Analytical Periodic Solutions for Non-Homogenous Integrable Dispersionless Equations Using a Modified Harmonic Balance Method

Muhammad Irfan Khan, Yiu-Yin Lee () and Muhammad Danish Zia
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Muhammad Irfan Khan: Military College of Engineering Risalpur, National University of Sciences and Technology, Islamabad 44000, Pakistan
Yiu-Yin Lee: Department of Architecture and Civil Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Muhammad Danish Zia: Department of Basic Sciences and Humanities, National University of Sciences and Technology, Islamabad 44000, Pakistan

Mathematics, 2025, vol. 13, issue 15, 1-17

Abstract: In this study, we outline a modified harmonic balance method for solving non-homogenous integrable dispersionless equations and obtaining the corresponding periodic solutions, a research field which shows limited investigation. This study is the first to solve this nonlinear problem, based on a recently developed harmonic balance method combined with Vieta’s substitution technique. A set of analytical formulas are generated from the modified harmonic balance method and used to compute the approximate periodic solutions of the dispersionless equations. The main advantage of this method is that the computation effort required in the solution procedure can be smaller. The results of the modified harmonic balance method show reasonable agreement with those obtained using the classic harmonic balance method. Our proposed solution method can decouple the nonlinear algebraic equations generated in the harmonic balance process. We also investigated the effects of various parameters on nonlinear periodic responses and harmonic convergence.

Keywords: Vieta’s substitution; integrable-dispersionless equation; harmonic balance; periodic solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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