Journal Vol – 18 No -5, May 2023



Md. Ishaque Ali, B M Ikramul Haque, M. M. Ayub Hossain



Haque’s Approach with Mickens’ Iteration Method is used to find the exact analytic solution of the nonlinear equation involving velocity times acceleration squared. A truncated Fourier series is used in different rhythms with different repetition steps. Our results are very close to the exact results and our results are comparatively closer to the exact results than others. Our solution method is obtained around the second-order angular frequency using Newton's method. For some third-order (jerk) differential equations with cubic nonlinearities and nonlinear second-order differential equations; Mickens' iteration method is used to determine the exact analytical approximate periodic solution. A numerical experiment of general differential equations with third-order, one-dimensional, autonomous, quadratic, and cubic nonlinearity has uncovered several algebraically simple equations involving the shaking of time-dependent acceleration that contain chaotic solutions.


Jerk equation,Truncated Fourier series,Newton’s method,Angular frequency,Haque’s Approach with Mickens’ Iteration Method,Autonomous,Chaotic solutions,


I. Gottlieb, H. P. W. (2004). Harmonic Balance Approach to Periodic Solutions of Non-linear Jerk Equations. Journal of Sound and Vibration, 271(3-5), 671-683. 10.1016/s0022-460x(03)00299-2

II. Haque, B. I., & Hossain, M. A. (2021). An Effective Solution of the Cube-root Truly Nonlinear Oscillator: Extended Iteration Procedure. International Journal of Differential Equations, 2021, 1-11. 10.1155/2021/7819209

III. Haque, B. I., & Hossain, M. I. (2021). An Analytical Approach for Solving the Nonlinear Jerk Oscillator Containing Velocity Times Acceleration-squared by an Extended Iteration Method. Journal of Mechanics of Continua and Mathematical Sciences, 16(2), 35-47. 10.26782/jmcms.2021.02.00004

IV. Haque, B. I., Rahman, M. Z., & Hossain, M. I. (2021). Periodic Solution of the Nonlinear Jerk Oscillator Containing Velocity Times Acceleration-Squared: An Iteration Approach, Journal of Mechanics of Continua and Mathematical Sciences, 15(6), 419-433. 10.26782/jmcms.2020.06.00033

V. Haque, B. I., & Flora, S. A. (2020). On the analytical approximation of the quadratic nonlinear oscillator by modified extended iteration. Method, Applied Mathematics and Nonlinear Sciences, June 15th 2020.1-10.
VI. Haque, B. I. (2014). A New Approach of Mickens’ Extended Iteration Method for Solving Some Nonlinear Jerk Equations. British Journal of Mathematics & Computer Science, 4(22), 3146.

VII. Haque, B. I. (2013). A new approach of Mickens’ iteration method for solving some nonlinear jerk equations. Global Journal of Sciences Frontier Research Mathematics and Decision Science, 13(11), 87-98.
VIII. Hossain, M. A., & Haque, B. I. (2021). A Solitary Convergent Periodic Solution of the Inverse Truly Nonlinear Oscillator by Modified Mickens’ Extended Iteration Procedure, Journal of Mechanics of Continua and Mathematical Sciences, 16(8), 1-9. 10.26782/jmcms.2021.08.00001
IX. Hossain, M. A., & Haque, B. I. (2022). Fixation of the Relation between Frequency and Amplitude for Nonlinear Oscillator Having Fractional Term Applying Modified Mickens’ Extended Iteration Method. Journal of Mechanics of Continua and Mathematical Sciences, 17(1), 88-103. 10.26782/jmcms.2022.01.00007
X. Hossain, M. A., & Haque, B. I. (2023). An Analytic Solution for the Helmholtz-Duffing Oscillator by Modified Mickens’ Extended Iteration Procedure. In Mathematics and Computing: ICMC 2022, Vellore, India, January 6–8 (pp. 689-700). Singapore: Springer Nature Singapore. 10.1007/978-981-19-9307-7_53

XI. Hu, H. (2008). Perturbation Method for Periodic Solutions of Nonlinear Jerk Equations. Physics letters A, 372(23), 4205-4209. 10.1016/j.physleta.2008.03.027

XII. Hu, H., Zheng, M. Y., & Guo, Y. J. (2010). Iteration Calculations of Periodic Solutions to Nonlinear Jerk Equations. Acta mechanica, 209(3-4), 269-274. 10.1007/s00707-009-0179-y

XIII. Leung, A. Y. T., & Guo, Z. (2011). Residue harmonic balance approach to limit cycles of non-linear jerk equations. International Journal of Non-Linear Mechanics, 46(6), 898-906. 10.1016/j.ijnonlinmec.2011.03.018

XIV. Ma, X., Wei, L., & Guo, Z. (2008). He’s homotopy perturbation method to periodic solutions of nonlinear Jerk equations. Journal of Sound and Vibration, 314(1-2), 217-227.
XV. Mickens, R. E. (2010). Truly nonlinear oscillations: harmonic balance, parameter expansions, iteration, and averaging methods. World Scientific.
XVI. Mickens, R. E. (1987). Iteration Procedure for Determining Approximate Solutions to Non-linear Oscillator Equations. Journal of Sound Vibration, 116(1), 185-187. 10.1016/s0022-460x(87)81330-5

XVII. Ramos, J. I. (2010). Approximate Methods Based on Order Reduction for the Periodic Solutions of Nonlinear Third-order Ordinary Differential Equations. Applied mathematics and computation, 215(12), 4304-4319. 10.1016/j.amc.2009.12.057

XVIII. Ramos, J. I., & Garcı, C. M. (2010). A Volterra Integral Formulation for Determining the Periodic Solutions of Some Autonomous, Nonlinear, Third-order Ordinary Differential Equations. Applied mathematics and computation, 216(9), 2635-2644. 10.1016/j.amc.2010.03.108

XIX. Ramos, J. I. (2010). Analytical and Approximate Solutions to Autonomous, Nonlinear, Third-order Ordinary Differential Equations. Nonlinear Analysis: Real World Applications, 11(3), 1613-1626. 10.1016/j.nonrwa.2009.03.023

XX. Wu, B. S., Lim, C. W., & Sun, W. P. (2006). Improved Harmonic Balance Approach to Periodic Solutions of Non-linear Jerk Equations. Physics Letters A, 354(1-2), 95-100.

