Determination of order in linear fractional differential equations
-
,
Abstract
The order of fractional differential equations (FDEs) has been proved to be of great importance in an accurate simulation of the system under study. In this paper, the orders of some classes of linear FDEs are determined by using the asymptotic behaviour of their solutions. Specifically, it is demonstrated that the decay rate of the solutions is influenced by the order of fractional derivatives. Numerical investigations are conducted into the proven formulas.
Acknowledgements
The authors are grateful to Professor Masahiro Yamamoto for pointing out the papers [13], [25], [7].
References
[1] R.P. Agarwal, A propos d'une note de M. Pierre Humbert. CR Acad. Sci. Paris 236, No 21 (1953), 2031–2032.Search in Google Scholar
[2] O.P. Agrawal, O. Defterli, D. Baleanu, Fractional optimal control problems with several state and control variables. J. Vib. Control. 16, No 13 (2010), 1967–1976.10.1177/1077546309353361Search in Google Scholar
[3] D. Baleanu, Z.B. Gü;venç, J.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications. Springer, New York (2010).10.1007/978-90-481-3293-5Search in Google Scholar
[4] M. Caputo, Linear models of dissipation whose Q is almost frequency independent–II. Geophys. J. Int. 13, No 5 (1967), 529–539.10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar
[5] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, No 2 (2015), 1–13.Search in Google Scholar
[6] A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997).10.1007/978-3-7091-2664-6Search in Google Scholar
[7] J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl. 25, No 11 (2009), # 115002.10.1088/0266-5611/25/11/115002Search in Google Scholar
[8] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Heidelberg (2010).10.1007/978-3-642-14574-2Search in Google Scholar
[9] M. D'Ovidio, E. Orsingher, B. Toaldo, Time-changed processes governed by space-time fractional telegraph equations. Stoch. Anal. Appl. 32, No 6 (2014), 1009–1045.10.1080/07362994.2014.962046Search in Google Scholar
[10] R. Garrappa, The Mittag-Leffler function, MATLAB Central File Exchange (2014).Search in Google Scholar
[11] R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM J. Numer. Anal. 53, No 3 (2015), 1350–1369.10.1137/140971191Search in Google Scholar
[12] R. Gorenflo, A.A. Kilbas, F. Mainardi, and S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer Monographs in Mathematics, Springer-Verlag, Berlin-Heidelberg (2014).10.1007/978-3-662-43930-2Search in Google Scholar
[13] Y. Hatano, J. Nakagawa, S. Wang, M. Yamamoto, Determination of order in fractional diffusion equation. J. Math-for-Ind. 5A (2013), 51–57.Search in Google Scholar
[14] P. Humbert, Quelques résultats relatifs à la fonction de Mittag-Leffler. Comptes rendus hebdomadaires des seances de l'academie des sciences 236, No 15 (1953), 1467–1468.Search in Google Scholar
[15] R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems. Math. Methods Appl. Sci. 37, No 11 (2014), 1668–1686.10.1002/mma.2928Search in Google Scholar
[16] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar
[17] S.D. Lin, C.H. Lu, Laplace transform for solving some families of fractional differential equations and its applications. Adv. Difference Equ. 2013, No 1 (2013), 137.10.1186/1687-1847-2013-137Search in Google Scholar
[18] H. Lopushanska, V. Rapita, Inverse coefficient problem for semi-linear fractional telegraph equation. Electr. J. Differ. Equ. 2015, No 153 (2015), 1–13.Search in Google Scholar
[19] A.O. Lopushansky, H.P. Lopushanska, One inverse problem for the diffusion-wave equation in bounded domain. Ukr. Math. J. 66, No 5 (2014), 743–757.10.1007/s11253-014-0969-9Search in Google Scholar
[20] R.L. Magin, Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59, No 5 (2010), 1586–1593.10.1016/j.camwa.2009.08.039Search in Google Scholar
[21] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010).10.1142/p614Search in Google Scholar
[22] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1998).Search in Google Scholar
[23] Y.A. Rossikhin, M.V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63, No 1 (2010), # 010801.10.1115/1.4000563Search in Google Scholar
[24] J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado, Advances in Fractional Calculus. Springer, Dordrecht (2007).10.1007/978-1-4020-6042-7Search in Google Scholar
[25] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, No 1 (2011), 426–447.10.1016/j.jmaa.2011.04.058Search in Google Scholar
[26] D. Valério, J.T. Machado, V. Kiryakova, Some pioneers of the applications of fractional calculus. Fract. Calc. Appl. Anal. 17, No 2 (2014), 552–578; 10.2478/s13540-014-0185-1https://www.degruyter.com/view/j/fca.2014.17.issue-2/issue-files/fca.2014.17.issue-2.xml.Search in Google Scholar
[27] A. Wiman, Über den fundamentalsatz in der teorie der funktionen Ea(x). Acta Math. 29, No 1 (1905), 191–201.10.1007/BF02403202Search in Google Scholar
[28] Y. Zhang, X. Xu, Inverse source problem for a fractional diffusion equation. Inverse Probl. 27, No 3 (2011), Art. # 035010.10.1088/0266-5611/27/3/035010Search in Google Scholar
© 2018 Diogenes Co., Sofia
