Solutions of the nonlinear evolution problems and their applications

• Autorzy: Amir Muhammad , Haider Jamil Abbas , Rahman Jamshaid Ul , Ashraf Asifa

Identyfikatory

DOI

10.2478/ama-2023-0040

• Języki publikacji: EN

Abstrakty

EN

In this article, a well-known technique, the variational iterative method with the Laplace transform, is used to solve nonlinear evolution problems of a simple pendulum and mass spring oscillator, which represents the duffing equation. In the variational iteration method (VIM), finding the Lagrange multiplier is an important step, and the variational theory is often used for this purpose. This paper shows how the Laplace transform can be used to find the multiplier in a simpler way. This method gives an easy approach for scientists and engineers who deal with a wide range of nonlinear problems. Duffing equation is solved by different analytic methods, but we tackle this for the first time to solve the duffing equation and the nonlinear oscillator by using the Laplace-based VIM. In the majority of cases, Laplace variational iteration method (LVIM) just needs one iteration to attain high accuracy of the answer for linearization anddiscretization, or intensive computational work is needed. The convergence criteria of this method are efficient as compared with the VIM. Comparing the analytical VIM by Laplace transform with MATLAB’s built-in command Simulink that confirms the method’s suitability for solving nonlinear evolution problems will be helpful. In future, we will be able to find the solution of highly nonlinear oscillators.

Słowa kluczowe

EN

Laplace variational iteration method; nonlinear problems; duffing equation; simple pendulum; mass and spring oscillator; Simulink

• Wydawca: Oficyna Wydawnicza Politechniki Białostockiej
• Czasopismo: Acta Mechanica et Automatica
• Rocznik: 2023
• Tom: Vol. 17, no. 3
• Strony: 357--363
• Opis fizyczny: Bibliogr. 33 poz., rys., tab., wykr.

Twórcy

autor

Amir Muhammad

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

• https://orcid.org/0009-0002-4871-4312

autor

Haider Jamil Abbas

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

• https://orcid.org/0000-0002-7008-8576

autor

Rahman Jamshaid Ul

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

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

autor

Ashraf Asifa

• Department of Mathematics, University of Management and Technology Lahore, Pakistan

