Modified semi-analytical approach for duffing equation

• Autorzy: Amara Um E , Rehman Shahida , Abbas Mujahid , Rehman Jamshaid Ul

Identyfikatory

DOI

10.2478/ama-2024-0033

• Języki publikacji: EN

Abstrakty

EN

This research endeavour-investigates the enhanced adaptation of the Laplace-based variational iteration method (VIM) tailored specifically for tackling the Duffing Equation. This is accomplished by incorporating the Lagrange multiplier as a strategic tool to effectively address the inherent natural frequency within the Duffing Equation. Using a meticulous comparative analysis, here are juxtapose the ana-lytical outcomes generated by the modified VIM approach with the numerical solution obtained through the application of the renowned Runge-Kutta Fehlberg method (RKF45), implemented by using the powerful mathematical software, MAPLE. Furthermore, by exploring the

Słowa kluczowe

EN

nonlinear dynamics; modified variational iterative method; Simulink model; mechanical vibration

• Wydawca: Oficyna Wydawnicza Politechniki Białostockiej
• Czasopismo: Acta Mechanica et Automatica
• Rocznik: 2024
• Tom: Vol. 18, no. 2
• Strony: 300--306
• Opis fizyczny: Bibliogr. 37 poz., tab., wykr.

Twórcy

autor

Amara Um E

• Department of Mathematics, Government College University, Lahore-54600, Pakistan

• https://orcid.org/0009-0009-0572-7368

autor

Rehman Shahida

• Abdus Salam School of Mathematical Sciences, Government College University, Lahore-54600, Pakistan

• https://orcid.org/0000-0001-7137-9079

autor

Abbas Mujahid

• Department of Mathematics, Government College University, Lahore-54600, Pakistan
• Abdus Salam School of Mathematical Sciences, Government College University, Lahore-54600, Pakistan

• https://orcid.org/0000-0001-5528-1207

autor

Rehman Jamshaid Ul

• Abdus Salam School of Mathematical Sciences, Government College University, Lahore-54600, Pakistan

