### THE UP-GRATING RANK APPROACH TO SOLVE THE FORCED FRACTAL DUFFING OSCILLATOR BY NON- PERTURBATIVE TECHNIQUE

Yusry El-Dib, Nasser S. Elgazery, Haifa A. Alyousef

DOI Number
10.22190/FUME230605035E
First page
Last page

#### Abstract

The current research studies a fractal Duffing oscillator in the presence of periodic force. To find an analytic solution for this oscillator, the aspects explained in the following are considered. First, we obtain an alternative unforced fractal fourth-order equation and then convert it into a continuous space. Therefore, the non-perturbative (NP) approach is used to calculate the analytic solution for the alternate equation in the second-order form after reducing its rank. It is seen that the analytical and numerical solutions agree very well. The computations reveal that for every value of the fraction parameter, the approximation and numerical solutions are identical. The present study gives reliability in the technique of reducing the order of differential equations. Furthermore, the required periodic solution is also obtained by Galerkin’s technique. In contrast to the traditional technique, which works to transform the variable and is valid only in the absence of external forces, if there is an external force, it leads to significant mathematical difficulties. The current technique works on the operator, which is simple and effective when investigating fractal oscillators with external forces, easy to obtain analytic solutions, and doesn't lead to any mathematical difficulties.

#### Keywords

Forced fractal Duffing oscillator, Non-perturbative (NP) technique, Fractal space, Galerkin’s approach

PDF

#### References

Zmeskal, O., Dzik, P., Vesely M., 2013, Entropy of fractal systems, Computers & Mathematics with Applications, 66(2), pp. 135-146.

Edgar , G. A., 2008, Measure, Topology, and Fractal Geometry, Edition Number 2, Springer New York, NY.

Chen, M., Yu, B., Xu, P., Chen, J., 2007, A new deterministic complex network model with hierarchical structure, Physica A: Statistical Mechanics and its Applications, 385(2), pp. 707-717.

Yang, S., Fu, H., Yu, B., 2017, Fractal analysis of flow resistance in tree-like branching networks with roughened microchannels, Fractals, 25(1), 1750008.

Miao, T., Chen, A., Xu, Y., Yang, S., Yu, B., 2016, Optimal structure of damaged tree-like branching networks for the equivalent thermal conductivity, International Journal of Thermal Sciences, 102, pp. 89–99.

Liang, M., Gao, Y., Yang, S., Xiao, B., Wang, Z., Li, Y., 2018, An analytical model for two-phase relative permeability with Jamin effect in porous media, Fractals, 26(3), 1850037.

He, J.-H., Kou, S.-J., He, C.-H., Zhang, Z.-W., Gepreel, K. A., 2021, Fractal oscillation and its frequency-amplitude property, Fractals, 29(4), 2150105.

Wang, K.-L., 2022, Exact solitary wave solution for fractal shallow water wave model by He’s variational method, Modern Physics Letters B, 36(7), 2150602.

Ain, Q.T., He, J.-H., 2019, On two-scale dimension and its applications, Thermal Science, 23(3) Part B, pp. 1707-1712.

Li, X., Liu, Z., He, J.-H., 2020, A fractal two-phase flow model for the fiber motion in a polymer filling process, Fractals, 28(5), 2050093.

Zuo, Y., 2021, A gecko-like fractal receptor of a three-dimensional printing technology: a fractal oscillator, Journal of Mathematical Chemistry, 59(3), pp. 735-744.

Liu, C., 2021, Periodic solution of fractal Phi-4 equation, Thermal Science, 25(2) Part B, pp. 1345–1350.

He, C.-H., Liu, C., He, J.-H., Gepreel, K.A., 2021, Low frequency property of a fractal vibration model for a concrete beam, Fractals, 29(5), 2150117.

Anjum, N., He, C.-H., He, J.-H., 2021, Two-scale fractal theory for the population dynamics, Fractals, 29(7), 2150182.

Anjum, N., Ain, Q.T., Li, X.X., 2021, Two-scale mathematical model for tsunami wave, GEM - International Journal on Geomathematics, 12(1), 10.

Nayfeh, A.H., 2011, Introduction to Perturbation Techniques, John Wiley & Sons.

He, J.-H., 1999, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178(3-4), pp. 257–262.

Wang, K.-L., 2021, A new fractal transform frequency formulation for fractal nonlinear oscillators, Fractals, 29(3), 2150062.

Wang, K.-L., 2021, He’s frequency formulation for fractal nonlinear oscillator arising in a microgravity space, Numerical Methods for Partial Differential Equations, 37(2), pp. 1374-1384.

Wang, K.-L., Wei, C.-F., 2021, A powerful and simple frequency formula to nonlinear fractal oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 40(3), pp. 1373-1379

Lu, J., Chen, L., 2022, Numerical analysis of a fractal modification of Yao-Cheng oscillator, Results in Physics, 105602.

Elías-Zúñiga, A., Martínez-Romero, O., Olvera-Trejo, D., Palacios-Pineda, L.M., 2021, An efficient ancient Chinese algorithm to investigate the dynamics response of a fractal microgravity forced oscillator, Fractals, 29(6), 2150144.

Elías-Zúñiga, A., Martínez-Romero, O., Olvera-Trejo, D., Palacios-Pineda, L.M., 2021, Investigation of the steady-state solution of the fractal forced Duffing’s oscillator using an ancient Chinese algorithm, Fractals, 29(6), 2150133.

Sheng, M., Li, G., Tian, S., Huang, Z., Chen, L., 2016, A fractal permeability model for shale matrix with multi-scale porous structure, Fractals, 24(1), 1650002.

Miao, T., Chen, A., Xu, Y., Cheng, S., Yu, B., 2019, A fractal permeability model for porous–fracture media with the transfer of fluids from porous matrix to fracture, Fractals, 27(6), 1950121.

