Fractional calculus and its applications

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A.3712013003720130037http://doi.org/10.1098/rsta.2013.0037SectionYou have accessIntroductionFractional calculus and its applications Changpin Li Changpin Li Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China [email protected] Google Scholar Find this author on PubMed Search for more papers by this author , YangQuan Chen YangQuan Chen School of Engineering, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA Google Scholar Find this author on PubMed Search for more papers by this author and Jürgen Kurths Jürgen Kurths Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany Google Scholar Find this author on PubMed Search for more papers by this author Changpin Li Changpin Li Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China [email protected] Google Scholar Find this author on PubMed Search for more papers by this author , YangQuan Chen YangQuan Chen School of Engineering, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA Google Scholar Find this author on PubMed Search for more papers by this author and Jürgen Kurths Jürgen Kurths Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany Google Scholar Find this author on PubMed Search for more papers by this author Published:13 May 2013https://doi.org/10.1098/rsta.2013.0037Fractional calculus was formulated in 1695, shortly after the development of classical calculus. The earliest systematic studies were attributed to Liouville, Riemann, Leibniz, etc. [1,2]. For a long time, fractional calculus has been regarded as a pure mathematical realm without real applications. But, in recent decades, such a state of affairs has been changed. It has been found that fractional calculus can be useful and even powerful, and an outline of the simple history about fractional calculus, especially with applications, can be found in Machado et al. [3].Now, fractional calculus and its applications is undergoing rapid developments with more and more convincing applications in the real world [4,5]. This Theme Issue, including one review article and 12 research papers, can be regarded as a continuation of our first special issue of European Physical Journal Special Topics in 2011 [4], and our second special issue of International Journal of Bifurcation and Chaos in 2012 [5]. These selected papers were mostly reported in The Fifth Symposium on Fractional Derivatives and Their Applications (FDTA'11) that was held in Washington DC, USA in 2011.The first paper presents an overview of chaos synchronization of coupled fractional differential systems. A list of coupling schemes are presented, including one-way coupling, Pecora–Carroll coupling, active–passive decomposition coupling, bidirectional coupling and other unidirectional coupling configurations. Meanwhile, several extended concepts of synchronizations are introduced, namely projective synchronization, hybrid projective synchronization, function projective synchroni- zation, generalized synchronization and generalized projective synchronization. Corresponding to different kinds of synchronization schemes, various analysis methods are presented and discussed [6]. The rest of the papers can be roughly grouped into three parts: three papers for fundamental theories of fractional calculus [7–9], five papers for fractional modelling with applications [10–14] and four papers for numerical approaches [15–18].In the theory part, three papers focus on the existence of the solutions to the considered classes of nonlinear fractional systems, the equivalence system of the multiple-rational-order fractional system, and the reflection symmetry with applications to the Euler–Lagrange equations [7–9]. Baleanu et al. [7] use fixed-point theorems to prove the existence and uniqueness of the solutions to a class of nonlinear fractional differential equations with different boundary-value conditions. Li et al. [8] apply the properties of the fractional derivatives to change the multiple-rational-order system into the fractional system with the same order. Such a reduction makes it convenient for stability analysis and numerical simulations. The reflection symmetry and its applications to the Euler–Lagrange equations in fractional mechanics are investigated in Klimek [9], where an illustrative example is presented.The part on fractional modelling with applications consists of five papers [10–14]. Chen et al. [10] establish a fractional variational optical flow model for motion estimation from video sequences, where the experiments demonstrate the validity of the generalization of derivative order. Another fractional modelling in heat transfer with heterogeneous media is studied in Sierociuk et al. [11]. In the following paper, two-particle dispersion is explored in the context of the anomalous diffusion, where two modelling approaches related to time subordination are considered and unified in the framework of self-similar stochastic processes [12]. The last two papers in this part emphasize the applications of fractional calculus [13,14], where a novel method for the solution of linear constant coefficient fractional differential equations of any commensurate order is introduced in the former paper, and where the CRONE control-system design toolbox for the control engineering community is presented in the latter paper.The last four papers in part three are attributed to numerical approaches [15–18]. Sun et al. [15] construct a semi-discrete finite-element method for a class of temporal-fractional diffusion