Recent Trends in Wave Mechanics and Vibrations pp 393-406 | Cite as

# Artificial Neural Network Based Solution of Fractional Vibration Model

## Abstract

The purpose of the investigation is to handle the fractional vibration problem using the multilayer artificial neural network (ANN) method. Fractional calculus has found several applications in different fields of physical systems, viz., viscoelasticity, dynamics, and anomalous diffusion transport. Fractional derivatives are practically described viscoelasticity features in structural dynamics. In general, damping models involve ordinary integer differential operators that are relatively easy to handle. On the other hand, fractional derivatives give better models with respect to the vibration systems in comparison to classical integer-order models. Here, the fractional order in the damping coefficient has been considered. We have employed the multilayer feed-forward neural architecture and error back-propagation algorithm with unsupervised learning for minimizing the error function and modification of the parameters (weights and biases). The results obtained by the present method are compared with the analytical results and are found to be in good agreement.

## Keywords

Fractional differential equation Vibration problem Bagley–Torvik problem Artificial neural network model Feed-forward structure Back-propagation algorithm## Notes

### Acknowledgements

The first author is thankful to the Department of Science and Technology (DST), Government of India for financial support under Women Scientist Scheme-A.

## References

- 1.Podlubny I (1999) Fractional differential equations. Academic PressGoogle Scholar
- 2.Dumitru B, Kai D, Enrico S (2012) Fractional calculus: models and numerical methods. World ScientificGoogle Scholar
- 3.Uchaikin V (2013) Fractional derivatives for physicists and engineers. Springer, BerlinCrossRefGoogle Scholar
- 4.Atanackovic TM, Pilipovic S, Stankovic B, Zorica D (2014) Fractional calculus with applications in mechanics: from the cell to the ecosystem. Wiley-ISTEGoogle Scholar
- 5.Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27:201–210CrossRefGoogle Scholar
- 6.Rossikhin YA, Shitikova MV (1997) Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems. Acta Mech 120(1–4):109–125MathSciNetCrossRefGoogle Scholar
- 7.Bohannan GW (2008) Analog fractional order controller in temperature and motor control applications. J Vib Control 14:1487–1489MathSciNetCrossRefGoogle Scholar
- 8.Bansal MK, Jain R (2016) Analytical solution of Bagley–Torvik equation by generalize differential transform. Inter J pure Appl Math 110(2):265–273CrossRefGoogle Scholar
- 9.Dai H, Zhibao Z, Wang W (2017) On generalized fractional vibration equation. Chaos, Solitons Fractals 95:48–51MathSciNetCrossRefGoogle Scholar
- 10.Mohyud-Din ST, Yildirim A (2012) An algorithm for solving the fractional vibration equation. Comput Math Model 23:228–237MathSciNetCrossRefGoogle Scholar
- 11.Galucio AC, Deu JF, Ohayon RA (2005) Fractional derivative viscoelastic model for hybrid active-passive damping treatments in time domain—application to sandwich beams. J Intell Mater Syst Struct 16:33–45CrossRefGoogle Scholar
- 12.Singh H (2018) Approximate solution of fractional vibration equation using Jacobi polynomials. Appl Math Comput 317:85–100MathSciNetzbMATHGoogle Scholar
- 13.Wang ZH, Wang X (2010) General solution of the Bagley–Torvik equation with fractional-order derievative. Commun Nonlinear Sci Numer Simulat 15:1279–1285CrossRefGoogle Scholar
- 14.Palfalvi A (2010) Efficient solution of a vibration equation invovlving fractional derivative. Int J Non-Linear Mech 45:169–175CrossRefGoogle Scholar
- 15.Gulsu M, Ozturk Y, Anapali A (2017) Numerical solution of the fractional Bagley–Torvik equation arising in fluid mechanics. Int J Comput Math 94:173–184CrossRefGoogle Scholar
- 16.Saloma C (1993) Computation complexity and observations of physical signals. J Appl Phys 74:5314–5319CrossRefGoogle Scholar
- 17.Zurada JM (1994) Introduction to artificial neural network. West Publ, CoGoogle Scholar
- 18.Graupe D (2007) Principle of artificial neural networks, 2nd edn. World Scientific PublishingGoogle Scholar
- 19.Mall S, Chakraverty S (2016) Application of Legendre neural network for solving ordinary differential equations. Appl Soft Comput 43:347–356CrossRefGoogle Scholar
- 20.Mall S, Chakraverty S (2016) Hermite functional link neural network for solving the Van der Pol-Duffing oscillator equation. Neural Comput 28(8):1574–1598MathSciNetCrossRefGoogle Scholar
- 21.Mall S, Chakraverty S (2015) Numerical solution of nonlinear singular initial value problems of Emden-Fowler type using Chebyshev neural network method. Neurocomputing 149:975–982CrossRefGoogle Scholar
- 22.Mall S, Chakraverty S (2014) Chebyshev neural network based model for solving Lane-Emden type equations. Appl Math Comput 247:100–114MathSciNetzbMATHGoogle Scholar
- 23.Chakraverty S, Mall S (2014) Regression based weight generation algorithm in neural network for solution of initial and boundary value problems. Neural Comput Appl 25:585–594CrossRefGoogle Scholar
- 24.Chakraverty S, Mall S (2017) Artificial neural networks for engineers and scientists: solving ordinary differential equations. CRC Press/Taylor & Francis GroupGoogle Scholar
- 25.Miller KS, Ross B (1993) An Introduction to the fractional calculus and fractional differential equations. Wiley-Interscience Publication, Wiley, New YorkzbMATHGoogle Scholar
- 26.Merdan M (2012) On the solutions fractional riccati differential equation with modified Riemann–Liouville derivative. Int J Differ Eqn 2012:1–17MathSciNetzbMATHGoogle Scholar
- 27.Diethelm K, Luchko Y (2004) Numerical solution of linear multi-term initial value problems of fractional order. J Comput Anal Appl 6:243–263MathSciNetzbMATHGoogle Scholar
- 28.Khalil R, Al Horani M, Yousef A, Sababheh M (2014) A new definition of fractional derivative. J Comput Appl Math 264:65–70MathSciNetCrossRefGoogle Scholar
- 29.Emrah U, Gokdogan A (2017) Solution of conformable fractional ordinary differential equations via differential transform method. Optik-Int J Light Electron Optics 128:264–273CrossRefGoogle Scholar