## Analytical approximations for a conservative nonlinear singular oscillator in plasma physics.(English)Zbl 1267.34030

Summary: A modified variational approach and the coupled homotopy perturbation method with variational formulation are exerted to obtain periodic solutions of a conservative nonlinear singular oscillator in plasma physics. The frequency-amplitude relations for the oscillator which the restoring force is inversely proportional to the dependent variable are achieved analytically. The approximate frequency obtained using the coupled method is more accurate than the modified variational approach and ones obtained using other approximate methods and the discrepancy between the approximate frequency using this coupled method and the exact one is lower than 0.31% for the whole range of values of oscillation amplitude. The coupled method provides a very good accuracy and is a promising technique to a lot of practical engineering and physical problems.

### MSC:

 34A45 Theoretical approximation of solutions to ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 65L99 Numerical methods for ordinary differential equations
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### References:

 [1] Melvin, P.J., On the construction of poincare-Lindstedt solutions: the nonlinear oscillator equation, SIAM journal on applied mathematics, 33, 161-194, (1977) · Zbl 0358.65066 [2] Casal, A.; Freedman, M., A Poincaré-Lindstedt approach to bifurcation problems for differential-delay equations, IEEE transactions on automatic control, 25, 967-973, (1980) · Zbl 0437.34058 [3] Nayfeh, A.H.; Mook, D.T., Nonlinear oscillations, (1979), John Wiley & Sons [4] Ahmadian, H.; Azizi, H., Stability analysis of a nonlinear jointed beam under distributed follower force, Journal of vibration and control, 17, 27-38, (2011) · Zbl 1271.74212 [5] Ahmadian, H.; Jalali, H., Identification of bolted lap joints parameters in assembled structures, Mechanical systems and signal processing, 21, 1041-1050, (2007) [6] Öziş, T.; Yıldırım, A., A study of nonlinear oscillators with x1/3 force by he’s variational iteration method, Journal of sound and vibrations, 306, 372-376, (2007) · Zbl 1242.74214 [7] He, J.H.; Wu, G.-C.; Austin, F., The variational iteration method which should be followed, Nonlinear science letters A, 1, 1-30, (2010) [8] He, J.H., Homotopy perturbation technique, Computer methods in applied mechanics and engineering, 178, 257-262, (1999) · Zbl 0956.70017 [9] Younesian, D.; Askari, H.; Saadatnia, Z.; KalamiYazdi, M., Free vibration analysis of strongly nonlinear generalized Duffing oscillators using he’s variational approach & homotopy perturbation method, Nonlinear science letters A, 2, 11-16, (2011) [10] Belendez, A.; Pascual, C.; Ortuno, M.; Belendez, T.; Gallego, S., Application of modified he’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities, Nonlinear analysis: real world applications, 10, 601-610, (2009) · Zbl 1167.34327 [11] Cveitcanin, L., Homotopy-perturbation for pure nonlinear differential equation, Chaos solitons and fractals, 30, 1221-1230, (2006) · Zbl 1142.65418 [12] He, J.H., Preliminary report on the energy balance for nonlinear oscillations, Mechanics research communications, 29, 107-111, (2002) · Zbl 1048.70011 [13] KalamiYazdi, M.; Khan, Y.; Madani, M.; Askari, H.; Saadatnia, Z.; Yildirim, A., Analytical solutions for autonomous conservative nonlinear oscillator, International journal of nonlinear sciences and numerical simulation, 11, 979-984, (2010) [14] Younesian, D.; Askari, H.; Saadatnia, Z.; KalamiYazdi, M., Frequency analysis of strongly nonlinear generalized Duffing oscillators using he’s frequency-amplitude formulation and he’s energy balance