Kamran Malik,Muhammad Mujtaba Shaikh,Kashif Memon,Muhammad Saleem Chandio,Abdul Wasim Shaikh,




Cubature,Double integrals,Derivative-based schemes,Order of accuracy,computational cost,errors,Trapezoid,


In this work, double integration cubature schemes of Trapezoid type have been focused. Recently, some derivative-based Trapezoid-type schemes have been proposed in literature incorporating derivatives at means of the limits of integration. We carry out the exhaustive performance evaluation of the existing closed Newton-Cotes Trapezoidal (CNCT) double integral scheme with its derivative-based variants in recent literature. The derivative-free and derivative-based rules are discussed in basic forms with local error terms and composite forms with global error terms. The performance of the rules on some double integrals in the form of observed order of accuracy, computational costs and error drops demonstrates the encouraging performance of the derivative-based trapezoidal variants over the derivative-free scheme performing numerical experiments.


I. Babolian E., M. Masjed-Jamei and M. R. Eslahchi, On numerical improvement of Gauss-Legendre quadrature rules, Applied Mathematics and Computations, 160(2005) 779-789.
II. Bailey D. H. and J. M. Borwein, “High precision numerical integration: progress and challenges,” Journal of Symbolic Computation ,vol. 46, no. 7, pp. 741–754, 2011.
III. Bhatti, A. A., M.S. Chandio, R.A. Memon and M. M. Shaikh, (2019), “A Modified Algorithm for Reduction of Error in Combined Numerical Integration”, Sindh University Research Journal-SURJ (Science Series) 51(4): 745-750.
IV. Burden R. L., J. D. Faires, Numerical Analysis, Brooks/Cole, Boston, Mass, USA, 9th edition, 2011.
V. Burg. C. O. E., Derivative-based closed Newton-cotes numerical quadrature, Applied Mathematics and Computations, 218 (2012), 7052-7065.
VI. Dehghan M., M. Masjed-Jamei and M. R. Eslahchi, The semi-open Newton- Cotes quadrature rule and its numerical improvement, Applied Mathematics and Computations, 171 (2005) 1129-1140.
VII. Dehghan M., M. Masjed-Jamei, and M. R. Eslahchi, “On numerical improvement of closed Newton-Cotes quadrature rules,” Applied Mathematics and Computation, vol. 165, no. 2,pp. 251–260, 2005.
VIII. Dehghan M., M. Masjed-Jamei, and M. R. Eslahchi, “On numerical improvement of open Newton-Cotes quadrature rules,” Applied Mathematics and Computation, vol. 175, no. 1, pp.618–627, 2006.
IX. Jain M. K., S. R. K. Iyengar and R. K. Jain, Numerical Methods for Scientific and Computation, New Age International (P) Limited, Fifth Edition, 2007.
X. Kamran Malik, Muhammad Mujtaba Shaikh, Muhammad Saleem Chandio, Abdul Wasim Shaikh. : ‘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
XI. Memon K., M. M. Shaikh, M. S. Chandio, A. W. Shaikh, “A Modified Derivative-Based Scheme for the Riemann-Stieltjes Integral”, 52(01) 37-40 (2020).
XII. Pal M., Numerical Analysis for Scientists and Engineers: theory and C programs, Alpha Science, Oxford, UK, 2007.
XIII. Petrovskaya N., E. Venturino, “Numerical integration of sparsely sampled data,” Simulation Modelling Practice and Theory,vol. 19, no. 9, pp. 1860–1872, 2011.
XIV. Ramachandran T. (2016), D. Udayakumar and R. Parimala, “Comparison of Arithmetic Mean, Geometric Mean and Harmonic Mean Derivative-Based Closed Newton Cotes Quadrature“, Nonlinear Dynamics and Chaos Vol. 4, No. 1, 2016, 35-43 ISSN: 2321 – 9238.
XV. Sastry S.S., Introductory methods of numerical analysis, Prentice-Hall of India, 1997.
XVI. Shaikh, M. M., (2019), “Analysis of Polynomial Collocation and Uniformly Spaced Quadrature Methods for Second Kind Linear Fredholm Integral Equations – A Comparison”. Turkish Journal of Analysis and NumberTheory,7(4)91-97. doi: 10.12691/tjant-7-4-1.
XVII. Shaikh, M. M., M. S. Chandio and A. S. Soomro, (2016), “A Modified Four-point Closed Mid-point Derivative Based Quadrature Rule for Numerical Integration”, Sindh University Research Journal-SURJ (Science Series) 48(2): 389-392.
XVIII. Zafar F., S. Saleem and C. O. E. Burg, New derivative based open Newton-Cotes quadrature rules, Abstract and Applied Analysis, Volume 2014, Article ID 109138, 16 pages, 2014.
XIX. Zhao, W., and H. Li, (2013) “Midpoint Derivative- Based Closed Newton-Cotes Quadrature”, Abstract And Applied Analysis, Article ID 492507.
XX. Zhao, W., Z. Zhang, and Z. Ye, (2014), “Midpoint Derivative-Based Trapezoid Rule for the Riemann- Stieltjes Integral”, Italian Journal of Pure and Applied Mathematics, 33: 369-376.

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