Existence of Bounded Solutions to a Modified Version of the Bagley–Torvik Equation
Abstract
:1. Introduction
2. Fundamental Notions about Fractional Calculus and the Bagley–Torvik Equation
2.1. Fundamental Notions of Fractional Calculus
2.2. Short Survey of Special Functions
2.3. Deduction of the Bagley–Torvik Equation
3. Stability for the Motion of a Plate Immersed in a Newtonian Fluid
3.1. Assimilable Cases
3.1.1. The Classical Pure Harmonic Oscillator
3.1.2. The Classical Damped Harmonic Oscillator
3.2. The Immersed Plate in a Fluid
- Solutions fulfilling .
- Solutions fulfilling .
3.2.1. Case
3.2.2. Case
- Now, we have coefficients for the degrees zero and .
- The rest of the coefficients coincide with those of the previous case, but after a multiplication by .
3.3. An Example
- If , then . In this case, .
- If , then . In this case, .
- If , then . In this case, .
4. Sufficient Conditions for Bounded Solutions
- 1.
- 2.
- 3.
Author Contributions
Funding
Conflicts of Interest
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Cao Labora, D.; Tenreiro Machado, J.A. Existence of Bounded Solutions to a Modified Version of the Bagley–Torvik Equation. Mathematics 2020, 8, 289. https://doi.org/10.3390/math8020289
Cao Labora D, Tenreiro Machado JA. Existence of Bounded Solutions to a Modified Version of the Bagley–Torvik Equation. Mathematics. 2020; 8(2):289. https://doi.org/10.3390/math8020289
Chicago/Turabian StyleCao Labora, Daniel, and José António Tenreiro Machado. 2020. "Existence of Bounded Solutions to a Modified Version of the Bagley–Torvik Equation" Mathematics 8, no. 2: 289. https://doi.org/10.3390/math8020289
APA StyleCao Labora, D., & Tenreiro Machado, J. A. (2020). Existence of Bounded Solutions to a Modified Version of the Bagley–Torvik Equation. Mathematics, 8(2), 289. https://doi.org/10.3390/math8020289