Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation

References

• West BJ. Fractional calculus view of complexity: tomorrow's science. Boca Rato: CRC Press; 2016.
   Google Scholar
• Boyadzhiev D, Kiskinov H, Zahariev A. Integral representation of solutions of fractional system with distributed delays. Int Trans Spec Funct. 2018;29(9):725–744. doi: 10.1080/10652469.2018.1497025
   Web of Science ®Google Scholar
• Sun HG, Zhang Y, Baleanu D et al. A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul. 2018;64:213–231. doi: 10.1016/j.cnsns.2018.04.019
   Web of Science ®Google Scholar
• Wei YH, Sheng D, Chen YQ et al. Fractional order chattering-free robust adaptive backstepping control technique. Nonlinear Dyn. 2019;95(3):2383–2394. doi: 10.1007/s11071-018-4698-8
   Web of Science ®Google Scholar
• Liu DY, Tian Y, Boutat D, et al. An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation. Signal Processing. 2015;116:78–90. doi: 10.1016/j.sigpro.2015.04.017
   Web of Science ®Google Scholar
• Liu DY, Zheng G, Boutat D, et al. Non-asymptotic fractional order differentiator for a class of fractional order linear systems. Automatica. 2017;78:61–71. doi: 10.1016/j.automatica.2016.12.017
   Web of Science ®Google Scholar
• Srivastava HM, Yakubovich SB, Luchko YF. The convolution method for the development of new Leibniz rules involving fractional derivatives and of their integral analogues. Int Trans Spec Funct. 1993;1(2):119–134. doi: 10.1080/10652469308819014
   Google Scholar
• Tarasov VE. No violation of the leibniz rule, No fractional derivative. Commun Nonlinear Sci Numer Simul. 2013;18(11):2945–2948. doi: 10.1016/j.cnsns.2013.04.001
   Web of Science ®Google Scholar
• Tarasova VV, Tarasov VE. Elasticity for economic processes with memory: fractional differential calculus approach. Fract Differ Calculus. 2016;6(2):219–232. doi: 10.7153/fdc-06-14
   Google Scholar
• Tarasov VE. Leibniz rule and fractional derivatives of power functions. J Comput Nonlinear Dyn. 2016;11(3):ID.031014.
   Web of Science ®Google Scholar
• Chen DY, Zhang RF, Liu XZ, et al. Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks. Commun Nonlinear Sci Numer Simul. 2014;19(12):4105–4121. doi: 10.1016/j.cnsns.2014.05.005
   Web of Science ®Google Scholar
• Mujumdar A, Tamhane B, Kurode S. Observer-based sliding mode control for a class of noncommensurate fractional-order systems. IEEE ASME Trans Mechatron. 2015;20(5):2504–2512. doi: 10.1109/TMECH.2014.2386914
   Web of Science ®Google Scholar
• Oldham K, Spanier J. The fractional calculus theory and applications of differentiation and integration to arbitrary order. San Diego: Elsevier; 1974.
   Google Scholar
• Sayevand K, Machado JT, Baleanu D. A new glance on the Leibniz rule for fractional derivatives. Commun Nonlinear Sci Numer Simul. 2018;62:244–249. doi: 10.1016/j.cnsns.2018.02.037
   Web of Science ®Google Scholar
• Osler TJ. The fractional derivative of a composite function. SIAM J Math Anal. 1970;1(2):288–293. doi: 10.1137/0501026
   Google Scholar
• Osler TJ. Fractional derivatives and Leibniz rule. Am Math Monthly. 1971;78(6):645–649. doi: 10.1080/00029890.1971.11992819
   Web of Science ®Google Scholar
• Osler TJ. The integral analog of the Leibniz rule. Math Comput. 1972;26(120):903–915.
