Convergence Analysis of the Straightforward Expansion Perturbation Method for Weakly Nonlinear Vibrations
Abstract
:1. Introduction
2. Materials and Methods
2.1. The Banach Space Unitary Algebra
- 1.
- There exists such that is convergent.
- 2.
- There exists such that is -convergent to g.
- 1.
- .
- 2.
- .
- 3.
- If f is invertible, then .
- 1.
- It is clear since
- 2.
- We will prove, by induction on , thatFor . It is clear that (3) holds if . Since is a Banach algebra endowed with the sup norm, we have thatFor . By using the product rule, we have that , thereforeBy induction hypothesis, , so we conclude thatFor . For the second derivative, we have , soBy induction hypothesis, and , thus we obtain thatNext, we will prove that by relying on our induction Hypothesis (4). Notice that it is sufficient to show that for all . If , by (4) we know thatIt only remains to show that . Observe that
- 3.
- Finally, if f is invertible, by applying Theorem 1(2),
2.2. Perturbation to Second-Order Linear Ordinary Differential Equations with Constant Coefficients
2.3. Functions of Polynomial Behavior
3. Results
Algorithm of Application for the Straightforward Expansion Perturbation Method
- is the unique solution of the second-order linear IVP
- is the unique solution of the second-order linear IVP
- is the unique solution of the second-order linear IVP
- And so on.
4. Discussion
Validation Example (Pendulum) for a Fixed Bounded Time Interval
- is the unique solution of the IVP
- is the unique solution of the IVP
- is the unique solution of the IVP
- And so on.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
SEPM | Straightforward Expansion Perturbation Method |
HAM | Homotopy Analysis Method |
ODE | Ordinary Differential Equation |
IVP | Initial Value Problem |
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Moreno-Pulido, S.; García-Pacheco, F.J.; Sánchez-Alzola, A.; Rincón-Casado, A. Convergence Analysis of the Straightforward Expansion Perturbation Method for Weakly Nonlinear Vibrations. Mathematics 2021, 9, 1036. https://doi.org/10.3390/math9091036
Moreno-Pulido S, García-Pacheco FJ, Sánchez-Alzola A, Rincón-Casado A. Convergence Analysis of the Straightforward Expansion Perturbation Method for Weakly Nonlinear Vibrations. Mathematics. 2021; 9(9):1036. https://doi.org/10.3390/math9091036
Chicago/Turabian StyleMoreno-Pulido, Soledad, Francisco Javier García-Pacheco, Alberto Sánchez-Alzola, and Alejandro Rincón-Casado. 2021. "Convergence Analysis of the Straightforward Expansion Perturbation Method for Weakly Nonlinear Vibrations" Mathematics 9, no. 9: 1036. https://doi.org/10.3390/math9091036