There are few numerical techniques available to solve the Bagley-Torvik equation which occurs considerably frequently in various offshoots of applied mathematics and mechanics. In this paper, we show that Chelyshkov-tau method is a very effective tool in numerically solving this equation. To show the accuracy and the efficiency of the method, several problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that Chelyshkov-tau method is superior to other existing ones and is highly accurate.
Luchko, Y. and Gorenflo, R. (1999) An Operational Method for Solving Fractional Differential Equations with the Caputo Derivatives. Acta Mathematica Vietnamica, 24, 207-233.
Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.
Meerschaert, T. and Tadjeran, C. (2006) Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations. Applied Numerical Mathematics, 56, 80-90. https://doi.org/10.1016/j.apnum.2005.02.008
Esmaeili, S. and Shamsia, M. and Luchko, Y. (2011) Numerical Solution of Fractional Differential Equations with a Collocation Method Based on Müntz Polynomials. Computers & Mathematics with Applications, 62, 918-929. https://doi.org/10.1016/j.camwa.2011.04.023
Vanani, S. and Aminataei, A. (2011) Tau Approximate Solution of Fractional Partial Differential Equations. Computers & Mathematics with Applications, 62, 1075-1083. https://doi.org/10.1016/j.camwa.2011.03.013
Momani, S. and Al-Khaled, K. (2005) Numerical Solutions for Systems of Fractional Differential Equations by the Decomposition Method. Computers & Mathematics with Applications, 162, 1351-1365. https://doi.org/10.1016/j.amc.2004.03.014
Ray, S. and Bera, R. (2005) Analytical Solution of the Bagley-Torvik Equation by Adomian Decomposition Method. Applied Mathematics and Computation, 168, 398-410. https://doi.org/10.1016/j.amc.2004.09.006
Odibat, Z. and Momani, S. (2006) Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order. International Journal of Nonlinear Sciences and Numerical Simulation, 70, 27-34. https://doi.org/10.1515/IJNSNS.2006.7.1.27
Li, Y. and Sun, N. (2011) Numerical Solution of Fractional Differential Equations Using the Generalized Block Pulse Operational Matrix. Computers & Mathematics with Applications, 62, 1046-1054. https://doi.org/10.1016/j.camwa.2011.03.032
Odibat, Z. (2008) Compact and Noncompact Structures for Nonlinear Fractional Evolution Equations. Physics Letters A, 372, 1219-1227. https://doi.org/10.1016/j.physleta.2007.09.022
Ganji, Z., Jafari, H. and Rostamian, M. (2008) Application of the Homotopy Perturbation Method to Coupled System of Partial Differential Equations with Time Fractional Derivatives. Topological Methods in Nonlinear Analysis, 31, 341-348.
Odibat, Z. and Odibat, S. (2008) Generalized Differential Transform Method for Linear Partial Differential Equations of Fractional Order. Applied Mathematics Letters, 31, 194-199. https://doi.org/10.1016/j.aml.2007.02.022
El-Gamel, M. and Abd El-Hady, M. (2017) Numerical Solution of the Bagley-Torvik Equation by Legendre-Collocation Method. SeMA Journal, 74, 371-383. https://doi.org/10.1007/s40324-016-0089-6
Uddin, M. and Ahmad S. (2017) On the Numerical Solution of Bagley-Torvik Equation via the Laplace Transform. Tbilisi Mathematical Journal, 10, 279-284. https://doi.org/10.1515/tmj-2017-0017
Gaul, L., Klein, P. and Kemple, S. (1991) Damping Description Involving Fractional Operators. Mechanical Systems and Signal Processing, 5, 81-88. https://doi.org/10.1016/0888-3270(91)90016-X
Suarez, L., Shokooh, A. and Kemple, S. (1991) An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives. Journal of Applied Mechanics, 64, 629-635. https://doi.org/10.1115/1.2788939
Arikoglu, A. and Ozkol, I. (2009) Solution of Fractional Integro-Differential Equations by Using Fractional Differential Transform Method. Chaos, Solitons & Fractals, 40, 521-529. https://doi.org/10.1016/j.chaos.2007.08.001
Mekkaouii, T. and Hammouch, Z. (2012) Application of Generalized Differential Transform Method to Multi-Order Fractional Differential Equations. Mathematics in Computer Science, 39, 251-256.
