Theorems on some families of fractional differential equations and their applications. (English) Zbl 07144728

Bozkurt, Gülçin; Albayrak, Durmuş; Dernek, Neşe

Appl. Math., Praha 64, No. 5, 557-579 (2019).

The paper is concerned with solving some fractional differential equations of Riemann-Liouville and Caputo types using Laplace transform methods. The authors begin by providing some background to exact and numerical solution of fractional differential equations. They go on to consider several different families of fractional differential equation and they give results that show how the Laplace transform approach provides an exact formula for the solution. The paper ends with some worked examples that apply their method.

Reviewer: Neville J. Ford (Chester)

MSC:

26A33	Fractional derivatives and integrals
34A08	Fractional ordinary differential equations
44A10	Laplace transform
44A15	Special integral transforms (Legendre, Hilbert, etc.)

Keywords:

fractional calculus; fractional differential equation; Caputo derivative; Laplace transform; Riemann-Liouville derivative; Mittag-Leffler function

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
[2]	Bansal, M.; Jain, R., Analytical solution of Bagley Torvik equation by generalize differential transform, Int. J. Pure Appl. Math. 110 (2016), 265-273
[3]	Chung, W. S.; Jung, M., Fractional damped oscillators and fractional forced oscillators, J. Korean Phys. Soc. 64 (2014), 186-191
[4]	Debnath, L., Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 2003 (2003), 3413-3442 · Zbl 1036.26004
[5]	Debnath, L.; Bhatta, D., Integral Transforms and Their Applications, Chapman & Hall/CRC, Boca Raton (2007) · Zbl 1113.44001
[6]	Diethelm, K.; Ford, N. J., Numerical solution of the Bagley-Torvik equation, BIT 42 (2002), 490-507 · 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) · 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 · 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 · 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 · 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) · 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 · Zbl 1360.74035
[15]	Pálfalvi, A., Efficient solution of a vibration equation involving fractional derivatives, Int. J. Non-Linear Mech. 45 (2010), 169-175
[16]	Prabhakar, T. R., A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7-15 · 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 · Zbl 1382.34010