The Periodicity of the Accuracy of Numerical Integration Methods for the Solution of Different Engineering Problems
DOI:
https://doi.org/10.38032/jea.2021.04.006Keywords:
Numerical Integration Accuracy, Trapezoidal Rule, Simpson’s 1/3 Rule, Simpson’s 3/8 Rule, Weddle’s RuleAbstract
Newton-Cotes integration formulae have been researched for a long time, but the topic is still of interest since the correctness of the techniques has not yet been explicitly defined in a sequence for diverse engineering situations. The purpose of this paper is to give the readers an overview of the four numerical integration methods derived from Newton-Cotes formula, namely the Trapezoidal rule, Simpson's 1/3rd rule, Simpson's 3/8th rule, and Weddle's rule, as well as to demonstrate the periodicity of the most accurate methods for solving each engineering integral equation by varying the number of sub-divisions. The exact expressions by solving the numerical integral equations have been determined by Maple program and comparisons have been done using Python version 3.8.
References
Sastry, S.S., 2007. Introductory methods of numerical analysis. PHI Learning Pvt. Ltd..
Burden, R.L., 2007. Numerical analysis, 7th Edition, International Thomson Publishing Company.
Mathews, J.H., 2000. Numerical methods for mathematics, science and engineering. Prentice-Hall International., 2nd Edition, Prentice Hall of India Private Limited.
Ausin, M.C., 2007. An introduction to quadrature and other numerical integration techniques. Encyclopedia of Statistics in Quality and reliability. Chichester, England. DOI: https://doi.org/10.1002/9780470061572.eqr438
Smith G.K., 2004. Numerical Integration, Encyclopedia of Biostatistics. 2nd edition, Vol-6. DOI: https://doi.org/10.1002/0470011815.b2a14026
Sinha, R.K. and Kumar, R., 2010. Numerical method for evaluating the integrable function on a finite interval. International Journal of Engineering Science and Technology, 2(6), pp.2200-2206.
Sozio, G., 2009. Numerical integration. Australian Senior Mathematics Journal, 23(1), pp.43-50.
Oliver, J., 1971. The evaluation of definite integrals using high-order formulae. The Computer Journal, 14(3), pp.301-306. DOI: https://doi.org/10.1093/comjnl/14.3.301
Natarajan, A. and Mohankumar, N., 1995. A comparison of some quadrature methods for approximating Cauchy principal value integrals. Journal of Computational Physics, 116(2), pp.365-368. DOI: https://doi.org/10.1006/jcph.1995.1034
Liu, D., Wu, J. and Yu, D., 2010. The superconvergence of the Newton–Cotes rule for Cauchy principal value integrals. Journal of computational and applied mathematics, 235(3), pp.696-707. DOI: https://doi.org/10.1016/j.cam.2010.06.023
Muthumalai, R.K., 2012. Some Formulae for Numerical Differentiation through Divided Difference. International Journal of Mathematical Archive, 3.
Khan, M.M.U.R., Hossain, M.R. and Parvin, S., 2017. Numerical integration schemes for unequal data spacing. American Journal of Applied Mathematics, 5(2), pp.48-56. DOI: https://doi.org/10.11648/j.ajam.20170502.12
Bailey, D.H. and Borwein, J.M., 2011. High-precision numerical integration: Progress and challenges. Journal of Symbolic Computation, 46(7), pp.741-754. DOI: https://doi.org/10.1016/j.jsc.2010.08.010
White, G.C. and Cooch, E.G., 2017. Population abundance estimation with heterogeneous encounter probabilities using numerical integration. The Journal of Wildlife Management, 81(2), pp.322-336. DOI: https://doi.org/10.1002/jwmg.21199
Wei, W., Zhou, B., Połap, D. and Woźniak, M., 2019. A regional adaptive variational PDE model for computed tomography image reconstruction. Pattern Recognition, 92, pp.64-81. DOI: https://doi.org/10.1016/j.patcog.2019.03.009
Liang, M. and Simos, T.E., 2016. A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation. Journal of Mathematical Chemistry, 54(5), pp.1187-1211. DOI: https://doi.org/10.1007/s10910-016-0615-x
Chapra, S.C., 2008. Applied numerical methods with MATLAB for engineers and scientists. McGraw-Hill Higher Education. 4th edition. ISBN-13: 978-0073397962.
Fausett, L.V., 2002. Numerical methods using MathCAD. Pearson.
Siauw, T. and Bayen, A., 2014. An introduction to MATLAB® programming and numerical methods for engineers. Academic Press, Elsevier, USA, 2015.
Davis, P.J. and Rabinowitz, P., 2007. Methods of numerical integration, Dover Publications, Mineola, New York, 2007.
Chapra, S.C. and Canale, R.P., 2010. Numerical methods for engineers, McGraw-Hill. Inc., New York.
Caligaris, M., Rodríguez, G. and Laugero, L., 2015. Designing tools for numerical integration. Procedia-Social and Behavioral Sciences, 176, pp.270-275. DOI: https://doi.org/10.1016/j.sbspro.2015.01.471
Kiusalaas, J., 2005. Numerical methods in engineering with MATLAB®. Cambridge university press. DOI: https://doi.org/10.1017/CBO9780511614682
Jaluria, Y., 2011. Computer methods for engineering with MATLAB applications. CRC Press. DOI: https://doi.org/10.1201/9781439897270
Chapra, S.C., 2008. Applied numerical methods with MATLAB for engineers and scientists (pp. 335-359). McGraw-Hill Higher Education.
Hsu, T.R., 2017. Applied engineering analysis. John Wiley & Sons.
Budynas, R.G. and Nisbett, J.K., 2011. Shigley's mechanical engineering design (Vol. 9). New York: McGraw-Hill.
Hibbeler, R.C., 2004. Engineering mechanics: dynamics. Pearson.
Downloads
Published
- Abstract view516
How to Cite
Issue
Section
License
Copyright (c) 2021 Toukir Ahmed Chowdhury, Towhedul Islam, Ahmad Abdullah Mujahid, Md. Bayazid Ahmed
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
