On two improved numerical algorithms for vibration analysis of systems involving fractional derivatives
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https://doi.org/10.15625/0866-7136/15758Keywords:
vibration, fractional differential equation, numerical algorithm, dynamical systemsAbstract
Zhang and Shimizu (1998) proposed a numerical algorithm based on Newmark method to calculate the dynamic response of mechanical systems involving fractional derivatives. On the basis of Runge-Kutta-Nyström method and Newmark method, the present study proposes two new numerical algorithms, namely, the improved Newmark algorithm using the second order derivative and the improved Runge-Kutta-Nyström algorithm using the second order derivative to solve the fractional differential equations of vibration systems. The accuracy of new algorithms is investigated in detail by numerical simulation. The simulation result demonstrated that the Runge-Kutta-Nyström algorithm using the second order derivative for the vibration analysis of systems involving fractional derivatives is more effective than the Newmark algorithm of Zhang and Shimizu.
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References
K. B. Oldham and J. Spanier. The fractional calculus. Academic Press, Boston, New York, (1974).
K. S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations. JohnWiley & Sons, New York, (1993).
I. Podlubnv. Fractional differential equations. Academic Press, Boston, New York, (1999).
D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo. Fractional calculus, models and numerical methods. World Scientific Publishing, Singapore, (2011).
A. Kochubei and Y. Luchko. Handbook of fractional calculus with applications, Volume 2: Fractional differential equations. De Gruyter, Berlin/Boston, (2019).
V. E. Tarasov. Handbook of fractional calculus with applications, Volume 4: Applications in physics, Part A. De Gruyter, Berlin/Boston, (2019).
V. E. Tarasov. Handbook of fractional calculus with applications, Volume 5: Applications in physics, Part B. De Gruyter, Berlin/Boston, (2019).
D. Baleanu and A. M. Lopes. Handbook of fractional calculus with applications, Volume 7: Applications in engineering, life and social sciences, Part A. De Gruyter, Berlin/Boston, (2019).
D. Baleanu and A. M. Lopes. Handbook of fractional calculus with applications, Volume 8: Applications in engineering, life and social sciences, Part B. De Gruyter, Berlin/Boston, (2019).
G. Adomian. A new approach to nonlinear partial differential equations. Journal of Mathematical Analysis and Applications, 102, (1984), pp. 420–434.
G. Adomian. A review of the decomposition method and some recent results for nonlinear equations. Mathematical and Computer Modelling, 13, (7), (1990), pp. 17–43.
S. S. Ray, B. P. Poddar, and R. K. Bera. Analytical solution of a dynamic system containing fractional derivative of order one-half by adomian decomposition method. Journal of Applied Mechanics, 72, (2005), pp. 290–295.
S. S. Ray, K. S. Chaudhuri, and R. K. Bera. Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method. Applied Mathematics and Computation, 182, (2006), pp. 544–552.
H. Jafari and V. Daftardar-Gejji. Revised Adomian decomposition method for solving systems of ordinary and fractional differential equations. Applied Mathematics and Computation, 181, (2006), pp. 598–608.
Q. Wang. Numerical solutions for fractional KdV–Burgers equation by Adomian decomposition method. Applied Mathematics and Computation, 182, (2006), pp. 1048–1055.
K. Diethelm. An algorithm for the numerical solution of differential equations of fractional order. Electronic Transactions on Numerical Analysis, 5, (1), (1997), pp. 1–6.
K. Diethelm and N. J. Ford. Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265, (2002), pp. 229–248.
K. Diethelm and J. Ford. Numerical solution of the Bagley-Torvik equation. BIT Numerical Mathematics, 42, (3), (2002), pp. 490–507.
K. Diethelm, N. J. Ford, and A. D. Freed. A predictor-corrector approach for thenumerical solution of fractional differential equations. Nonlinear Dynamics, 29, (1/4), (2002), pp. 3–22.
K. Diethelm and N. J. Ford. Numerical solution of linear and non-linear fractional differential equations involving fractional derivatives of several orders. Numerical Analysis Report No. 379, Manchester Center for Computational Mathematics, Manchester, England, (2003).
J. T. Edwards, N. J. Ford, and A. C. Simpson. The numerical solution of linear multi-term fractional differential equations: Systems of equations. Journal of Computational and Applied Mathematics, 148, (2002), pp. 401–418.
L. Yuan and O. P. Agrawal. A numerical scheme for dynamic systems containing fractional derivatives. Journal of Vibration and Acoustics, 124, (2002), pp. 321–324.
A. Schmidt and L. Gaul. On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems. Mechanics Research Communications, 33, (2006), pp. 99–107.
J.-Z. Wang and et al. Coiflets bases method in solution of nonlinear of dynamic systems containing fractional derivative. In Proceedings of Fourth international Conference on Nonlinear Mechanics, (2002).
T. M. Atanackovic and B. Stankovic. On a numerical scheme for solving differential equations of fractional order. Mechanics Research Communications, 35, (2008), pp. 429–438.
A. Pálfalvi. Efficient solution of a vibration equation involving fractional derivatives. International Journal of Non-Linear Mechanics, 45, (2010), pp. 169–175.
J. T. Machado. Numerical calculation of the left and right fractional derivatives. Journal of Computational Physics, 293, (2015), pp. 96–103.
W. Zhang and N. Shimizu. Numerical algorithm for dynamic problems involving fractional operators. JSME International Journal Series C, 41, (3), (1998), pp. 364–370.
N. M. Newmark. A method of computation for structural dynamics. Journal of the Engineering Mechanics Division, 85, (1959), pp. 67–94.
L. Collatz. Numerische behandlung von differentialgleichungen. Springer-Verlag, Berlin, (1951).
S. Banerjee, S. Shaw, and B. Mukhopadhyay. A modified series solution method for fractional integro-differential equations. International Research Journal of Engineering and Technology (IRJET), 3, (8), (2016), pp. 1966–1973.
D. V. Lac. Calculating vibration of systems involving fractional derivatives. Engineering Graduation Project, Hanoi University of Science and Technology, (2014). (in Vietnamese).
D. V. Lac. Development of Runge-Kutta-Nystr¨om method for calculating vibration of systems involving fractional derivatives. Master Science Thesis, Hanoi University of Science and Technology, (2016). (in Vietnamese).
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National Foundation for Science and Technology Development
Grant numbers 107.04-2020.28