method, Computers and mathematics with applications, 59, 3222-3228, (2010) · Zbl 1193.65152 [15] Askari, H.; KalamiYazdi, M.; Saadatnia, Z., Frequency analysis of nonlinear oscillators with rational restoring force via he’s energy balance method and he’s variational approach, Nonlinear science letters A, 1, 425-430, (2010) [16] Kakutani, T.; Sugimoto, N., Krylov-Bogoliubov-mitropolsky method for nonlinear wave modulation, Physics of fluids, 17, 1617-1625, (1974) [17] Hassan, A., KBM derivative expansion method is equivalent to the multipletimescales method, Journal of sound and vibration, 200, 433-440, (1997) · Zbl 1235.34139 [18] Beléndez, A.; Méndez, D.I.; Beléndez, T.; Hernández, A.; Alvarez, M.L., Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable, Journal of sound and vibration, 314, 775-782, (2008) [19] Beléndez, A.; Pascual, C., Harmonic balance approach to the periodic solutions of the (an)harmonic relativistic oscillator, Physics letters A, 371, 291-299, (2007) [20] Beléndez, A.; Gimeno, E.; Fernández, E.; Méndez, D.I.; Alvarez, M.L., Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable, Physica scripta, 77, 065004, (2008) · Zbl 1145.70012 [21] Hu, H.; Tang, J.H., Solution of a Duffing-harmonic oscillator by the method of harmonic balance, Journal of sound and vibration, 294, 637-639, (2006) · Zbl 1243.34049 [22] Mickens, R.E., Harmonic balance and iteration calculations of periodic solutions to $$\ddot{y} + y^{- 1} = 0$$, Journal of sound and vibration, 306, 968-972, (2007) [23] Ganji, S.S.; Barari, A.; Ganji, D.D., Approximate analyses of two mass-spring systems and buckling of a column, Computers and mathematics with applications, 61, 1088-1095, (2011) · Zbl 1217.74050 [24] Ibsen, L.B.; Barari, A.; Kimiaeifar, A., Analysis of highly nonlinear oscillation systems using he’s MAX-MIN method and comparison with homotopy analysis and energy balance methods, Sadhana, 35, 1-16, (2010) · Zbl 1206.34023 [25] Kalami-Yazdi, M.; Ahmadian, H.; Mirzabeigy, A.; Yıldırım, A., Dynamic analysis of vibrating systems with nonlinearities, Communications in theoretical physics, 57, 183-187, (2012) · Zbl 1247.34054 [26] He, J.H., Some asymptotic methods for strongly nonlinear equations, International journal of modern physics B, 20, 1141-1199, (2006) · Zbl 1102.34039 [27] He, J.H., Hamiltonian approach to nonlinear oscillators, Physics letters A, 374, 2312-2314, (2010) · Zbl 1237.70036 [28] Kalami-Yazdi, M.; Mirzabeigy, A.; Abdollahi, H., Nonlinear oscillators with non-polynomial and discontinuous elastic restoring forces, Nonlinear science letters A, 3, 48-53, (2012) [29] Khan, Y.; Wu, Q.; Askari, H.; Saadatnia, Z.; KalamiYazdi, M., Nonlinear vibration analysis of a rigid rod on a circular surface via Hamiltonian approach, Mathematical and computational applications, 15, 974-977, (2010) · Zbl 1371.74125 [30] Cveticanin, L.; KalamiYazdi, M.; Saadatnia, Z.; Askari, H., Application of Hamiltonian approach to the generalized nonlinear oscillator with fractional power, International journal of nonlinear sciences and numerical simulation, 11, 997-1002, (2010) [31] He, J.H., Variational approach for nonlinear oscillators, Chaos solitons and fractals, 34, 1430-1439, (2007) · Zbl 1152.34327 [32] Akbarzede, M.; Langari, J.; Ganji, D.D., A coupled homotopy-variational method and variational formulation applied to nonlinear oscillators with and without discontinuities, Journal of vibration and acoustics, 133, 044501, (2011) [33] Farzaneh, Y.; Tootoonchi, A.A., Global error minimization method for solving strongly nonlinear oscillator differential equations, Computers and mathematics with applications, 59, 2887-2895, (2010) · Zbl 1193.65146
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