Mekkaouii, T. and Hammouch, Z. (2012) Approximate Analytical Solutions to the Bagley-Torvik Equation by the Fractional Iteration Method. Mathematics in Computer Science, 38, 251-256.
Odibat, Z. and Shawagfeh, N. (2007) Generalized Taylor Formula. Applied Mathematics and Computation, 186, 286-293. https://doi.org/10.1016/j.amc.2006.07.102
Yzba, S. and Shawagfeh, N. (2013) Numerical Solution of the Bagley-Torvik Equation by the Bessel Collocation Method. Mathematical Methods in the Applied Sciences, 36, 300-312. https://doi.org/10.1002/mma.2588
Wu, J.L. (2013) A Wavelet Operational Method for Solving Fractional Partial Differential Equations Numerically. Applied Mathematics and Computation, 214, 31-40. https://doi.org/10.1016/j.amc.2009.03.066
Lepik, H. (2009) Solving Fractional Integral Equations by the Haar Wavelet Method. Applied Mathematics and Computation, 214, 468-678. https://doi.org/10.1016/j.amc.2009.04.015
Oguz, C. and Sezer, M. (2015) Chelyshkov Collocation Method for a Class of Mixed Functional Integro-Differential Equations. Applied Mathematics and Computation, 259, 943-954. https://doi.org/10.1016/j.amc.2015.03.024
Oguz, C., Sezer, M. and Denk Oguz, A. (2015) Chelyshkov Collocation Approach to Solve the Systems of Linear Functional Differential Equations. NTMSCI, 3, 83-97.
Hilfer, R. (2000) Applications of Fractional Calculus in Physics. World Scientific Publishing Company, Singapore. https://doi.org/10.1142/3779
Diethelm, K. (1982) The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin Heidelberg.
Caputo, M. and Sezer, M. (1967) Linear Models of Dissipation Whose Q Is Almost Frequency Independent. Part II. Geophysical Journal of the Royal Astronomical Society, 13, 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
Chelyshkov, V. and Sezer, M. (1974) Stability of the Rest Position of the Inner Cylinder in a Couette Flow. Fluid Dynamics, 6, 1003-1005.
Chelyshkov, V. (1982) Three-Dimensional Self-Oscillatory Regimes of Fluid Flows. Hydromech, 4, 53-57.
Chelyshkov, V. (1986) Using the Method of Integration with Respect to a Small Parameter for Calculating a Laminar Boundary Layer on a Cylinder. USSR Computational Mathematics and Mathematical Physics, 26, 94-96. https://doi.org/10.1016/0041-5553(86)90047-9
Chelyshkov, V. (2006) Alternative Orthogonal Polynomials and Quadratures. ETNA, 25, 17-26.
Chelyshkov, V. (1994) A Variant of Spectral Method in the Theory of Hydrodynamic Stability. Hydromech, 68, 105-109.
Canuto, C., Hussaini, M. and Quarteroni, A. (1988) Spectral Method in Fluid Dynamic. Prentice-HLL, Englewood Cliffs. https://doi.org/10.1007/978-3-642-84108-8
?enesiz, Y., Keskin, Y. and Kurnaz, A. (2010) The Solution of the Bagley-Torvik Equation with the Generalized Taylor Collocation Method. Journal of the Franklin Institute, 347, 452-466. https://doi.org/10.1016/j.jfranklin.2009.10.007
Setia, A., Liu, Y. and Vatsala, A. (2014) The Solution of the Bagley-Torvik Equation by Using Second Kind Chebyshev Wavelet. 11th International Conference on Information Technology.
Bagly, R. and Torvik, P. (1984) On the Appearance of the Fractionl Derivative in the Behavior of Real Materials. Journal of Applied Mechanics, 51, 294-298. https://doi.org/10.1115/1.3167615
Rehman, M. and Ali Khan, R. (2012) A Numerical Method for Solving Boundary Value Problems for Fractionl Differential Equation. Applied Mathematical Modelling, 36, 894-907. https://doi.org/10.1016/j.apm.2011.07.045