AccScience Publishing / IJOCTA / Volume 9 / Issue 3 / DOI: 10.11121/ijocta.01.2019.00638
RESEARCH ARTICLE

On the explicit solutions of fractional Bagley-Torvik equation arises in engineering

Zehra Pinar1*
Show Less
1 Department of Mathematics, Tekirdağ Namık Kemal University, Turke
Submitted: 19 July 2018 | Accepted: 10 December 2018 | Published: 2 May 2019
© 2019 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Abstract

In this work, Bagley-Torvik equation is considered with conformable derivatives. The analytical solutions will be obtained via Sine-Gordon expansion method and Bernouli equation method for the two cases of Bagley-Torvik equation. We will illustrate and discuss about the methodology and solutions therefore the proposed equation has meaning in different areas of science and engineering.

Keywords
Bagley-Torvik Equation
Explicit solutions
Conformable derivative
Conflict of interest
The authors declare they have no competing interests.
References

[1] Baleanu, D. (2009). About fractional quantization and fractional variational principles. Communications in Nonlinear Science and Numerical Simulation, 14(6), 2520–2523.

[2] Gülsu, M., Öztürk, Y. & Anapali, A. (2017). Numerical solution the fractional Bagley–Torvik equation arising in fluid mechanics. International Journal of Computer Mathematics, 94(1), 173-184.

[3] Podlubny, I. (1999). Fractional differential equations. Academic Press, San Diego.

[4] Bagley, R.L. & Torvik, P.J. (1984). On the appearance of the fractional derivative in the behavior of real materials. Journal of Applied Mechanics, 51(2), 294–302.

[5] Bagley, R.L. & Torvik, P.J. (1983). Fractional calculus – a different approach to the analysis of viscoe1astically damped structures. AIAA Journal, 21(5), 741–749.

[6] Ray, S.S. & Bera, R.K. (2005). Analytical solution of the Bagley–Torvik equation by Adomian decomposition method. Applied Mathematics and Computation, 168(1), 398–410.

[7] Wang, Z.H. & Wang, X. (2010). General solution of the Bagley–Torvik equation with fractional-order derivative. Communications in Nonlinear Science and Numerical Simulation, 15, 1279–1285.

[8] Gülsu, M., Öztürk,Y. & Anapali,A. (2017). Numerical solution of the fractional Bagley–Torvik equation arising in fluid mechanics. International Journal of Computer Mathematics, 94(1), 173-184.

[9] Yüzbasi, S. (2013). Numerical solution of the Bagley–Torvik equation by the bessel collocation method, Mathematical Methods in the Applied Sciences, 36, 300–312.

[10] Mohammadi, F. (2014). Numerical solution of Bagley–Torvik equation using Chebyshev wavelet operational matrix of fractional derivative. International Journal of Advances in Applied Mathematics and Mechanics, 2, 83–91 .

[11] Mashayekhi, S. & Razzaghi, M. (2016). Numerical solution of the fractional Bagley–Torvik equation by using hybrid functions approximation. Mathematical Methods in the Applied Sciences, 39, 353–365 .

[12] Ashyralyev, A., Dal, F. & Pınar, Z. (2011). A note on the fractional hyperbolic differential and difference equations. Applied Mathematics and Computation, 217(9), 4654-4664.

[13] Rui, W. (2017). Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs. Communications in Nonlinear Science and Numerical Simulation,47, 253–266.

[14] Sahadevan, R. & Prakash, P. (2016). Exact solution of certain time fractional nonlinear partial differential equations. Nonlinear Dynamics,1–15.

[15] Fitt, A.D., Goodwin, A.R.H., Ronaldson, K.A. & Wakeham, W.A. (2009). A fractional differential equation for a MEMS viscometer used in the oil industry. Journal of Computational and Applied Mathematics, 229, 373–381 .

[16] Ashyralyev, A., Dal, F. & Pinar, Z. (2009). On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations. Mathematical Problems in Engineering, 1-11.

[17] Zahra, W.K. & Van Daele, M. (2017). Discrete spline methods for solving two point fractional Bagley–Torvik equation. Applied Mathematics and Computation, 296, 42–56

[18] Zahra, W.K. & Elkholy, S.M. (2012). Quadratic spline solution for boundary value problem of fractional order. Numerical Algorithms, 59, 373– 391.

[19] Bakkyaraj, T. & Sahadevan, R. (2016). Approximate analytical solution of two coupled time fractional nonlinear schrodinger equations. International Journal of Applied and Computational Mathematics, 2(1), 113–35.

[20] Daftardar-Gejji, V. & Jafari H. (2005). Adomian decomposition: a tool for solving a system of fractional differential equations. Journal of Mathematical Analysis and Applications, 301(2), 508–18 .

[21] Bakkyaraj, T. & Sahadevan, R. (2014). An approximate solution to some classes of fractional nonlinear partial differential difference equation using adomian decomposition method. Journal of Fractional Calculus and Applications, 5(1), 37–52 .

[22] Eslami, M., Vajargah, B.F., Mirzazadeh, M. & Biswas, A. (2014). Applications of first integral method to fractional partial differential equations. Indian Journal of Physics, 88(2), 177–84 .

[23] Bakkyaraj, T. &Sahadevan, R. (2014). On solutions of two coupled fractional time derivative hirota equations. Nonlinear Dynamics, 77(4), 1309–1331 .

[24] Sahadevan, R., Bakkyaraj, T. (2012). Invariant analysis of time fractional generalized burgers and korteweg-de vries equations. Journal of Mathematical Analysis and Applications, 393(2), 341–348 .

