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



Cubature,Double integrals,Derivative-based schemes,Precision,Order of accuracy,Trapezoid,


In this research work, some new derivative-based numerical cubature schemes have been proposed for the accurate evaluation of double integrals under finite range. The proposed modifications are based on the Trapezoidal-type quadrature and cubature rules. The proposed schemes are important to numerically evaluate the complex double integrals, where the exact value is not available but the approximate values can only be obtained. The proposed derivative-based double integral schemes provide efficient results with regards to higher precision and order of accuracy. The proposed schemes, in basic and composite forms, with local and global error terms are presented with necessary proofs with their performance evaluation against conventional Trapezoid rule through some numerical experiments. The consequent observed error distributions of the proposed schemes are found to be lower than the conventional Trapezoidal cubature scheme in composite form


I. A. Harshavardhan, Syed Nawaz Pasha, Sallauddin Md, D. Ramesh, “TECHNIQUES USED FOR CLUSTERING DATA AND INTEGRATING CLUSTER ANALYSIS WITHIN MATHEMATICAL PROGRAMMING”, J.Mech.Cont.& Math. Sci., Vol.-14, No.-6, November – December (2019) pp 546-557
II. 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.
III. 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.
IV. 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.
V. Burden R. L., J. D. Faires, Numerical Analysis, Brooks/Cole, Boston, Mass, USA, 9th edition, 2011.
VI. Burg. C. O. E., Derivative-based closed Newton-cotes numerical quadrature, Applied Mathematics and Computations, 218 (2012), 7052-7065.
VII. 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.
VIII. 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.
IX. 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.
X. 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.
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. MOHAMMED M. Fayyadh, R. Kandasamy, RADIAH Mohammed, JAAFAR Abdul Abbas Abbood, “THE PERFORMANCE OF Al2 O3 Crude Oil ON NONLINEAR STRETCHING SHEET”, J. Mech. Cont. & Math. Sci., Vol.-13, No.-5, November-December (2018) Page 263-279
XIII. Pal M., Numerical Analysis for Scientists and Engineers: theory and C programs, Alpha Science, Oxford, UK, 2007.
XIV. Petrovskaya N., E. Venturino, “Numerical integration of sparsely sampled data,” Simulation Modelling Practice and Theory,vol. 19, no. 9, pp. 1860–1872, 2011.
XV. 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.
XVI. Sastry S.S., Introductory methods of numerical analysis, Prentice-Hall of India, 1997.
XVII. 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.
XVIII. 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.
XIX. 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.
XX. Zhao, W., and H. Li, (2013) “Midpoint Derivative- Based Closed Newton-Cotes Quadrature”, Abstract And Applied Analysis, Article ID 492507.
XXI. 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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