A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems
Abstract
:1. Introduction
2. Definition and Properties of the Ψ-Functions and the Ψ-Functions Series Method
3. Ψ-Multistep Methods
3.1. Explicit Ψ-Multistep Method of p Steps for Perturbed and Damped Systems
3.2. Implicit Ψ-Multistep Method of p Steps for Perturbed and Damped Systems
3.3. Predictor–Corrector Ψ-Multistep Method for Perturbed and Damped Systems
4. Recurrent Calculus of the Matrices and and a New Definition of the Multistep Methods
4.1. Recurrent Calculus and a New Definition of the Explicit Ψ-Multistep Method
4.2. Recurrent Calculus and New Definition of the Implicit Ψ-Multistep Method
4.3. New Predictor–Corrector Ψ-Multistep Method
5. Numerical Experiments
- LSODE methods cause a numerical solution to be found using the Livermore Stiff ODE solver.
- GEAR finds a numerical solution using a Gear single-step extrapolation method.
- ROSENBROCK finds a numerical solution using an Implicit Rosenbrock third–fourth-order Runge–Kutta method with third-degree interpolant.
- DVERK78 finds a numerical solution using a seventh–eighth-order continuous Runge–Kutta method.
5.1. Problem 1
5.2. Problem 2
5.3. Problem 3
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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García-Alonso, F.; Reyes, J.A.; Cortés-Molina, M. A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems. Mathematics 2024, 12, 2018. https://doi.org/10.3390/math12132018
García-Alonso F, Reyes JA, Cortés-Molina M. A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems. Mathematics. 2024; 12(13):2018. https://doi.org/10.3390/math12132018
Chicago/Turabian StyleGarcía-Alonso, Fernando, José Antonio Reyes, and Mónica Cortés-Molina. 2024. "A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems" Mathematics 12, no. 13: 2018. https://doi.org/10.3390/math12132018
APA StyleGarcía-Alonso, F., Reyes, J. A., & Cortés-Molina, M. (2024). A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems. Mathematics, 12(13), 2018. https://doi.org/10.3390/math12132018