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A modified solution of the nonlinear singular oscillator has been obtained based on the extended iteration procedure. We have used an appropriate truncation of the obtained Fourier series in each step of iterations to determine the approximate analytic solution of the oscillator. The third approximate frequency of the nonlinear singular oscillator shows a good agreement with its exact values. Earlier different authors presented the analytic solution of the oscillator by using various types of methods. We have compared the results obtained by the modified technique with some of the existing results. We see that some of their techniques deviate from higher-order approximations and the present technique performs comparatively better. The rate of change of percentage of error of the presented modified solution shows the validity of convergence.
Nayfeh AH. Introduction to perturbation techniques. 1st Ed. New York: Wiley; 1981.
Belendez A, Pascual C, Gallego S, Ortuno M, Neipp C. Application of a modified He’s homotopy perturbation method to obtain higher-order approximations of a force nonlinear oscillator. Physics Latter A. 2007;371:421-426.
Belendez A, Pascual C, Ortuno M, Belendez T, Gallego S. Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities. Real World Applications. 2009;10(2):601-610.
Haque BMI, Alom MA, Rahman MH. Perturbation solution of fourth order critically damped oscillatory nonlinear systems. International Journal of Basic and Applied Science. 2011;11:68-77.
He JH. Modified Lindstedt-Poincare methods for some non-linear oscillations. Part III: Double series expansion. Int. J. Nonlinear Sci. Numer. Simul. 2001;2:317-320.
Mickens RE. Comments on the method of harmonic balance. J. Sound Vib. 1984;94:456-460.
Gottlieb HPW. Harmonic balance approach to limit cycle for nonlinear jerk equation. J. Sound Vib. 2006;297:243-250.
Mickens RE. Harmonic balance and iteration calculations of periodic solutions to . J. Sound and Vib. 2007;306:968-972.
Belendez A, Gimeno E, Alvarez ML, Mendez DI. Nonlinear oscillator with discontinuity by generalized harmonic balanced method. J. Computers and Math. with App. 2009;58:2117-2123.
Leung AYT, Zhongjin G. Residue harmonic balance approach to limit cycles of non-linear jerk equations. Int. J. Nonlinear Mech.; 2011.
Xu H, Cang J. Analysis of a time fractional wave-like equation with the homotopy analysis method. Phys. Lett. A. 2008;372:1250-1255.
Mickens RE. Iteration procedure for determining approximate solutions to nonlinear oscillator equation. J. Sound and Vib. 1987;116:185-188.
Hu H. Solutions of Duffing-harmonic oscillator by an iteration procedure. J. Sound Vib. 2006;298:446-452.
Hu H. Solutions of a quadratic nonlinear oscillator: Iteration procedure. J. Sound Vib. 2006;298:1159-1165.
Hu H, Tang JH. A classical iteration procedure valid for certain strongly nonlinear oscillator. J. Sound Vib. 2006;299:397-402.
Chen YM, Liu JK. A modified Mickens iteration procedure for nonlinear oscillators. J. Sound and Vib. 2008;314:465-473.
Mickens RE. Truly nonlinear oscillations. 1st Ed. World Scientific: Singapore; 2010.
Haque BMI, Alam MS, Rahmam MM. Modified solutions of some oscillators by iteration procedure. J. Egyptian Math. Soci. 2013;21:68-73.
Haque BMI. A new approach of Mickens’ iteration method for solving some nonlinear jerk equations. Global Journal of Sciences Frontier Research Mathematics and Decision Science. 2014;13(11): 87-98.
Haque BMI, Alam MS, Rahman MM, Yeasmin IA. Iterative technique of periodic solutions to a class of non-linear conservative systems. Int. J. Conceptions on Computation and Information Technology. 2014;2(1):92-97.
Haque BMI. A new approach of modified Mickens’ iteration method for solving some nonlinear jerk equations. British Journal of Mathematics & Computer Science. 2014;4(22).
Haque BMI, Hossain MR. An analytic investigation of the quadratic nonlinear oscillator by an iteration method. Journal of Advances in Mathematics and Computer Science. 2016;13(1):1-8.
Haque BMI, Reza AKMS, Rahman MM. On the analytical approximation of the nonlinear cubic oscillator by an iteration method. 2019;33(3):1-9.
Lim CW, Wu BS. A modified procedure for certain non-linear oscillators. J. Sound and Vib. 2002;257:202-206.
Wu BS, Sun WP, Lim CW. An analytical approximate technique for a class of strongly nonlinear oscillator. Int. J. Nonlinear Mech. 2006;41:766-774.
Hu H. A modified method of equivalent linearization that works even when the non-linearity is not small. J. Sound and Vib. 2004;276:1145-1149.
González-Gaxiola O. Periodic solution for strongly nonlinear oscillators by He’s new amplitude–frequency relationship. Int. J. Appl. Comput. Math. 2017;3:S1249–S1259.
Yazdi MK, Tehrani PH. The energy balance to nonlinear oscillations via Jacobi collocation method. Alexandria Engineering Journal. 2015;54:99-103.