Numerical simulation of nonlinear equations by converting quadrature rule to iterative technique

Inderjeet; Bhardwaj, Rashmi

doi:10.47974/JIM-2138

Open Access Research Article

Numerical simulation of nonlinear equations by converting quadrature rule to iterative technique

Inderjeetyadavinderjeet386@gmail.comUniversity School of Basic and Applied Sciences (USBAS)Guru Gobind Singh Indraprastha University (GGSIPU)Dwarka, Delhi, 110078, IndiaView full profile → , ^*Rashmi BhardwajCorresponding authorrashmib@ipu.ac.inNonlinear Dynamics Research LabUniversity School of Basic and Applied Sciences (USBAS)Guru Gobind Singh Indraprastha University (GGSIPU)Dwarka, Delhi, 110078, India0000-0002-0502-762XView full profile →

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pp. 1837–1845Vol. 28Issue 5August 2025DOI: 10.47974/JIM-2138XML

Received:: 10 Jul 2024
Published Online:: 21 Aug 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-2138
Pages:: 1837–1845

Abstract

In computational mathematics, numerical integration is an essential method that yields effective approximations of definite integrals. The trapezoidal technique is one of the most popular numerical integration techniques because of its ease of use and efficiency. Unfortunately, when working with complicated functions or intervals that exhibit erratic behavior, its applicability is restricted. The goal of this research study is to examine the effectiveness and practical implications of converting the trapezoidal rule to iterative procedures through the use of fundamental theorem of calculus for numerical integration. In addition, we provide case studies and numerical experiments to show the usefulness and efficiency of the proposed iterative techniques in comparison with the conventional Trapezoidal Rule. These experiments demonstrate situations in which iterative approaches perform better than the conventional method, indicating the potential of these approaches in practical integration problems. The research findings provide insights into the transformation of simple methods such as the Trapezoidal Rule into advanced iterative approaches, hence contributing to the improvement of numerical integration methods. This study provides more dependable numerical solutions in a variety of scientific and engineering applications by increasing the accuracy and efficiency of integration techniques.

Keywords

Convergence analysis Number of iterations Nonlinear problems Quadrature rules

Subject Classifications

Primary 65LxxSecondary 65L05

References

[1] S. Abbasbandy, “Homotopy analysis method for the Kawahara-equations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 307–312 (2010).
[2] M. W. Alomari and S. S. Dragomir, “Various error estimations for several Newton-Cotes Quadrature formulae in terms of at most first derivative and applications in numerical integration,” Jordan Journal of Mathematics and Statistics, vol. 7, no. 2, pp. 89–108 (2014).
[3] G. Amos and V. Subramanian, Numerical Methods for Engineers and Scientists, 3rd ed., Wiley (2013), ISBN-10: 1118554930, ISBN-13: 978-1118554937.
[4] C. O. E. Burg, “Derivative based closed Newton Cotes numerical quadrature,” Applied Mathematics and Computation, vol. 218, no. 13, pp. 7052–7065 (2012).
[5] C. S. Chapra, Applied Mathematical Methods with MATLAB for Engineers and Scientists, 3rd ed., McGraw-Hill (2006), ISBN 978-0-07-340110-2, MHID 0-07-340110-2.
[6] M. K. Jain, S. R. K. Iyengar, and R. K. Jain, Numerical Methods, 2nd ed., New Age International Publisher (2004), ISBN: 9788122424263.
[7] A. Kharab and R. B. Guenther, An Introduction to Numerical Methods – A MATLAB Approach, CRC Press, Boca Raton, FL (2001).
[8] E. Kreyszig, Advanced Engineering Mathematics, 9th ed., Wiley (2006), ISBN-13: 978-0-471-72897-9.
[9] D. Levy, Introduction to Numerical Analysis, [Online]. Available: https://www.scribd.com/document/331408400/Introduction-to-Numerical-Analysis (2010).
[10] W. Malfliet, “The Tanh method: A tool for solving certain classes of nonlinear evaluations and wave equations,” Computational and Applied Mathematics, vol. 164-165, pp. 529–541 (2004).
[11] J. H. Mathews and K. F. Kurtis, Numerical Methods Using MATLAB, 4th ed., New Jersey, USA (2004), ISBN: 0-13-065248-2.
[12] S. Mechee and J. K. Mshachal, “Derivation of embedded explicit RK type methods for directly solving class of seventh order ordinary differential equations,” Journal of Interdisciplinary Mathematics, vol. 22, no. 8, pp. 1451–1456 (2019), DOI: 10.1080/09720502.2019.1700936.
[13] N. E. Tatar, “(L) type activation functions in a system of ordinary differential equations,” Journal of Information and Optimization Sciences, vol. 39, no. 8, pp. 1637–1645 (2018), DOI: 10.1080/02522667.2017.1383661.
[14] Q. I. Rahman and G. Schmeisser, “Characterization of the speed of convergence of the trapezoidal rule,” Numerische Mathematik, vol. 57, no. 1, pp. 123–138 (1990).
[15] V. Dukkipati Rao, Numerical Methods, New Age International Publisher (2010), ISBN 10: 8122428134, ISBN 13: 9788122428131.
[16] S. S. Sastry, Introductory Methods of Numerical Analysis, 5th ed., PHI (2012), ISBN 10: 8120345924, ISBN 13: 9788120345928.
[17] A. M. Wazwaz, “New compact and noncompact solutions for two variants of a modified Camassa-Holm equation,” Applied Mathematics and Computation, vol. 163, pp. 1165–1179 (2005).
[18] A. M. Wazwaz, “A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 485–501 (2005).
[19] J. A. C. Weidman, “Numerical integration of periodic functions: A few examples,” The American Mathematical Monthly, vol. 109, no. 1, pp. 21–36 (2002).
[20] Y. L. Chou, “Generalization and new proof for almost everywhere convergence to imply local convergence in measure,” Journal of Interdisciplinary Mathematics, vol. 24, no. 8, pp. 2195–2199 (2021).

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