   Web of Science ®Google Scholar
• Tremblay R, Gaboury S, Fugere BJ. A new Leibniz rule and its integral analogue for fractional derivatives. Inte Trans Spec Funct. 2013;24(2):111–128. doi: 10.1080/10652469.2012.668904
   Web of Science ®Google Scholar
• Aguila-Camacho N, Duarte-Mermoud MA, Gallegos JA. Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul. 2014;19(9):2951–2957. doi: 10.1016/j.cnsns.2014.01.022
   Web of Science ®Google Scholar
• Aguila-Camacho N, Duarte-Mermoud MA. Comments on ‘Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks’. Commun Nonlinear Sci Numer Simul. 2015;25(1):145–148. doi: 10.1016/j.cnsns.2015.01.013
   Web of Science ®Google Scholar
• Hua CC, Ning JH, Zhao GL, et al. Output feedback NN tracking control for fractional-order nonlinear systems with time-delay and input quantization. Neurocomputing. 2018;290:229–237. doi: 10.1016/j.neucom.2018.02.047
   Web of Science ®Google Scholar
• Ishteva M, Boyadjiev L, Scherer R. On the Caputo operator of fractional calculus and C-Laguerre functions. Math Sci Res J. 2005;9(6):161–174.
   Google Scholar
• Freed A, Diethelm K. Caputo derivatives in viscoelasticity: a non-linear finite-deformation theory for tissue. Fract Calculus Appl Anal. 2007;10(3):219–248.
   Google Scholar
• Diethelm K. The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Heidelberg: Springer; 2010.
   Google Scholar
• Ortigueira MD, Machado JAT. What is a fractional derivative?. J Comput Phys. 2015;293:4–13. doi: 10.1016/j.jcp.2014.07.019
   Web of Science ®Google Scholar
• Teodoro GS, Machado JT, De Oliveira EC. A review of definitions of fractional derivatives and other operators. J Comput Phys. 2019;388:195–208. doi: 10.1016/j.jcp.2019.03.008
   Web of Science ®Google Scholar
• Zhou X, Wei YH, Liang S, et al. Robust fast controller design via nonlinear fractional differential equations. ISA Trans. 2017;69:20–30. doi: 10.1016/j.isatra.2017.03.010
   PubMed Web of Science ®Google Scholar
• Wu GC, Baleanu D, Luo WH. Lyapunov functions for Riemann–Liouville-like fractional difference equations. Appl Math Comput. 2017;314:228–236.
   Web of Science ®Google Scholar
• Lundberg KH, Miller HR, Trumper DL. Initial conditions, generalized functions, and the laplace transform troubles at the origin. IEEE Control Syst Magaz. 2007;27(1):22–35. doi: 10.1109/MCS.2007.284506
   Web of Science ®Google Scholar
• Belkhatir Z, Laleg-Kirati TM. Parameters and fractional differentiation orders estimation for linear continuous-time non-commensurate fractional order systems. Syst Control Lett. 2018;115:26–33. doi: 10.1016/j.sysconle.2018.02.012
   Web of Science ®Google Scholar
• Williams P. Fractional calculus on time scales with Taylor's theorem. Fract Calculus Appl Anal. 2012;15(4):616–638.
   Web of Science ®Google Scholar
• Wei YH, Chen YQ, Cheng SS, et al. A note on short memory principle of fractional calculus. Fract Calculus Appl Anal. 2017;20(6):1382–1404.
   Web of Science ®Google Scholar
• Wei YH, Chen YQ, Gao Q, et al. Infinite series representation of fractional calculus: theory and applications. arXiv preprint. 2018;ArXiv:1901.11134.
   Google Scholar
• Wei YH, Gao Q, Liu DY, et al. On the series representation of nabla discrete fractional calculus. Commun Nonlinear Sci Numer Simul. 2019;69:198–218. doi: 10.1016/j.cnsns.2018.09.024
   Web of Science ®Google Scholar
• Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential eqnations, to methods of their solution and some of their applications. San Diego: Academic Press; 1999.
   Google Scholar
• Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: theory and applications. Amsterdam: Gordon and Breach Science Publishers; 1993.
   Google Scholar
• Luque R, Galué L. The application of a generalized Leibniz rule to infinite sums. Int Trans Spec Funct. 1999;8(1–2):65–76. doi: 10.1080/10652469908819217
   Web of Science ®Google Scholar
• Babakhani A, Yadollahzadeh M, Neamaty A. Some properties of pseudo-fractional operators. J Pseudo-Differ Oper Appl. 2017;9(3):677–700. doi: 10.1007/s11868-017-0206-z
   Web of Science ®Google Scholar
• Sousa J, de Oliveira EC. Leibniz type rule: Ψ-Hilfer fractional derivative. arXiv. 2018; Preprint arXiv: 1811.02717.
   Google Scholar