equations. On the other hand, an implicit numerical algorithm for the spatial- and temporal-fractional Bloch–Torrey equation is established, where stability and convergence are also considered [16]. In Fukunaga & Shimizu [17], a high-speed scheme for the numerical approach of fractional differentiation and fractional integration is proposed. In the last paper, Podlubny et al. [18] further develop Podlubny's matrix approach to discretization of non-integer derivatives and integrals, where non-equidistant grids, variable step lengths and distributed orders are considered.We try our best to organize this Theme Issue in order to offer fresh stimuli for the fractional calculus community to further promote and develop cutting-edge research on fractional calculus and its applications.AcknowledgementsWe thank all the authors for their contributions and reviewers for their efforts. We also specially thank the staff of Phil. Trans. R. Soc. A, particularly Suzanne Abbott, for their careful reading and providing constructive suggestions. The Lead Guest Editor C.L. acknowledges the financial support of the National Natural Science Foundation of China (grant no. 10872119), the Shanghai Leading Academic Discipline Project (grant no. S30104) and the Key Program of Shanghai Municipal Education Commission (grant no. 12ZZ084).FootnotesOne contribution of 14 to a Theme Issue ‘Fractional calculus and its applications’.© 2013 The Author(s) Published by the Royal Society. All rights reserved.References1Oldham KB& Spanier J.. 1974The fractional calculus.New York, NY: Academic Press. Google Scholar2Samko SG, Kilbas AA& Marichev OI.. 1993Fractional integrals and derivatives: theory and applications.Amsterdam, The Netherlands: Gordon and Breach. Google Scholar3Machado JT, Kiryakova V& Mainardi F.. 2011Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16, 1140–1153.doi:10.1016/j.cnsns.2010.05.027 (doi:10.1016/j.cnsns.2010.05.027). Crossref, ISI, Google Scholar4Li CP& Mainardi F.. 2011Editorial. Eur. Phys. J. Special Top. 193, 1–4.doi:10.1140/epjst/e2011-01377-3 (doi:10.1140/epjst/e2011-01377-3). Crossref, ISI, Google Scholar5Li CP, Chen YQ, Vinagre BM& Podlubny I.. 2012Introduction. Int. J. Bifurcation Chaos 22, 1202002.doi:10.1142/S0218127412020026 (doi:10.1142/S0218127412020026). Crossref, ISI, Google Scholar6Zhang F, Chen G, Li C& Kurths J.. 2013Chaos synchronization in fractional differential systems. Phil. Trans. R. Soc. A 371, 20120155.doi:10.1098/rsta.2012.0155 (doi:10.1098/rsta.2012.0155). Link, ISI, Google Scholar7Baleanu D, Rezapour S& Mohammadi H.. 2013Some existence results on nonlinear fractional differential equations. Phil. Trans. R. Soc. A 371, 20120144.doi:10.1098/rsta.2012.0144 (doi:10.1098/rsta.2012.0144). Link, ISI, Google Scholar8Li C, Zhang F, Kurths J& Zeng F.. 2013Equivalent system for a multiple-rational-order fractional differential system. Phil. Trans. R. Soc. A 371, 20120156.doi:10.1098/rsta.2012.0156 (doi:10.1098/rsta.2012.0156). Link, ISI, Google Scholar9Klimek M.. 2013On reflection symmetry and its application to the Euler–Lagrange equations in fractional mechanics. Phil. Trans. R. Soc. A 371, 20120145.doi:10.1098/rsta.2012.0145 (doi:10.1098/rsta.2012.0145). Link, ISI, Google Scholar10Chen D, Sheng H, Chen YQ& Xue D.. 2013Fractional-order variational optical flow model for motion estimation. Phil. Trans. R. Soc. A 371, 20120148.doi:10.01098/rsta.2012.0148 (doi:10.01098/rsta.2012.0148). Link, Google Scholar11Sierociuk D, Dzieliński A, Sarwas G, Petras I, Podlubny I& Skovranek T.. 2013Modelling heat transfer in heterogeneous media using fractional calculus. Phil. Trans. R. Soc. A 371, 20120146.doi:10.1098/rsta.2012.0146 (doi:10.1098/rsta.2012.0146). Link, ISI, Google Scholar12Pagnini G, Mura A& Mainardi F.. 2013Two-particle anomalous diffusion: probability density functions and self-similar stochastic processes. Phil. Trans. R. Soc. A 371, 20120154.doi:10.1098/rsta.2012.0154 (doi:10.1098/rsta.2012.0154). Link, ISI, Google Scholar13Lorenzo CF, Hartley TT& Malti R.. 2013Application of the principal fractional meta-trigonometric functions for the solution of linear commensurate-order time-invariant fractional differential equations. Phil. Trans. R. Soc. A 371, 20120151.doi:10.1098/rsta.2012.0151 (doi:10.1098/rsta.2012.0151). Link, ISI, Google Scholar14Lanusse P, Malti R& Melchior P.. 2013CRONE control system design toolbox for the control engineering community: tutorial and a case study. Phil. Trans. R. Soc. A 371, 20120149.doi:10.1098/rsta.2012.0149 (doi:10.1098/rsta.2012.0149). Link, ISI, Google Scholar15Sun HG, Chen W& Sze KY.. 2013A semi-discrete finite element method for a class of time-fractional diffusion equations. Phil. Trans. R. Soc. A 371, 20120268.doi:10.1098/rsta.2012.0268 (doi:10.1098/rsta.2012.0268). Link, ISI, Google Scholar16Yu Q, Liu F, Turner I& Burrage K.. 2013Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation. Phil. Trans. R. Soc. A 371, 20120150.doi:10.1098/rsta.2012.0150 (doi:10.1098/rsta.2012.0150). Link, ISI, Google Scholar17Fukunaga M& Shimizu N.. 2013A high-speed algorithm for computation of fractional differentiation and fractional integration. Phil. Trans. R. Soc. A 371, 20120152.doi:10.1098/rsta.2012.0152 (doi:10.1098/rsta.2012.0152). Link, ISI, Google Scholar18Podlubny I, Skovranek T, Vinagre Jara BM, Petras I, Verbitsky V& Chen YQ.. 2013Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders. Phil. Trans. R. Soc. A 371, 20120153.doi:10.1098/rsta.2012.0153 (doi:10.1098/rsta.2012.0153). 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