Abstract.
The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.
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References
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
K. Diethelm, The Analysis of Fractional Differential Equations (Springer Press, Heidelberg, 2010)
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)
Yu.A. Rossikhin, M.V. Shitikova, Appl. Mech. Rev. 50, 15 (1997)
L. Gaul, Mech. Syst. Signal Process. 13, 1 (1999)
N. Shimizu, W. Zhang, JSME Int. J., Ser. C 42, 825 (1999)
Yu.A. Rossikhin, M.V. Shitikova, Shock Vib. Dig. 36, 3 (2004)
M. Xu, W. Tan, Sci. China, Ser. G 49, 257 (2006)
F. Mainardi, R. Gorenflo, Fractional Calculus Appl. Anal. 10, 369 (2007)
V.M. Zelenev, S.I. Meshkov, Y.A. Rossikhin, J. Appl. Mech. Tech. Phys. 11, 290 (1970)
R.L. Bagley, P.J. Torvik, ASME J. Appl. Mech. 51, 294 (1984)
Yu.A. Rossikhin, M.V. Shitikova, Appl. Mech. Rev. 63, 010801-01-010801-52 (2010)
W.H. Deng, Ph.D. Thesis (Shanghai University, 2007) (in Chinese)
P.G. Nutting, J. Franklin Inst. 191, 679 (1921)
P.G. Nutting, J. Franklin Inst. 235, 513 (1943)
G.W. Scott Blair, M. Reiner, Appl. Sci. Res. 2, 225 (1951)
C.F. Lorenzo, T.T. Hartley, NASA TP-1998-208415 (1998)
C.F. Lorenzo, T.T. Hartley, Int. J. Appl. Math Comput. Sci. 3, 249 (2000)
C.F. Lorenzo, T.T. Hartley, ASME J. Comput. Nonlinear Dyn. 3, 021101-1 (2008)
J. Sabatier, M. Merveillaut, R. Malti, A. Oustaloup, Commun. Nonlinear. Sci. Numer. Simulat. 15, 1318 (2010)
K. Diethelm, N.J. Ford, A.D. Freed, Nonlinear Dyn. 29, 3 (2002)
J.L. Besson, E. Streicher, T. Chartier, P. Goursat, J. Mater. Sci. Lett. 5, 803 (1986)
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Du, M., Wang, Z. Initialized fractional differential equations with Riemann-Liouville fractional-order derivative. Eur. Phys. J. Spec. Top. 193, 49–60 (2011). https://doi.org/10.1140/epjst/e2011-01380-8
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DOI: https://doi.org/10.1140/epjst/e2011-01380-8