XXI. Zheng, M. Y., Zhang, B. J., Zhang, N., Shao, X. X., & Sun, G. Y. (2013). Comparison of Two Iteration Procedures for a Class of Nonlinear Jerk Equations. Acta Mechanica, 224(1), 231-239. 10.1007/s00707-012-0723-z

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Arunava Bhattacharyya



Using TOAD based switch we have designed a parallel half-adder. The approach to designing all-optical arithmetic circuits not only enhances the computational speed but is also capable of synthesizing light as input to produce the desired output. The main advantage of a parallel circuit is the synchronization of input is not required. All the circuits are designed theoretically and verified through numerical simulations.


Terahertz optical asymmetric demultiplexer,semiconductor optical amplifier,half adder,optical logic,


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II. A. Bhattacharyya, D. K. Gayen, and T. Chattopadhyay, “Alternative All-optical Circuit of Binary to BCD Converter Using Terahertz Asymmetric Demultiplexer Based Interferometric Switch”, Proceedings of 1st International Conference on Computation and Communication Advancement (IC3A-2013).
III. A. Bhattacharyya, D. K. Gayen, ALL-OPTICAL N-BIT BINARY TO TWO’S COMPLEMENT CONVERTER WITH THE HELP OF SEMICONDUCTOR OPTICAL AMPLIFIER-ASSISTED SAGNAC SWITCH. J. Mech. Cont. & Math. Sci., Vol.-17, No.-1, January (2022) pp 117-125. 10.26782/jmcms.2022.01.00009.
IV. A. M. Melo, J. L. S. Lima, R. S. de Oliveira, and A. S. B. Sombra, “Photonic time Division Multiplexing (OTDM) using Ultra-short Picosecond Pulses in a Terahertz Optical Asymmetric Demultiplexer (TOAD)”, Optics Communications, 205(4-6), 299-312 (2002). 10.1109/SBMOMO.2001.1008820
V. B. Wang, V. Baby, W. Tong, L. Xu, M. Friedman, R. Runser, I. Glesk, and P. Prucnal, “A novel fast optical switch based on two cascaded terahertz optical asymmetric demultiplexers (TOAD)”, Optics Express, 10(1), 15-23 (2002).
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VIII. G. Li, “Recent advances in coherent optical communication”, Advances in Optics and Photonics, 1(2), 279-307 (2009).
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X. J. P. Sokoloff, P. R. Prucnal, I. Glesk, and M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD)”, IEEE Photonics Technology Letters, 5(7), 787-790 (1993).
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XII. J. P. Sokoloff, I. Glesk, P. R. Prucnal, and R. K. Boneck, “Performance of a 50 Gbit/s optical time domain multiplexed system using a terahertz optical asymmetric demultiplexer”, IEEE Photonics Technology Letters, 6(1), 98-100 (1994).
XIII. K. E. Zoiros, J. Vardakas, T. Houbavlis, and M. Moyssidis, “Investigation of SOA-assisted Sagnac recirculating shift register switching characteristics”, International Journal for Light and Electron Optics, 116(11), 527-541 (2005). 10.1016/j.ijleo.2005.03.005
XIV. M. Eiselt, W. Pieper, and H. G. Weber, “SLALOM: Semiconductor laser amplifier in a loop mirror”, Journal of Lightwave Technology, 13(10), 2099-2112 (1995). 10.1109/50.469721
XV. M Suzuki, H. Uenohara, “Invesigation of all-optical error detection circuitusing SOA-MZI based XOR gates at 10 Gbit/s”, Electron. Lett. 45 (4), (2009).

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Murtaza Ali Khooharo, Muhammad Mujtaba Shaikh, Ashfaque Ahmed Hashmani



As the most energy-intensive machines on the planet, induction motors are the subject of an ongoing study to increase their effectiveness. In this respect, new energy-efficient motors (NEEMs) are being developed. For increasing energy conservation, motors with efficiencies considerably higher than traditional standard motors (TSMs) and energy-efficient motors (EEMs) have been suggested. NEEMs have the potential to save a significant quantity of energy as well as operating costs. A comparative study is conducted in this paper to show how much energy and cost can be saved if TSMs in various industries in Pakistan are replaced with NEEMs, as well as their payback period. A data sample of 23 motors of different ratings has been collected in this pilot study and 90 percent confidence limits are calculated using a t-distribution. The energy conservation benefits of the NEEMs are found encouraging


Energy-efficient motors,energy conservation,payback,cost saving,energy saving,


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XXIV. Z. Zhang, “Analysis of Reluctance Torque in Interior Permanent Magnet Synchronous Machines With Fractional Slot Concentrated Windings,” in 2019 4th International Conference on Intelligent Green Building and Smart Grid (IGBSG), Sep. 2019, pp. 158–163. doi: 10.1109/IGBSG.2019.8886289.

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