• https://orcid.org/0000-0001-8642-0660

Bibliografia

• 1. He JH. Variational iteration method–a kind of non-linear analytical technique: some examples. International journal of non-linear me-chanics. 1999 Jul 1;34(4):699-708.
• 2. He JH. A short remark on fractional variational iteration method. Physics Letters A. 2011 Sep 5;375(38):3362-4.
• 3. Suleman M, Lu D, Yue C, Ul Rahman J, Anjum N. He–Laplace method for general nonlinear periodic solitary solution of vibration equations. Journal of Low Frequency Noise, Vibration and Active Control. 2019 Dec;38(3-4):1297-304.
• 4. He JH. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons & Fractals. 2004 Mar 1;19(4):847-51.
• 5. He JH. Variational approach to (2+1)-dimensional dispersive long water equations. Physics Letters A. 2005 Feb 7;335(2-3):182-4.
• 6. Ul Rahman J, Mohyuddin MR, Anjum N, Zahoor S. Mathematical modelling & simulation of mixing of salt in 3-interconnected tanks. Journal of Advances in Civil Engineering. 2015;1(1):1-6.
• 7. Anjum N, He JH. Analysis of nonlinear vibration of nano/microelectromechanical system switch induced by electromag-netic force under zero initial conditions. Alexandria Engineering Jour-nal. 2020 Dec 1;59(6):4343-52.
• 8. Ain QT, Anjum N, He CH. An analysis of time-fractional heat transfer problem using two-scale approach. GEM-International Journal on Geomathematics. 2021 Dec;12:1-0.
• 9. He, J. H., & El-Dib, Y. O. (2020). Homotopy perturbation method for Fangzhu oscillator. Journal of Mathematical Chemistry, 58, 2245-2253.
• 10. Ul Rahman J, Lu D, Suleman M, He JH, Ramzan M. He–Elzaki method for spatial diffusion of biological population. Fractals. 2019 Aug 13;27(05):1950069.
• 11. Suleman M, Lu D, He JH, Farooq U, Hui YS, Rahman JU. Numerical investigation of fractional HIV model using Elzaki projected differen-tial transform method. Fractals. 2018 Oct 5;26(05):1850062.
• 12. He CH, Liu C, He JH, Gepreel KA. Low frequency property of a fractal vibration model for a concrete beam. Fractals. 2021 Aug 25;29(05):2150117.
• 13. Anjum N, He JH. Higher-order homotopy perturbation method for conservative nonlinear oscillators generally and microelectromechan-ical systems’ oscillators particularly. International Journal of Modern Physics B. 2020 Dec 30;34(32):2050313.
• 14. Tian D, Ain QT, Anjum N, He CH, Cheng B. Fractal N/MEMS: from pull-in instability to pull-in stability. Fractals. 2021 Mar 10;29(02):2150030.
• 15. Ain QT, Anjum N, He CH. An analysis of time-fractional heat transfer problem using two-scale approach. GEM-International Journal on Geomathematics. 2021 Dec;12:1-0.
• 16. Ain QT, He JH, Anjum N, Ali M. The fractional complex transform: A novel approach to the time-fractional Schrödinger equation. Fractals. 2020 Nov 2;28(07):2050141.
• 17. Rehman S, Hussain A, Rahman JU, Anjum N, Munir T. Modified Laplace based variational iteration method for the mechanical vibra-tions and its applications. acta mechanica et automatica. 2022;16(2):98-102.
• 18. He JH. Some asymptotic methods for strongly nonlinear equations. International journal of Modern physics B. 2006 Apr 20;20(10): 1141-99.
• 19. Noor MA, Mohyud-Din ST. Variational iteration method for solving higher-order nonlinear boundary value problems using He's polyno-mials. International Journal of Nonlinear Sciences and Numerical Simulation. 2008 Jun;9(2):141-56.
• 20. He JH. Generalized equilibrium equations for shell derived from a generalized variational principle. Applied Mathematics Letters. 2017 Feb 1;64:94-100.
• 21. He JH. An alternative approach to establishment of a variational principle for the torsional problem of piezoelastic beams. Applied Mathematics Letters. 2016 Feb 1;52:1-3.
• 22. Wu Y, He JH. A remark on Samuelson’s variational principle in economics. Applied Mathematics Letters. 2018 Oct 1;84:143-7.
• 23. He JH. Variational iteration method—some recent results and new interpretations. Journal of computational and applied mathematics. 2007 Oct 1;207(1):3-17.
• 24. He JH, Wu XH. Variational iteration method: new development and applications. Computers & Mathematics with Applications. 2007 Oct 1;54(7-8):881-94.
• 25. He JH. Variational iteration method for autonomous ordinary differen-tial systems. Applied mathematics and computation. 2000 Sep 11;114(2-3):115-23.
• 26. He JH. Variational theory for linear magneto-electro-elasticity. Inter-national Journal of Nonlinear Sciences and Numerical Simulation. 2001 Dec;2(4):309-16.
• 27. He J. Variational iteration method for delay differential equations. Communications in Nonlinear Science and Numerical Simulation. 1997 Dec 1;2(4):235-6.
• 28. Petrova Z, Puleva T. Mathematical modeling of the equation of Duffing with applications for master degree students—Part I. InAIP Conference Proceedings 2018 Dec 10 (Vol. 2048, No. 1). AIP Pub-lishing.
• 29. Kanamaru T. Duffing oscillator. Scholarpedia. 2008 Mar 25; 3(3):6327.
• 30. Savov VN, Georgiev ZD, Todorov TG. Analysis and synthesis of perturbed Duffing oscillators. International journal of circuit theory and applications. 2006 May;34(3):281-306.
• 31. Tao H, Anjum N, Yang YJ. The Aboodh transformation-based ho-motopy perturbation method: new hope for fractional calculus. Fron-tiers in Physics. 2023 Apr 27;11:1168795.
• 32. Anjum N, He JH. Laplace transform: making the variational iteration method easier. Applied Mathematics Letters. 2019 Jun 1;92:134-8.
• 33. Anjum N, Suleman M, Lu D, He JH, Ramzan M. Numerical iteration for nonlinear oscillators by Elzaki transform. Journal of Low Frequen-cy Noise, Vibration and Active Control. 2020 Dec;39(4):879-84.
• 34. Yildirim A, Saadatnia Z, Askari H, Khan Y, KalamiYazdi M. Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach. Applied Mathematics Letters. 2011 Dec 1;24(12):2042-51.
• 35. Rehman S, Muhammad N. Mathematical analysis of nonlinear mod-els and their role in dynamics. Modern Physics Letters B. 2023 Oct 31:2450097.
• 36. Shah NA, Rehman S, Vieru D, Yook SJ. Unsteady flows of micropo-lar fluids parallel to the axis of an annular domain with a porous layer. Alexandria Engineering Journal. 2023 Aug 1;76:275-87.
• 37. Rehman S, Muhammad N, Alshehri M, Alkarni S, Eldin SM, Shah NA. Analysis of a viscoelastic fluid flow with Cattaneo–Christov heat flux and Soret–Dufour effects. Case Studies in Thermal Engineering. 2023 Sep 1;49:103223.