• https://orcid.org/0009-0005-4786-7757

Bibliografia

• 1. Asghar S, Haider JA, Muhammad N. The modified KdV equation for a nonlinear evolution problem with perturbation technique. Inter-national Journal of Modern Physics B. 2022 Sep 30;36(24):2250160.
• 2. Amir M, Awais M, Ashraf A, Ali R, Ali Shah SA. Analytical Method for Solving Inviscid Burger Equation. Punjab University Journal of Math-ematics. 2023 Dec 3;55(1).
• 3. Ali M, Anjum N, Ain QT, He JH. Homotopy perturbation method for the attachment oscillator arising in nanotechnology. Fibers and Pol-ymers. 2021 Jun;22:1601-6.
• 4. Samadi H, Mohammadi NS, Shamoushaki M, Asadi Z, Ganji DD. An analytical investigation and comparison of oscillating systems with nonlinear behavior using AGM and HPM. Alexandria Engineering Journal. 2022 Nov 1;61(11):8987-96.
• 5. Koochi A, Goharimanesh M. Nonlinear oscillations of CNT nano-resonator based on nonlocal elasticity: The energy balance method. Reports in Mechanical Engineering. 2021 Feb 20;2(1):41-50.
• 6. 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.
• 7. Lu J, Ma L. The VIM-Pade technique for strongly nonlinear oscillators with cubic and harmonic restoring force. Journal of Low Frequency Noise, Vibration and Active Control. 2019 Dec;38(3-4):1272-8.
• 8. Ren ZF, Yao SW, He JH. He’s multiple scales method for nonlinear vibrations. Journal of Low Frequency Noise, Vibration and Active Control. 2019 Dec;38(3-4):1708-12.
• 9. Xu L. Application of He's parameter-expansion method to an oscilla-tion of a mass attached to a stretched elastic wire. Physics Letters A. 2007 Aug 20;368(3-4):259-62.
• 10. Aljahdaly NH, El-Tantawy SA. On the multistage differential trans-formation method for analyzing damping Duffing oscillator and its ap-plications to plasma physics. Mathematics. 2021 Feb 22;9(4):432.
• 11. Seadawy AR, Cheemaa N. Some new families of spiky solitary waves of one-dimensional higher-order K-dV equation with power law nonlinearity in plasma physics. Indian Journal of Physics. 2020 Jan;94(1):117-26.
• 12. Seadawy AR. Stability analysis for Zakharov–Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Computers & Mathematics with Applications. 2014 Jan 1;67(1):172-80.
• 13. Rizvi ST, Seadawy AR, Ashraf F, Younis M, Iqbal H, Baleanu D. Lump and interaction solutions of a geophysical Korteweg–de Vries equation. Results in Physics. 2020 Dec 1;19:103661.
• 14. Marin M, Seadawy A, Vlase S, Chirila A. On mixed problem in ther-moelasticity of type III for Cosserat media. Journal of Taibah Univer-sity for Science. 2022 Dec 31;16(1):1264-74.
• 15. Ali I, Seadawy AR, Rizvi ST, Younis M, Ali K. Conserved quantities along with Painleve analysis and Optical solitons for the nonlinear dynamics of Heisenberg ferromagnetic spin chains model. Interna-tional Journal of Modern Physics B. 2020 Dec 10;34(30):2050283.
• 16. Ebrahimi F, Seyfi A, Dabbagh A. A novel porosity-dependent ho-mogenization procedure for wave dispersion in nonlocal strain gradi-ent inhomogeneous nanobeams. The European Physical Journal Plus. 2019 May;134:1-1.
• 17. 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.
• 18. Rahman JU, Mannan A, Ghoneim ME, Yassen MF, Haider JA. Insight into the study of some nonlinear evolution problems: Applica-tions based on Variation Iteration Method with Laplace. International Journal of Modern Physics B. 2023 Jan 30;37(03):2350030.
• 19. He JH, Latifizadeh H. A general numerical algorithm for nonlinear differential equations by the variational iteration method. International Journal of Numerical Methods for Heat & Fluid Flow. 2020 Feb 20.
• 20. Haider JA, Muhammad N. Computation of thermal energy in a rec-tangular cavity with a heated top wall. International Journal of Mod-ern Physics B. 2022 Nov 20;36(29):2250212.
• 21. Haider JA, Ahmad S. Dynamics of the Rabinowitsch fluid in a re-duced form of elliptic duct using finite volume method. International Journal of Modern Physics B. 2022 Dec 10;36(30):2250217.
• 22. Anjum N, He JH. Laplace transform: making the variational iteration method easier. Applied Mathematics Letters. 2019 Jun 1;92:134-8.
• 23. Bush A. Perturbation methods for engineers and scientists. Routledge; 2018 May 4.
• 24. Beléndez A, Beléndez T, Martínez FJ, Pascual C, Alvarez ML, Arri-bas E. Exact solution for the unforced Duffing oscillator with cubic and quintic nonlinearities. Nonlinear Dynamics. 2016 Nov;86:1687-700.
• 25. Suleman M, Wu Q. Comparative solution of nonlinear quintic cubic oscillator using modified homotopy perturbation method. Advances in Mathematical Physics. 2015 Jan 1;2015.
• 26. Ganji DD, Gorji M, Soleimani S, Esmaeilpour M. Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method. Journal of Zhejiang University-Science A. 2009 Sep;10:1263-8.
• 27. Daeichin M, Ahmadpoor MA, Askari H, Yildirim A. Rational energy balance method to nonlinear oscillators with cubic term. Asian-European Journal of Mathematics. 2013 Jun 17;6(02):1350019.
• 28. He JH, Jiao ML, Gepreel KA, Khan Y. Homotopy perturbation meth-od for strongly nonlinear oscillators. Mathematics and Computers in Simulation. 2023 Feb 1;204:243-58.
• 29. Wang KJ, Wang GD. Gamma function method for the nonlinear cubic-quintic Duffing oscillators. Journal of Low Frequency Noise, Vi-bration and Active Control. 2022 Mar;41(1):216-22.
• 30. Durmaz S, Kaya MO. High-order energy balance method to nonlinear oscillators. Journal of Applied Mathematics. 2012 Jan 1;2012.
• 31. Durmaz S, Demirbağ SA, Kaya MO. High order He's energy balance method based on collocation method. International Journal of Non-linear Sciences and Numerical Simulation. 2010 Dec 1;11 (Supple-ment):1-6.
• 32. Wu BS, Sun WP, Lim CW. An analytical approximate technique for a class of strongly non-linear oscillators. International Journal of Non-Linear Mechanics. 2006 Jul 1;41(6-7):766-74.
• 33. Ali M, Anjum N, Ain QT, He JH. Homotopy perturbation method for the attachment oscillator arising in nanotechnology. Fibers and Pol-ymers. 2021 Jun;22:1601-6.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).