# Article

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Keywords:
fractional calculus; fractional differential equation; Caputo derivative; Laplace transform
Summary:
We use the Laplace transform method to solve certain families of fractional order differential equations. Fractional derivatives that appear in these equations are defined in the sense of Caputo fractional derivative or the Riemann-Liouville fractional derivative. We first state and prove our main results regarding the solutions of some families of fractional order differential equations, and then give examples to illustrate these results. In particular, we give the exact solutions for the vibration equation with fractional damping and the Bagley-Torvik equation.
References:
[1] Albadarneh, R. B., Batiha, I. M., Zurigat, M.: Numerical solutions for linear fractional differential equations of order $1<\alpha<2$ using finite difference method (ffdm). J. Math. Comput. Sci. 16 (2016), 102-111. DOI 10.22436/jmcs.016.01.11
[2] Bansal, M., Jain, R.: Analytical solution of Bagley Torvik equation by generalize differential transform. Int. J. Pure Appl. Math. 110 (2016), 265-273. DOI 10.12732/ijpam.v110i2.3
[3] Chung, W. S., Jung, M.: Fractional damped oscillators and fractional forced oscillators. J. Korean Phys. Soc. 64 (2014), 186-191. DOI 10.3938/jkps.64.186
[4] Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003 (2003), 3413-3442. DOI 10.1155/s0161171203301486 | MR 2025566 | Zbl 1036.26004
[5] Debnath, L., Bhatta, D.: Integral Transforms and Their Applications. Chapman & Hall/CRC, Boca Raton (2007). MR 2253985 | Zbl 1113.44001
[6] Diethelm, K., Ford, N. J.: Numerical solution of the Bagley-Torvik equation. BIT 42 (2002), 490-507. DOI 10.1023/A:1021973025166 | MR 1931882 | Zbl 1035.65067
[7] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Tables of Integral Transforms. Vol. I. Bateman Manuscript Project. California Institute of Technology, McGraw-Hill, New York (1954). MR 0061695 | Zbl 0055.36401
[8] Hu, S., Chen, W., Gou, X.: Modal analysis of fractional derivative damping model of frequency-dependent viscoelastic soft matter. Advances in Vibration Engineering 10 (2011), 187-196.
[9] Kazem, S.: Exact solution of some linear fractional differential equations by Laplace transform. Int. J. Nonlinear Sci. 16 (2013), 3-11. MR 3100782 | Zbl 1394.34015
[10] Kumar, H., Pathan, M. A.: On the distribution of non-zero zeros of generalized Mittag-Leffler functions. J. Eng. Res. Appl. 1 (2016), 66-71.
[11] Li, C., Qian, D., Chen, Y.: On Riemann-Liouville and Caputo derivatives. Discrete Dyn. Nat. Soc. 2011 (2011), Article ID 562494, 15 pages. DOI 10.1155/2011/562494 | MR 2782260 | Zbl 1213.26008
[12] Lin, S.-D., Lu, C.-H.: Laplace transform for solving some families of fractional differential equations and its applications. Adv. Difference Equ. 2013 (2013), Article ID 137, 9 pages. DOI 10.1186/1687-1847-2013-137 | MR 3068648 | Zbl 1390.34025
[13] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York (1993). MR 1219954 | Zbl 0789.26002
[14] Nolte, B., Kempfle, S., Schäfer, I.: Does a real material behave fractionally? Applications of fractional differential operators to the damped structure borne sound in viscoelastic solids. J. Comput. Acoust. 11 (2003), 451-489. DOI 10.1142/s0218396x03002024 | MR 2013238 | Zbl 1360.74035
[15] Pálfalvi, A.: Efficient solution of a vibration equation involving fractional derivatives. Int. J. Non-Linear Mech. 45 (2010), 169-175. DOI 10.1016/j.ijnonlinmec.2009.10.006
[16] Prabhakar, T. R.: A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Math. J. 19 (1971), 7-15. MR 0293349 | Zbl 0221.45003
[17] Raja, M. A. Z., Khan, J. A., Qureshi, I. M.: Solution of fractional order system of Bagley-Torvik equation using evolutionary computational intelligence. Math. Probl. Eng. 2011 (2011), Article ID 675075, 18 pages. DOI 10.1155/2011/675075 | MR 2781559 | Zbl 1382.34010

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