thiagarajan, Parimala .R, Centroidal mean derivative–based closed Newton cotes quadrature, International Journal of Science and Research, 5, 338-343, August 2016.
XVIII. Shaikh, MM., MS Chandio and AS Soomro, A Modified Four-point Closed Mid-point Derivative Based Quadrature Rule for Numerical Integration, Sindh University Research Journal-SURJ (Science Series), 48.2, 2016.
XIX. Shaikh, Muhammad Mujtaba, Shafiq-ur-Rehman Massan, and Asim Imdad Wagan. “A new explicit approximation to Colebrook’s friction factor in roughpipes under highly turbulent cases.” International Journal of Heat and MassTransfer 88 (2015): 538-543.
XX. Shaikh, Muhammad Mujtaba, Shafiq-ur-Rehman Massan, and Asim Imdad Wagan. “A sixteen decimal places’ accurate Darcy friction factor database using non-linear Colebrook’s equation with a million nodes: A way forward to the soft computing techniques.” Data in brief 27 (2019): 104733.
XXI. Shaikh, Muhammad Mujtaba. “Analysis of Polynomial Collocation and Uniformly Spaced Quadrature Methods for Second Kind Linear Fredholm Integral Equations–A Comparison.” Turkish Journal of Analysis and Number Theory 7.4 (2019): 91-97
XXII. Umar, Sehrish, Muhammad Mujtaba Shaikh, and Abdul Wasim Shaikh. A new quadrature-based iterative method for scalar nonlinear equations. : J. Mech. Cont. & Math. Sci., 15 (10): 79-93, 2020.
XXIII. Zhao, W., and H. Li, Midpoint Derivative-Based Closed Newton-Cotes Quadrature, Abstract And Applied Analysis, Article ID 492507, (2013).
XXIV. 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.
XXV. 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.