{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:42:49Z","timestamp":1760035369505,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T00:00:00Z","timestamp":1751846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University","award":["IMSIU-DDRSP2502"],"award-info":[{"award-number":["IMSIU-DDRSP2502"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study investigates the solution structure of the nonlinear Rayleigh oscillator equation through two widely used semi-analytical techniques: the Laplace\u2013Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM). The Rayleigh oscillator exhibits inherent asymmetry in its nonlinear damping term, which disrupts the time-reversal symmetry present in linear oscillatory systems. Applying the LADM and HPM, we derive approximate solutions for the Rayleigh oscillator. Due to the absence of exact analytical solutions in the literature, these approximations are benchmarked against high-precision numerical results obtained using Mathematica\u2019s NDSolve function. We perform a detailed error analysis across different damping parameter values \u03b5 and time intervals. Our results reveal how the asymmetric damping influences the accuracy and convergence behavior of each method. This study highlights the role of nonlinear asymmetry in shaping the solution dynamics and provides insight into the suitability of the LADM and HPM under varying conditions.<\/jats:p>","DOI":"10.3390\/sym17071081","type":"journal-article","created":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T06:03:13Z","timestamp":1751868193000},"page":"1081","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Semi-Analytical Solutions of the Rayleigh Oscillator Using Laplace\u2013Adomian Decomposition and Homotopy Perturbation Methods: Insights into Symmetric and Asymmetric Dynamics"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5335-7060","authenticated-orcid":false,"given":"Emad K.","family":"Jaradat","sequence":"first","affiliation":[{"name":"Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1145-0256","authenticated-orcid":false,"given":"Omar","family":"Alomari","sequence":"additional","affiliation":[{"name":"College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait"}]},{"given":"Audai A.","family":"Al-Zgool","sequence":"additional","affiliation":[{"name":"Department of Physics, Faculty of Science, Mutah University, Al-Karak 61710, Jordan"}]},{"given":"Omar K.","family":"Jaradat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mutah University, Al-Karak 61710, Jordan"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,7]]},"reference":[{"key":"ref_1","unstructured":"Baron Rayleigh, J.W.S. 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