International Scientific Journal


In this study, the numerical solution of the ordinary kind of differential equation for a simple pendulum with large-angle of oscillation was introduced to obtain the time period. The analytical solution is obtained in terms of elliptic functions, and numerical solution of the problem was achieved by using two numerical quadrature methods, namely, Simpson’s 3/8 and Boole’s method. The period of a simple pendulum with large angle is presented. A comparison has been carried out between the analytical solution and the numerical integration results. In the case of error analysis, absolute and relative errors of the problem have been presented. A numerical algorithm has been developed by MATLAB software 2013R and used for analyzing the result. It is established that the results of the comparison guaranty the ability and the accuracy of the present method.
PAPER REVISED: 2020-05-15
PAPER ACCEPTED: 2020-05-23
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THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE Supplement 1, PAGES [S25 - S30]
  1. Beléndez, A., et al., Approximation for a large-angle simple pendulum period, European Journal of Physics, 30, (2009), 2, pp. L25-L28
  2. Antman, S. S., The simple pendulum is not so simple, SIAM review, 40, (1998), 4, pp. 927-930
  3. Lima, F. M. S., Arun. P., An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime, American Journal of Physics, 74, (2006), 10, pp. 892-895
  4. Beléndez, A., et al., Reply to Comment on Approximation for the large-angle simple pendulum period, European Journal of Physics, 30, (2009), 5, pp. L83-L86
  5. Parwani, R. R., An approximate expression for the large angle period of a simple pendulum, European journal of physics, 25, (2003), 1, pp.37-39
  6. Beléndez, A., et al., Approximate solutions for the nonlinear pendulum equation using a rational harmonic representation, Computers & Mathematics with Applications, 64, (2012), 6, pp.1602-1611.
  7. Gradshteyn, I. S., Ryzhik, I. M., Table of integrals, series, and products, Academic press, California, USA, 2014
  8. Carvalhaes, C. G., Suppes, P., Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, American Journal of Physics, 76, (2008), 12, pp.1150-1154
  9. Gil, S., et al., Measuring anharmonicity in a large amplitude pendulum, American Journal of Physics 76, (2008), 9, pp. 843-847.
  10. Big-Alabo, A., Approximate period for large-amplitude oscillations of a simple pendulum based on quintication of the restoring force, European Journal of Physics, 41, (2019), 1, pp.015001-015010
  11. Hite, G. E., Approximations for the period of a simple pendulum, The Physics Teacher, 43, (2005), 5 pp.290-292
  12. Kidd, R. B., Fogg, S. L., A simple formula for the large-angle pendulum period, The Physics Teacher, 40, (2002), 2, pp. 81-83
  13. Ganley, W. P., Simple pendulum approximation, American Journal of Physics, 53, (1985), 1, pp.73-76.
  14. Cromer, A., Many oscillations of a rigid rod, American Journal of Physics, 63, (1995), 2, pp.112-121
  15. Akgül, Ali., New reproducing kernel functions, Mathematical Problems in Engineering. 2015, (2015), pp.1-10,
  16. Hasan, P. M., et al., The existence and uniqueness of solution for linear system of mixed Volterra-Fredholm integral equations in Banach space, AIMS Mathematics, 5, (2019), 1, pp.226-235
  17. Theocaris, P. S., Ioakimidis, N. I., Numerical integration methods for the solution of singular integral equations, Quarterly of Applied Mathematics, 35, (1977), 1, pp.173-183
  18. Haselgrove, C. B., A method for numerical integration, Mathematics of computation, 15, (1961), 76, pp.323-337
  19. Fröberg, C.E., Introduction to numerical analysis, Reading Massachusetts: Addison-Wesley Publishing Company, USA, 1969
  20. Milne, W. E., Numerical integration of ordinary differential equations, The American Mathematical Monthly, 33, (1926), 9, pp. 455-460
  21. Ubale, P. V., Numerical Solution of Boole's rule in Numerical Integration By Using General Quadrature Formula, The Bulletin of Society for Mathematical Services and Standards, 2, (2012), 1, pp.1-4
  22. Burden, R. L., Faires, J. D., Numerical Analysis, Brooks/Cole, Boston, USA, 2011
  23. Johannessen, K., An approximate solution to the equation of motion for large-angle oscillations of the simple pendulum with initial velocity, European journal of physics, 31, (2010), 3, pp. 511-518

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