editorial analysis of fractional dynamic...

EditorialAnalysis of Fractional Dynamic Systems

Fawang Liu,1 Richard Magin,2 Changpin Li,3 Alla Sikorskii,4 and Santos Bravo Yuste5

1 School of Mathematical Sciences, Queensland University of Technology, P.O. Box 2434, Brisbane, QLD 4001, Australia2 Department of Bioengineering, University of Illinois, 851 South Morgan Street, Chicago, IL 60607, USA3Department of Mathematics, Shanghai University, Shanghai 200444, China4Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA5Departamento de Fı́sica, Universidad de Extremadura, Avenida de Elvas s/n, 06071 Badajoz, Spain

Correspondence should be addressed to Fawang Liu; [email protected]

Received 9 March 2014; Accepted 9 March 2014; Published 23 April 2014

Copyright © 2014 Fawang Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Due to the extensive applications of fractional differentialequations (FDEs) in engineering and science, research in thisarea has grown significantly. Fractional Dynamic Systemsare described by FDEs, and this special issue consists of 8original articles covering various aspects of FDEs and theirapplications written by prominent researchers in the field.

Paper titled “Sinc-Chebyshev collocation method for aclass of fractional diffusion-wave equations” by Z. Mao et al.investigates the numerical solution for a class of fractionaldiffusion-wave equations with a variable coefficient. Theapproach is based on the collocation technique where theshiftedChebyshev polynomials in time and the sinc functionsin space are utilized, respectively.

Paper titled “Leapfrog/finite element method of fractionaldiffusion equation” by Z. Zhao and Y. Zheng analyzes afully discrete leapfrog/Galerkin finite element method for thenumerical solution of the space fractional order diffusion.The fractional diffusion equations are discretized in space bythe finite elementmethod and in time by the explicit leap-frogscheme. For the resulting fully discrete, conditionally stablescheme the authors use an L2-error bound of finite elementaccuracy and of second order in time.

Paper titled “Determination of coefficients of high-orderschemes for Riemann-Liouville derivative” byR.Wu et al. stud-ies and develops recursion formulas to compute the coeffi-cients in the higher-order schemes approximating Riemann-Liouville integrals and derivatives. The fractional Runge’sphenomena are observed when the pth-order numericalscheme (p = 7, 8, 9, 10) is used, which means that pth-orderalgorithms (𝑝 ≥ 7) for Riemann-Liouville derivative seemnot to be appropriate.

Paper titled “Numerical solution of some types of fractionaloptimal control problems” byN.H. Sweilam et al. develops twodifferent approaches based on the spectral method for sometypes of fractional optimal control problems. The spectralmethod given here is based on the Chebyshev polynomi-als to approximate the unknown functions. Necessary andsufficient optimality conditions are obtained in the firstapproach, where the Hamiltonian functional is defined. Inthe second approach the Clenshaw and Curtis procedurefor the numerical integration of nonsingular functions andRayleigh-Ritz method are used to evaluate both the state andcontrol variables.

Paper titled “Stability analysis of distributed order frac-tional Chen system” by H. Aminikhah et al. derives thesufficient and necessary conditions of stability of nonlineardistributed order fractional system and then the integer-order Chen system is generalized into the distributed orderfractional domain. Based on the asymptotic stability theoryof nonlinear distributed order fractional systems, the stabilityof distributed order fractional Chen system is discussed.

Paper titled “Monotonicity, concavity, and convexity offractional derivative of functions” byX.-F. Zhou et al. discussesthe monotonicity, the concavity, and the convexity of somefunctions arising in solutions of fractional differential equa-tions. The results can be used in describing properties of thesolutions.

Paper titled “A directed continuous time random walkmodel with jump length depending onwaiting time” by L. Shi etal. proposes a coupled directed continuous time randomwalkmodel, where the randomwalker jumps toward one directionand the waiting time between jumps affects the subsequent

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 760634, 2 pageshttp://dx.doi.org/10.1155/2014/760634

2 The Scientific World Journal

jump. The limit distribution of the continuous time randomwalk and the corresponding evolving equations are derived.

Finally, paper titled “A parallel algorithm for the two-dimensional time fractional diffusion equation with implicitdifference method” by C. Gong et al. introduces and discussesa parallel algorithm for a two-dimensional time fractionaldifferential equation (2D-TFDE). A task distribution modelwith virtual boundary is designed for this parallel algorithm.

Thus, this special issue provides a wide spectrum ofcurrent research in the area of analysis of Fractional DynamicSystems, and we hope that experts in this and related fieldsfind it useful.

Fawang LiuRichard MaginChangpin LiAlla Sikorskii

Santos Bravo Yuste

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