ERROR ANALYSIS OF CLOSED NEWTON-COTES CUBATURE SCHEMES FOR DOUBLE INTEGRALS

Authors:

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

DOI NO:

https://doi.org/10.26782/jmcms.2020.11.00009

Keywords:

Cubature,Double integrals,closed Newton-Cotes,Precision,Order of accuracy,Local error,Computational cost,

Abstract

Numerical integration is one of the fundamental tools of numerical analysis to cope with the complex integrals which cannot be evaluated analytically, and for the cases where the integrand is not mathematically known in closed form. The quadrature rules are used for approximating single integrals, whereas cubature rules are used to evaluate integrals in higher dimensions. In this work, we consider the closed Newton-Cotes cubature schemes for double integrals and discuss consequent error analysis of these schemes in terms of the degree of precision, local error terms for the basic form approximations, composite forms and the global error terms. Besides, the computational cost of the implementation of these schemes is also presented. The theorems proved in this work area pioneering investigation on error analysis of such schemes in the literature.      

Refference:

I. 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.
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., and J. D. Faires, Numerical Analysis, Brooks/Cole,Boston, Mass, USA, 9th edition, 2011.
IV. Burg, C. O. E. Derivative-based closed Newton-cotes numerical quadrature, Applied Mathematics and Computations, 218 (2012) 7052-7065.
V. Burg, C. O. E., and E. Degny, Derivative-based midpoint quadrature rule, Applied Mathematics and Computations, 4 (2013) 228-234.
VI. 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.
VII. Dehghan, M., M. Masjed-Jamei, and M. R. Eslahchi, “On numerical improvement of closed Newton-Cotes quadraturerules,” Applied Mathematics and Computation, vol. 165, no. 2,pp. 251–260, 2005.
VIII. 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.
IX. Malik K., Shaikh, M. M., Chandio, M. S. and Shaikh, A. W. Some new and efficient derivative-based schemes for numerical cubature. Journal of Mechanics of Continua and Mechanical Sciences, 15 (10): 67-78, 2020.
X. 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.
XI. Memon K, Shaikh MM, Chandio MS and Shaikh AW, A new and efficient Simpson’s 1/3-type quadrature rule for Riemann-Stieltjes integral, Journal of Mechanics of Continua and Mechanical Sciences, 15 (11):, 2020.
XII. Memon, A. A., Shaikh, M. M., Bukhari, S. S. H., & Ro, J. S. (2020). 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.
XIII. Pal, M., Numerical Analysis for Scientists and Engineers: theory and C programs, Alpha Science, Oxford, UK, 2007.
XIV. Shaikh, M. M. “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. 2019, 7(4), 91-97.
XV. Shaikh, M. M., Chandio, M. S., Soomro, A. S. A Modified Four-point Closed Mid-point Derivative Based Quadrature Rule for Numerical Integration, Sindh Univ. Res. Jour. (Sci. Ser.) Vol. 48 (2) 389-392 2016.
XVI. Walter Guatschi, Numerical analysis second edition, Springer Science business, Media LLC 1997, 2012.
XVII. Weijing Zhao and Hongxing, “Midpoint Derivative-Based Closed Newton-Cotes Quadrature”, Abstract and Applied Analysis, vol.2013, Article ID 492507, 10 pages, 2013.
XVIII. Zafar, F., Saira Saleem and Clarence O.E.Burg, New derivative based open Newton-Cotes quadrature rules, Abstract and Applied Analysis, Volume 2014, Article ID 109138, 16 pages, 2014.

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