CLOSED NEWTON-COTES CUBATURE SCHEMES FOR TRIPLE INTEGRALS WITH ERROR ANALYSIS

Authors:

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

DOI NO:

https://doi.org/10.26782/jmcms.2021.01.00005

Keywords:

Cubature,Triple integrals,closed Newton-Cotes,Precision,Order of accuracy,Local error,Global error,

Abstract

Most of the problems in applied sciences in engineering contain integrals, not only in one dimension but also in higher dimensions. The complexity of integrands of functions in one variable or higher variables motivates the quadrature and cubature approximations. Much of the work is focused on the literature on single integral quadrature approximations and double integral cubature schemes. On the other hand, the work on triple integral schemes has been quite rarely focused. In this work, we propose the closed Newton-Cotes-type cubature schemes for triple integrals and discuss consequent error analysis of these schemes in terms of the degree of precision and local error terms for the basic form approximations. The results obtained for the proposed triple integral schemes are in line with the patterns observed in single and double integral schemes. The theorems proved in this work on the local error analysis will be a great aid in extending the work towards global error analysis of the schemes in the future.      

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., :J. Mech. Cont. & Math. Sci., Vol. 15, No.11, pp. 67-78, 2020.
X. 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, pp. 95-107, 2020.
XI. 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.
XII. Memon K, Shaikh MM, Chandio MS and Shaikh AW, A new and efficient Simpson’s 1/3-type quadrature rule for Riemann-Stieltjes integral, : J. Mech. Cont. & Math. Sci.,Vol. 15, No.11, pp. 132-148, 2020.
XIII. 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.
XIV. Pal, M., Numerical Analysis for Scientists and Engineers: theory and C programs, Alpha Science, Oxford, UK, 2007.
XV. 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.
XVI. 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.
XVII. Walter Guatschi, Numerical analysis second edition, Springer Science business, Media LLC 1997, 2012.
XVIII. Weijing Zhao and Hongxing, “Midpoint Derivative-Based Closed Newton-Cotes Quadrature”, Abstract and Applied Analysis, vol.2013, Article ID 492507, 10 pages, 2013.
XIX. 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.

View Download