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Fractional Approach to the Study of Damped Traveling Disturbances in a Vibrating Medium^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Solution of the Problem

## 4. Discussion of Results

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Some Useful Definitions

## Appendix B. Derivation of Expressions (11) and (12)

## References

- Gorenflo, R.; Iskenderov, A.; Luchko, Y. Mapping between solutions of fractional diffusion wave equations. Fract. Calc. Appl. Anal.
**2000**, 3, 75. [Google Scholar] - Olivar-Romero, F.; Rosas-Ortiz, O. Transition from the Wave Equation to Either the Heat or the Transport Equations through Fractional Differential Expressions. Symmetry
**2018**, 10, 524. [Google Scholar] [CrossRef] [Green Version] - Luchko, Y.J. Fractional wave equation and damped waves. Math. Phys.
**2013**, 54, 031505. [Google Scholar] [CrossRef] [Green Version] - Podlubny, I. Fractional Differential Equations; Academic Press: London, UK, 1999. [Google Scholar]
- Umarov, S. Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols; Springer: Basel, Switzerland, 2015. [Google Scholar]
- Widder, D.V. The Heat Equation; Academic Press: New York, NY, USA, 1975. [Google Scholar]
- Tikhonov, A.N.; Samarskii, A.A. Equations of Mathematical Physics; Pergamon Press: New York, NY, USA, 1963. [Google Scholar]
- Gustafson, K.E. Introduction to Partial Differential Equations and Hilbert Space Methods; Dover: New York, NY, USA, 1999. [Google Scholar]
- Duffy, D.G. Green’s Functions with Applications; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Borthwick, D. Introduction to Partial Differential Equations; Springer: Basel, Switzerland, 2018. [Google Scholar]
- Luchko, Y. The Wright function and its applications. In Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory; De Gruyte: Berlin, Germany; Boston, MA, USA, 2019; Chapter 10; pp. 241–268. [Google Scholar]
- Olivar-Romero, F.; Rosas-Ortiz, O. An integro-differential Equation of the Fractional Form: Cauchy Problem and Solution. In Integrability, Supersymmetry and Coherent States; A Volume in Honour of Professor Véronique Hussin, CRM Series in Mathematical Physics; Kuru, S., Negro, J., Nieto, L.M., Eds.; Springer: Berlin, Germany, 2019; Chapter 18; pp. 387–393. [Google Scholar]
- Olivar-Romero, F. Fractional Approach to the Study of Some Partial Differential and Integro-Differential Equations. J. Phys. Conf. Ser.
**2022**. accepted. [Google Scholar] - Srednicki, M. Quantum Field Theory; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Gravel, P.; Gauthier, C. Classical applications of the Klein-Gordon equation. Am. J. Phys.
**2011**, 79, 447. [Google Scholar] [CrossRef] - Bansu, H.; Kumar, S. Meshless method for the numerical solution of space and time fractional wave equation. In Proceedings of the International Workshop Numerical Solution of Fractional Differential Equations and Applications, Sozopol, Bulgaria, 8–13 June 2020; pp. 11–14. [Google Scholar]
- Bansu, H.; Kumar, S. Numerical Solution of Space and Time Fractional Telegraph Equation: A Meshless Approach. Int. J. Nonlinear Sci. Num. Simul.
**2019**, 20, 325. [Google Scholar] [CrossRef] - Bansu, H.; Kumar, S. Numerical Solution of Space-Time Fractional Klein-Gordon Equation by Radial Basis Functions and Chebyshev Polynomials. Int. J. Appl. Comput. Math.
**2021**, 7, 201. [Google Scholar] [CrossRef] - Kumar, S.; Piret, C. Numerical solution of space-time fractional PDEs using RBF-QR and Chebyshev polynomials. Appl. Numer. Math.
**2019**, 143, 300. [Google Scholar] [CrossRef] - Capelas de Oliveira, E.; Costa, F.; Vaz, J. The fractional Schrödinger equation for delta potentials. J. Math. Phys.
**2010**, 51, 123517. [Google Scholar] [CrossRef] [Green Version] - Kilbas, A.A. H-Transforms: Theory and Applications; CRC Press: New York, NY, USA, 2004. [Google Scholar]
- Mathai, A.M.; Saxena, R.K.; Haubold, H.J. The H-Function; Springer: New York, NY, USA, 2010. [Google Scholar]
- Saichev, A.; Zazlavsk, G. Fractional kinetic equations: Solutions and applications. Chaos
**1997**, 7, 753. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gorenflo, E.; Mainardi, F. Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal.
**1998**, 1, 167. [Google Scholar] - Gorenflo, R.; Mainardi, F. Approximation to Lévy-Feller diffusion by random walk. Z. Anal. Anwend.
**1999**, 18, 231. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Olivar-Romero, F.
Fractional Approach to the Study of Damped Traveling Disturbances in a Vibrating Medium. *Comput. Sci. Math. Forum* **2022**, *4*, 1.
https://doi.org/10.3390/cmsf2022004001

**AMA Style**

Olivar-Romero F.
Fractional Approach to the Study of Damped Traveling Disturbances in a Vibrating Medium. *Computer Sciences & Mathematics Forum*. 2022; 4(1):1.
https://doi.org/10.3390/cmsf2022004001

**Chicago/Turabian Style**

Olivar-Romero, Fernando.
2022. "Fractional Approach to the Study of Damped Traveling Disturbances in a Vibrating Medium" *Computer Sciences & Mathematics Forum* 4, no. 1: 1.
https://doi.org/10.3390/cmsf2022004001