Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative
Abstract
1. Introduction
2. Preliminaries
3. Separation of Variables Analytically Scheme
4. Laplace–Homotopy Perturbation Approximation Scheme
5. Orthogonality and Convergence Analysis
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1999. [Google Scholar]
- Davis, H.T. The Theory of Linear Operators; Myers Press: Bloomington, IN, USA, 2008. [Google Scholar]
- Sabatier, J.; Agrawal, O.P.; Machado, A.T. Advances in Fractional Calculus. In Theoretical Developments and Applications in Physics and Engineering; Springer: Dordrecht, The Netherlands, 2007. [Google Scholar]
- Aleroev, M.; Aleroev, H.; Aleroev, T. Proof of the completeness of the system of eigenfunctions for one boundary-value problem for the fractional differential equation. AIMS Math. 2019, 4, 714–720. [Google Scholar] [CrossRef]
- Caputo, M. Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 2001, 4, 421–442. [Google Scholar]
- Mainardi, F.; Luchko, Y.; Pagnini, G. The fundamental solution of the space time fractional diffusion equation. Fract. Calc. Appl. Anal. 2001, 4, 153–192. [Google Scholar]
- Aleroev, T.S.; Erokhin, S.V. Parametric Identification of the Fractional-Derivative Order in the Bagley–Torvik Model. Math. Models Comp. Simul. 2019, 2, 219–225. [Google Scholar] [CrossRef]
- Cohen, A.M. Numerical Methods for Laplace Transform Inversion; Springer: Berlin, Germany, 2007. [Google Scholar]
- Luchko, Y.; Gorenflo, R. An operationl method for solving fractional differential equations with the caputo derivatives. Acta Math. Vietnam. 1999, 2, 207–233. [Google Scholar]
- Davies, B.; Martin, B. Numerical inversion of Laplace transform: A survey and comparison of methods. J. Comput. Phys. 1979, 33, 1–32. [Google Scholar] [CrossRef]
- Madani, M.; Fathizadeh, M.; Khan, Y.; Yildirim, A. On the coupling of the homotopy perturbation method and Laplace transformation. Math. Comput. Model. 2011, 53, 1937–1945. [Google Scholar] [CrossRef]
- Javidi, M.; Ahmad, B. Numerical solution of fractional partial differential equations by numerical Laplace inversion technique. Adv. Diff. Eq. 2013, 375. [Google Scholar] [CrossRef]
- Aleroev, T.S. The analysis of the polymer concrete characteristics by fractional calculus. IOP Conf. Ser. J. Phys. Conf. Ser. 2019, 1425. [Google Scholar] [CrossRef]
- Aleroeva, H.; Aleroev, T.S. Some applications of fractional calculus. IOP Conf. Ser. Mater. Sci. Eng. 2020, 747. [Google Scholar] [CrossRef]
- Luchko, Y. Subordination principles for the multi-dimensional space-time fractional diffusion wave equations. Theor. Probab. Math. Stat. 2019, 98, 127–147. [Google Scholar] [CrossRef]
- Dimovski, I.H. Convolutional Calculus; Bulgarian Academy of Science: Sofia, Bulgaria, 1990. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Saxena, R.K.; Mathai, A.M.; Haubold, H.J. On generalized fractional kinetic equations. Phys. A Stat. Mech. Its Applic. 2004, 344, 657–664. [Google Scholar] [CrossRef]
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Aleroev, T.S.; Elsayed, A.M. Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative. Mathematics 2020, 8, 1154. https://doi.org/10.3390/math8071154
Aleroev TS, Elsayed AM. Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative. Mathematics. 2020; 8(7):1154. https://doi.org/10.3390/math8071154
Chicago/Turabian StyleAleroev, Temirkhan S., and Asmaa M. Elsayed. 2020. "Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative" Mathematics 8, no. 7: 1154. https://doi.org/10.3390/math8071154
APA StyleAleroev, T. S., & Elsayed, A. M. (2020). Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative. Mathematics, 8(7), 1154. https://doi.org/10.3390/math8071154