Authors:
B. M. Ikramul Haque,Md. Iqbal Hossain,DOI NO:
https://doi.org/10.26782/jmcms.2021.02.00004Keywords:
Jerk equation,Nonlinear oscillator,Extended iteration technique,Truncated Fourier series,Abstract
The technique to evade jerk from a dynamical system is to reduce the rate of acceleration or deceleration. It is an important issue for our real life. In motion control systems the term “jerk” is the main topic. The jerk equation containing velocity times acceleration-squared describes the characteristics of chaotic behaviour in many nonlinear phenomena, cosmological analysis, kinematical physics, pendulum analysis etc. Thus, the mentioned equation is important in its own right. An extended iteration method, based on Haque’s approach has been applied to find the analytical solution of the oscillator. The recently various method has been developed for finding analytical solutions of the nonlinear equation but; modified extended iteration method based on Haque’s approach is faster and straight forward than others.Refference:
I. Alam, M.S., Razzak M.A., Hosen M.A. and Parvez M.R., “The rapidly convergent solutions of strongly nonlinear oscillators” Spring. Plu., vol. 5, pp. 1258-1273, 2016.
II. Belendez A., Hernandez A., Belendez T., Fernandez E., Alvarez M.L. and Neipp C., “Application of He’s Homotopy perturbation method to Duffing-harmonic oscillator” Int. J. Nonlinear Sci. and Numer. Simul., vol. 8(1), pp. 79-88, 2007.
III. Gottlieb, H.P.W., “Question #38. What is the simplest jerk function that gives chaos?” American Journal of Physics, vol. 64, pp. 525, 1996.
IV. Gottlieb, H.P.W., “Harmonic balance approach to periodic solutions of nonlinear jerk equation” J. Sound Vib., vol. 271, pp. 671-683, 2004.
V. Gottlieb, H.P.W., “Harmonic balance approach to limit cycle for nonlinear jerk equation” J. Sound Vib., vol. 297, pp. 243-250, 2006.
VI. Hu, H., “Perturbation method for periodic solutions of nonlinear jerk equations” Phys. Lett. A, vol. 372, pp. 4205-4209, 2008.
VII. Hu, H., Zheng, M.Y. and Guo, Y.J., “Iteration calculations of periodic solutions to nonlinear jerk equations” Acta Mech., vol. 209, pp. 269-274, 2010.
VIII. Haque, B.M.I., Alam, M.S. and Majedur Rahmam, M., “Modified solutions of some oscillators by iteration procedure” J. Egyptian Math. Soci., vol. 21, pp. 68-73, 2013.
IX. Haque, B.M.I., A “New Approach of Iteration Method for Solving Some Nonlinear Jerk Equations” Global Journal of Science Frontier Research Mathematics and Decision Sciences, vol. 13, pp. 87-98, 2013.
X. Haque, B.M.I., “A New Approach of Mickens’ Extended Iteration Method for Solving Some Nonlinear Jerk Equations” British journal of Mathematics & Computer Science, vol. 4, pp. 3146-3162, 2014.
XI. Haque, B.M.I., Bayezid Bostami M., Ayub Hossain M.M., Hossain M.R. and Rahman M.M., “Mickens Iteration Like Method for Approximate Solution of the Inverse Cubic Nonlinear Oscillator” British journal of Mathematics & Computer Science, vol. 13, pp. 1-9, 2015.
XII. Haque, B.M.I., Ayub Hossain M.M., Bayezid Bostami M. and Hossain M.R., “Analytical Approximate Solutions to the Nonlinear Singular Oscillator: An Iteration Procedure” British journal of Mathematics & Computer Science, vol. 14, pp. 1-7, 2016.
XIII. Haque, B.M.I., Asifuzzaman M. and Kamrul Hasam M., “Improvement of analytical solution to the inverse truly nonlinear oscillator by extended iterative method” Communications in Computer and Information Science, vol. 655, pp. 412-421, 2017.
XIV. Haque, B.M.I., Selim Reza A.K.M. and Mominur Rahman M., “On the Analytical Approximation of the Nonlinear Cubic Oscillator by an Iteration Method” Journal of Advances in Mathematics and Computer Science, vol. 33, pp. 1-9, 2019.
XV. Haque, B.M.I. and Ayub Hossain M.M., “A Modified Solution of the Nonlinear Singular Oscillator by Extended Iteration Procedure” Journal of Advances in Mathematics and Computer Science, vol. 34, pp. 1-9, 2019.
XVI. Haque B M I, Zaidur Rahman M and Iqbal Hossain M, “Periodic solution of the nonlinear jerk oscillator containing velocity times acceleration-squared: an iteration approach”, J. Mech. Cont.& Math. Sci., Vol.-15, No.-6, June (2020) pp 493-433.
XVII. Leung, A.Y.T. and Guo, Z., “Residue harmonic balance approach to limit cycles of nonlinear jerk equations” Int. J. Nonlinear Mech., vol. 46, pp. 898-906, 2011.
XVIII. Ma, X., Wei, L. and Guo, Z., “He’s homotopy perturbation method to periodic solutions of nonlinear jerk equations” J. Sound Vib., vol. 314, pp. 217-227, 2008.
XIX. Mickens, R.E., “Comments on the method of harmonic balance” J. Sound Vib., vol. 94, pp. 456-460, 1984.
XX. Mickens, R.E., “Iteration Procedure for determining approximate solutions to nonlinear oscillator equation” J. Sound Vib., vol. 116, pp. 185-188, 1987.
XXI. Nayfeh, A.H., “Perturbation Method” John Wiley & Sons, New York, 1973.
XXII. Ramos, J.I. and Garcia-Lopez, “A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations” Appl. Math. Comput., vol. 216, pp. 2635-2644, 2010.
XXIII. Wu, B.S., Lim, C.W. and Sun, W.P., “Improved harmonic balance approach to periodic solutions of nonlinear jerk equations” Phys. Lett. A, vol. 354, pp. 95-100, 2006.
XXIV. Zheng, M.Y., Zhang, B.J., Zhang, N., Shao, X.X. and Sun, G.Y., “Comparison of two iteration procedures for a class of nonlinear jerk equations” Acta Mech. vol. 224, pp. 231-239, 2013.