Xu, J., Wu, K., Li, R., Li, Z., Li, J., Xu, Q., Chen, Z., 2018, Real gas transport in shale matrix with fractal structures, Fuel, 219, pp. 353-363.

Wang, S., Wu, T., Cao, X., Zheng, Q., Ai, M., 2017, A fractal model for gas apparent permeability in microfractures of tight/shale reservoirs, Fractals, 25(3), 1750036.

Wu, T., Wang, S., 2020, A fractal permeability model for real gas in shale reservoirs coupled with Knudsen diffusion and surface diffusion effects, Fractals, 28(1), 2050017.

Yang, C.-Y., Zhang, Y.-D., Yang, X.-J., 2016, Exact solutions for the differential equations in fractal heat transfer, Thermal Science, 20(suppl. 3), pp. 747-750.

El-Dib, Y.O., Elgazery, N.S., 2022, A novel pattern in a class of fractal models with the non-perturbative approach, Chaos, Solitons & Fractals, 164, 112694.

El-Dib, Y.O., Elgazery, N.S., 2023, An efficient approach to converting the damping fractal models to the traditional system, Communications in Nonlinear Science and Numerical Simulation, 118, 107036.

Selivanov, M.F., Chornoivan, Y.O., 2018, A semi-analytical solution method for problems of cohesive fracture and some of its applications, International Journal of Fracture, 212, pp. 113-121.

Khater, M.M.A., Mousa, A.A., El-Shorbagy, M.A., Attia, R.A.M., 2021, Analytical and semi-analytical solutions for Phi-four equation through three recent schemes, Results in Physics, 22, 103954.

Moshtaghi, N., Saadatmandi, A., 2021, Numerical solution of time fractional cable equation via the sinc-bernoulli collocation method, Journal of Applied and Computational Mechanics, 7(4), pp. 1916-1924.

Elías-Zúñiga, A., Martínez-Romero, O., Olvera-Trejo, D., Palacios-Pineda, L.M., 2021, Exact steady-state solution of fractals damped, and forced systems, Results in Physics, 28, 104580.

He, J.-H., 2019, The simplest approach to nonlinear oscillators, Results in Physics, 15(2019), 102546.

He, J.-H., 2019, The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4), pp. 1252-1260.

El-Dib, Y.O., 2023, Insightful and comprehensive formularization of frequency amplitude formula for strong or singular nonlinear oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 42(1), pp. 89-109.

El-Dib, Y.O., Alyousef, H.A., 2023, Successive approximate solutions for nonlinear oscillation and improvement of the solution accuracy with efficient non-perturbative technique, Journal of Low Frequency Noise, Vibration and Active Control, doi: 10.1177/14613484231161425.

Feng, G.-Q., 2021, He’s frequency formula to fractal undamped Duffing equation, Journal of Low Frequency Noise, Vibration and Active Control, 40(4), pp. 1671-1676.

Elías-Zúñiga, A., Palacios-Pineda, L.M., Jimenez-Cedeno, I. H., Martínez-Romero, O., Olvera-Trejo, D., 2021, Analytical solution of the fractal cubic–quintic Duffing equation, Fractals, 29(4), 2150080.

Elías-Zúñiga, A., Palacios-Pineda, L.M., Martínez-Romero, O., Olvera-Trejo, D., 2021, Dynamics response of the forced Fangzhu fractal device for water collection from air, Fractals, 29(7), 2150186.

Podlubny, I., 1998, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier.

Abbas, S., Benchohra, M., N'Guérékata, G.M., 2012, Topics in Fractional Differential Equations, Springer Science & Business Media.

Wang, K.-L., Liu, S.-Y., 2016, He’s fractional derivative for non-linear fractional heat transfer equation, Thermal Science, 20(3), pp. 793-796.

Wang, K.-L., Liu, S.-Y., 2017, He’s fractional derivative and its application for fractional Fornberg-Whitham equation, Thermal Science, 21(5), pp. 2049-2055.

Wu, Y., Liu, Y.-P., 2021, Residual calculation in He’s frequency–amplitude formulation, Journal of Low Frequency Noise, Vibration and Active Control, 40(2), pp. 1040-1047.

He, J.-H., Ain, Q.T., 2020, New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle, Thermal Science, 24(2), pp. 659-681.

El-Dib, Y.O., Elgazery, N.S., Alyousef, H.A., 2023, Galerkin’s method to solve a fractional time-delayed jerk oscillator, Archive of Applied Mechanics, 93, pp. 3597-3607.

El-Dib, Y.O., Elgazery, N.S., Alyousef, H.A., 2023, An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation, Journal of Low Frequency Noise, Vibration and Active Control, doi: 10.1177/14613484231185907.

He, J.-H., 2018, Fractal calculus and its geometrical explanation, Results in Physics, 10, pp. 272-276.

El-Dib, Y.O., 2022, An efficient approach to solving fractional Van der Pol–Duffing jerk oscillator, Communications in Theoretical Physics., 74, 105006.

El-Dib, Y.O., 2022, The simplest approach to solving the cubic nonlinear jerk oscillator with the non-perturbative method, Mathematical Methods in the Applied Sciences, 45(9), pp. 5165-5183.

El-Dib, Y.O., Elgazery, N.S., Khattab, Y.M., Alyousef, H.A., 2023, An innovative technique to solve a fractal damping Duffing-jerk oscillator, Communications in Theoretical Physics, 75, 055001.

### Refbacks

• There are currently no refbacks.

ISSN: 0354-2025 (Print)

ISSN: 2335-0164 (Online)

COBISS.SR-ID 98732551

ZDB-ID: 2766459-4