[25] Bakkyaraj, T. & Sahadevan, R. (2015). Invariant analysis of nonlinear fractional ordinary differential equations with Riemann–Liouville derivative. Nonlinear Dynamics, 80(1), 447–55 .

[26] Sahadevan, R. & Bakkyaraj, T . (2015). Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations. Fractional Calculus and Applied Analysis,18(1), 146–62 .

[27] Harris, P.A. & Garra, R. (2013). Analytic solution of nonlinear fractional burgers-type equation by invariant subspace method. Nonlinear Studies, 20(4), 471–81 .

[28] Odibat, Z.M. & Shaher, M. (2009) . The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Computers & Mathematics with Applications, 58, 2199–208 .

[29] Wu, G. & Lee, E.W.M. (2010). Fractional variational iteration method and its application. Physics Letters A, 374(25), 2506–9 .

[30] Momani, S.,& Zaid, O. (2007). Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differen- tial equations. Computers & Mathematics with Applications, 54(7), 910–19 .

[31] Eslami, M. (2017). Solutions for space-time fractional (2+1)-dimensional dispersive long wave equations. Iranian Journal of Science and Technology, Transactions A: Science, 41, 1027– 1032.

[32] Neirameh, A. (2016). New soliton solutions to the fractional perturbed nonlinear Schrodinger equation with power law nonlinearity. SeMA Journal, 73(4), 309-323.

[33] Neirameh, A. (2015). Topological soliton solutions to the coupled Schrodinger–Boussinesq equation by the SEM. Optik - International Journal for Light and Electron Optics, 126, 4179–4183.

[34] Neirameh, A. (2015). Binary simplest equation method to the generalized Sinh–Gordon equation. Optik - International Journal for Light and Electron Optics, 126, 4763–4770.

[35] Neirameh, A. (2016). New analytical solutions for the coupled nonlinear Maccari’s system. Alexandria Engineering Journal, 55, 2839–2847.

[36] Ma, W.X. (2012). A refined invariant subspace method and applications to evolution equations. Science China Mathematics, 55(9), 1–10.

[37] Ma, W.X. & Lee, J.H. (2009). A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation. Chaos, Solitons Fractals, 42(3), 1356–1363.

[38] Ma, W.X., Huang, T. & Zhang, Y. (2010). A multiple exp-function method for nonlinear differential equations and its application. Physica Scripta, 82(6), 065003.

[39] Kilbas, A.A., Srivastava, H.M. & Trujillo, J.J.(2006). Theory and application of fractional differential equations. First ed., Elsevier Science (North-Holland), Amsterdam.

[40] Miller, K.S. & Ross, B. (1993). An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York.

[41] Khalil, R., Al Horani, M., Yousef, A. & Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65–70.

[42] Atangana, A., Baleanu, D. & Alsaedi, A. (2015). New properties of conformable derivative. Open Mathematics, 13 (1), 1–10.

[43] Abdeljawad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279, 57–66.

[44] Wazwaz, A. M. (2002). Partial Differential Equations: Methods and Applications. Balkema, Leiden.

[45] Wazwaz, A. M. (2009). Partial Differential Equations and Solitary Wave Theory. Higher Education Press, Berlin Heidelberg.

[46] Younis, M. & Rizvi, S. T. R. (2015). Dispersive dark optical soliton in (2+1)-dimensions by (G′/G)- expansion with dual-power lawnonlinearity. Optik- International Journal for Light and Electron Optics, 126(24), 5812-5814.

[47] Parkes, E.J. (2010). Observations on the basic G/G'- expansion method for finding solutions to nonlinear evolution equations. Applied Mathematics and Computation, 217, 1759-1763.

[48] Vitanov, N. K. (2011). Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs. Communications in Nonlinear Science and Numerical Simulation,16, 1176–1185.

[49] Pınar, Z. & Öziş, T. (2013). An Observation on the Periodic Solutions to Nonlinear Physical models by means of the auxiliary equation with a sixth-degree nonlinear term. Communications in Nonlinear Science and Numerical Simulation, 18, 2177-2187.

[50] Pinar, Z. & Ozis, T. (2015). A remark on a variable-coefficient Bernoulli equation based on auxiliary equation method for nonlinear physical systems. arXiv:1511.02154 [math.AP]

[51] Yomba, E. (2005). The extended Fan’s sub- equation method and its application to KdV- MKdV, BKK and variant Boussinesq equations. Physics. Letters. A, 336, 463-476.

[52] Kurt, A., Tasbozan, O. & Baleanu, D. (2017). New solutions for conformable fractional Nizhnik– Novikov–Veselov system via G’/G expansion method and homotopy analysis methods. Optical and Quantum Electronics, 49, 333-384.

[53] Yépez-Martíneza, H., Gómez-Aguilarb, J.F. & Baleanu, D. (2018). Beta-derivative and sub-equation method applied to the optical solitons in medium with parabolic law nonlinearity and higher order dispersion. Optik, 155, 357–365.

[54] Pınar, Z. & Öziş, T. (2015). Observations on the class of ‘Balancing Principle’ for nonlinear PDEs that can be treated by the auxiliary equation method. Nonlinear Analysis: Real World Applications, 23, 9–16.

[55] Pinar, Z. & Kocak, H. (2018). Exact solutions for the third-order dispersive-Fisher equations. Nonlinear Dynamics, 91, 421–426.

[56] Evirgen, F. (2016). Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM. An International Journal of Optimization and Control: Theories & Applications, 6(2), 75-83.

[57] Yavuz, M. (2018). Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. An International Journal of Optimization and Control: Theories & Applications, 8(1), 1-7.

Share
Back to top
An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing