The Generalized Differential Transform Method for solution of a free vibration linear differential equation with fractional derivative damping

• Autorzy: Das Deepanjan

Identyfikatory

DOI

10.17512/jamcm.2019.2.02

• Języki publikacji: EN

Abstrakty

EN

In the present paper, the Generalized Differential Transform Method (GDTM) is used for obtaining the approximate analytic solutions of a free vibration linear differential equation of a single-degree-of-freedom (SDOF) system with fractional derivative damping. The fractional derivatives are described in the Caputo sense.

Słowa kluczowe

EN

fractional differential equations; Caputo fractional derivative; generalized; differential transform method; analytic solution

PL

ułamkowe równania różniczkowe; ułamkowa pochodna Caputo; metoda transformacji różnicowej; rozwiązanie analityczne; ułamkowe równanie różniczkowe

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2019
• Tom: Vol. 18, nr 2
• Strony: 19--29
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Das Deepanjan

• Department of Mathematics, Ghani Khan Choudhury Institute of Engineering and Technology Narayanpur, Malda, West Bengal-732141, India

Bibliografia

• [1] Zhou, J.K. (1986). Differential Transformation and its Applications for Electrical Circuits. Wuhan: Huazhong University Press.
• [2] Chen, C.K., & Ho, S.H. (1999). Solving partial differential equations by two dimensional differential transform method. Appl. Math. Comput., 106, 171-179.
• [3] Ayaz, F. (2004). Solutions of the systems of differential equations by differential transform method. Appl. Math. Comput., 147, 547-567.
• [4] Abazari, R., & Borhanifar, A. (2010). Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method. Comput. Math. Appl., 59, 2711-2722.
• [5] Chen, C.K. (1999). Solving partial differantial equations by two dimensional differential transformation method. Appl. Math. Comput., 106, 171-179.
• [6] Jang, M.J., & Chen, C.K. (2001). Two-dimensional differential transformation method for partial differantial equations. Appl. Math. Comput., 121, 261-270.
• [7] Kangalgil, F., & Ayaz, F. (2009). Solitary wave solutions for the KDV and mKDV equations by differential transformation method. Choas Solitons Fractals, 41, 464-472.
• [8] Arikoglu, A., & Ozkol, I. (2006). Solution of difference equations by using differential transformation method. Appl. Math. Comput., 174, 1216-1228.
• [9] Momani, S., Odibat, Z., & Hashim, I. (2008). Algorithms for nonlinear fractional partial differential equations: A selection of numerical methods. Topol. Method Nonlinear Anal., 31, 211-226.
• [10] Arikoglu, A., & Ozkol, I. (2007). Solution of fractional differential equations by using differential transformation method. Chaos Solitons Fractals, 34, 1473-1481.
• [11] Soltanalizadeh, B., & Zarebnia, M. (2011). Numerical analysis of the linear and nonlinear Kuramoto-Sivashinsky equation by using Differential Transformation Method. Inter. J. Appl. Math. Mechanics, 7(12), 63-72.
• [12] Tari, A., Rahimi, M.Y., Shahmoradb, S., & Talati, F. (2009). Solving a class of two-Dimensional linear and nonlinear Volterra integral equations by the differential transform method. J. Comput. Appl. Math., 228, 70-76.
• [13] Nazari, D., & Shahmorad, S. (2010). Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions. J. Comput. Appl. Math., 234 ,883-891.
• [14] Borhanifar, A., & Abazari, R. (2011). Exact solutions for non-linear Schr.dinger equations by differential transformation method. J. Appl. Math. Comput., 35, 37-51.
• [15] Borhanifar, A., & Abazari, R. (2010). Numerical study of nonlinear Schr.dinger and coupled Schr.dinger equations by differential transformation method. Optics Communications, 283, 2026-2031.
• [16] Momani, S., Odibat, Z., & Erturk, V.S. (2007). Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation. Physics Letters. A, 370(5-6), 379-387.
• [17] Odibat, Z., & Momani, S. (2008). A generalized differential transform method for linear partial Differential equations of fractional order”. Applied Mathematics Letters, 21(2), 194-199.
• [18] Odibat, Z., Momani, S., & Erturk, V.S. (2008). Generalized differential transform method: application to differential equations of fractional order. Applied Mathematics and Computation, 197(2), 467-477.
• [19] Ertiirka, V.S., & Momanib, S. (2010). On the generalized differential transform method: application to fractional integro-differential equations. Studies in Nonlinear Sciences, 1(4), 118-126.
• [20] Garg, M., Manohar, P., & Kalla, S.L. (2011). Generalized differential transform method to Space-time fractional telegraph equation. Int. J. of Differential Equations, Hindawi Publishing Corporation, vol. 2011, article id. 548982, 9 pages, DOI: 10.1155/2011/548982.
• [21] Bansal, M.K., & Jain, R. (2015). Application of generalized differential transform method to fractional order Riccati differential equation and numerical results. Int. J. of Pure and Appl. Math., 99(3), 355-366.
• [22] Cetinkaya, A., Kiymaz, O., & Camli, J. (2011). Solution of non linear PDE’s of fractional order with generalized differential transform method. Int. Mathematical Forum, 6(1), 39-47.
• [23] Das. S. (2008). Functional Fractional Calculus, Springer.
• [24] Miller, K.S., & Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Diff. Equations. John Wiley and Son.
• [25] Caputo, M. (1967). Linear models of dissipation whose q is almost frequency independent-ii. Geophys. J. R. Astron. Soc., 13, 529-539.
• [26] Podlubny, L. (1999). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press.
• [27] Almeida, R., & Torres, D.F. (2011). Necessary and sufficient conditions for the fractional calculus of variations with caputo derivatives. Communications in Nonlinear Science and Numerical Simulation, 16, 1490-1500

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).