The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
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J. A. Tenreiro Machado
Abstract
This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.
Acknowledgements
The author thanks the reviewers for the constructive comments that helped improving considerably the paper.
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© 2021 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 24–4–2021)
- Research Paper
- Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
- Tutorial paper
- The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
- Research Paper
- Fractional diffusion-wave equations: Hidden regularity for weak solutions
- Censored stable subordinators and fractional derivatives
- Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
- Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H∞ synthesis
- Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
- The rate of convergence on fractional power dissipative operator on compact manifolds
- Fractional Langevin type equations for white noise distributions
- Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
- Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function
- The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
- Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework
- Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems
Artikel in diesem Heft
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 24–4–2021)
- Research Paper
- Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
- Tutorial paper
- The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
- Research Paper
- Fractional diffusion-wave equations: Hidden regularity for weak solutions
- Censored stable subordinators and fractional derivatives
- Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
- Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H∞ synthesis
- Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
- The rate of convergence on fractional power dissipative operator on compact manifolds
- Fractional Langevin type equations for white noise distributions
- Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
- Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function
- The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
- Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework
- Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems