An Analytical Approach to an Elastic Circular Rod Equation

Year 2022, Volume: 10 Issue: 1, 45 - 49, 01.03.2022

Zehra Pinar

https://doi.org/10.36753/mathenot.799596

Abstract

The size-dependent longitudinal and torsional dynamic problems for small-scaled rods have importance in two-phase media. The special case of the elastic rod equation such as magneto-electro circular equation are seen in the literature commonly, but in this work, the generalized form of the nonlinear elastic circular equation, which was not studied in the literature, is considered. The exact solutions are obtained via Mathieu approximation method with a novel proposed ansatz. Obtained solutions are discussed and illustrated in details. We believe that the proposed results will be key part of further analytical and numerical studies for waves in the dispersive medium with reaction.

Keywords

Mathieu approximation method, the elastic rod equation, travelling wave solutions.

References

• [1] Abdou, M.A.: Exact travelling wave solutions in a nonlinear elastic rod equation. .Int. J. Nonlinear Sci. 7 (2), 167-173 (2009).
• [2] Zhuang, W.,Guiltong, Y.: The propagation of solitary wave in a nonlinear elastic rod. Applied Mathematics and Mechanics . 7, 615-626 (1986).
• [3] Zhuang, W., Zhang, S.Y.: The strain solitary wave in a nonlinear elastic rod. Acta Mech. Sin. 3 , 62-72 (1987).
• [4] Duan,W.S., Zhao, J.B.: Solitary waves in a cuadratic nonlinear elastic rod. Chaos Solitons and Fract. 11, 1265-1267 (2000).
• [5] Li, J., Zhang,Y.: Exact traveling wave solutions in a nonlinear elastic rod equation. Appl. Math. and Comp., 202 , 504-510 (2008).
• [6] Kabir,M.M.:Exacttravelngwavesolutionsfornonlinearelasticrodequation.JournalofKingSaudUniversity-Science, 31 (3), 390-397 (2019).
• [7] Gomez,C.A.,Garzon,H.G.,Hernandez,J.C.:OnElasticRodEquationwithForcingTerm:TravelingWaveSolutions. Contemporary Engineering Sciences, 11 (4), 173 – 181 (2018).
• [8] Dai,H.-H.,Kowloon,Kong,H.:ModelequationsfornonlineardispersivewavesinacompressibleMooney-Rivlinrod., Acta Mechanica 127, 193-207 (1998).
• [9] Li , L., Hu,Y., Li, X.: Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory. International Journal of Mechanical Sciences 115-116, 135–144 (2016).
• [10] Zhuang,W., Zhang,G.T.: The propagation of solitary waves in a nonlinear elastic rod. Appl. Math. Mech. 7, 615–626 (1986).
• [11] Lu,K.P.,Guo,P.,Zhang,L.,Yi,J.Q.,Duan,W.S.:Perturbationanalysisforwaveequationofnonlinearelasticrod.Appl. Math. Mech.27 , 1233–1238 (2006).
• [12] Dai,H.H., Huo,Y.: Solitary waves in an inhomogeneous rod composed of a general hyperelastic material. Acta Mech. 35, 55–69 (2002).
• [13] Porubov, A.V., Velarde, M.G.:Strain kinks in an elastic rod embedded in a viscoelastic medium. Wave Motion 35 , 189–204 (2002).
• [14] Pinar, Z., Kocak, H.:Exact solutions for the third-order dispersive-Fisher equations. Nonlinear Dynamics. 91 (1), 421-426(2018).
• [15] Kocak,H.,Pinar,Z.:Onsolutionsofthefifth-orderdispersiveequationswithporousmediumtypenon-linearity.Waves in Random and Complex Media. 28 (3), 516-522 (2018).
• [16] Pinar, Z., Özis ̧, T.:Observations on the class of “Balancing Principle” for nonlinear PDEs that can be treated by the auxiliary equation method. Nonlinear Analysis: Real World Applications. 23, 9–16(2015).
• [17] Baskonus,H.M.,Bulut,H.,Atangana,A.:OntheComplexandHyperbolicStructuresofLongitudinalWaveEquation in a Magneto-Electro-Elastic Circular Rod. Smart Materials and Structures. 25(3), (2016).
• [18] Baskonus, H. M., Gómez-Aguilar, J. F. :New Singular Soliton Solutions to the longitudinal Wave Equation in a Magneto Electro-elastic Circular Rod with Local M-derivative. Modern Physics Letters B. 33(21), 1–16(2019).
• [19] Ilhan,O.A.,Bulut,H.,Sulaiman,T.A.,Baskonus,H.M.:OnthenewwavebehavioroftheMagneto-Electro-Elastic (MEE) circular rod longitudinal wave equation. An International Journal of Optimization and Control: Theories and Applications 10(1), 1–8 (2020).
• [20] Bulut,H.,Sulaiman,T.A.,Baskonus,H.M.:OnthesolitarywavesolutionstothelongitudinalwaveequationinMEE circular rod. Opt. Quant. Electron. 50, (2018).