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Critical analysis for nonlinear oscillations by least square HPM.
Muhammad Rafiq1, Muhammad Kamran1, Hijaz Ahmad2,3,4,5
1Department of Mathematics, COMSATS University Islamabad, Wah Campus, Islamabad, Pakistan.
Scientific Reports
|January 16, 2024
Summary
A new Least Square Homotopy Perturbation Method (LSHPM) effectively solves nonlinear free vibration problems. This reliable and efficient technique shows promise for complex science and engineering initial value problems.
Area of Science:
- Mechanical Engineering
- Applied Mathematics
Background:
- Nonlinear phenomena in systems with one degree of freedom present significant analytical challenges.
- Existing methods like the Homotopy Perturbation Method (HPM) require adaptation for enhanced accuracy and efficiency.
Purpose of the Study:
- To introduce and evaluate a novel hybrid method, the Least Square Homotopy Perturbation Method (LSHPM).
- To assess the efficacy of LSHPM in solving nonlinear free vibration problems.
Main Methods:
- Integration of the Homotopy Perturbation Method (HPM) with a least-squares optimizer to form LSHPM.
- Application of LSHPM to benchmark nonlinear problems from existing literature.
- Comparative analysis with HPM, MATLAB's bvp5c, and the AG method.
Main Results:
- LSHPM demonstrated high reliability and efficiency across various nonlinear problems.
- The proposed method achieved comparable or superior results to existing techniques.
- Validation of LSHPM's performance against established numerical solvers.
Conclusions:
- LSHPM is a robust and efficient numerical method for addressing nonlinear free vibration.
- The technique is well-suited for solving more complex initial value problems in science and engineering.
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