Editorial Fractional Calculus and Its Applications in

EditorialFractional Calculus and Its Applications inApplied Mathematics and Other Sciences

Santanu Saha Ray,1 Abdon Atangana,2 S. C. Oukouomi Noutchie,3

Muhammet Kurulay,4 Necdet Bildik,5 and Adem Kilicman6

1Department of Mathematics, National Institute of Technology, Rourkela 769008, India2Department of Applied Mathematics, Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa3Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa4Department of Mathematics, Yildiz Technical University, 34220 Istanbul, Turkey5Department of Mathematics, Faculty of Art & Sciences, Celal Bayar University, Muradiye Campus, 45047 Manisa, Turkey6Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to Santanu Saha Ray; [email protected]

Received 9 September 2014; Accepted 9 September 2014; Published 21 December 2014

Copyright © 2014 Santanu Saha Ray et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The subject of fractional calculus has applications in diverseand widespread fields of engineering and science such aselectromagnetics, viscoelasticity, fluid mechanics, electro-chemistry, biological population models, optics, and signalsprocessing. It has been used to model physical and engi-neering processes that are found to be best described byfractional differential equations. The fractional derivativemodels are used for accurate modelling of those systemsthat require accurate modelling of damping. In these fields,various analytical and numerical methods including theirapplications to new problems have been proposed in recentyears. This special issue on “Fractional Calculus and itsApplications in Applied Mathematics and Other Sciences”is devoted to study the recent works in the above fieldsof fractional calculus done by the leading researchers. Thepapers for this special issue were selected after a careful andstudious peer-review process.

Mathematical modelling of real-life problems usuallyresults in fractional differential equations and various otherproblems involving special functions ofmathematical physicsas well as their extensions and generalizations in one ormore variables. In addition, most physical phenomena offluid dynamics, quantum mechanics, electricity, ecologicalsystems, and many other models are controlled within their

domain of validity by fractional order PDEs. Therefore, itbecomes increasingly important to be familiar with all tradi-tional and recently developed methods for solving fractionalorder PDEs and the implementations of these methods.

The aim of this special issue is to bring together theleading researchers of diverse fields of engineering includingapplied mathematicians and allow them to share their inno-vative research work. Analytical and numerical methods withadvanced mathematical modelling and recent developmentsof differential and integral equations of arbitrary order arisingin physical systems are included in themain focus of the issue.

Accordingly, various papers on fractional differentialequations have been included in this special issue aftercompleting a heedful, rigorous, and peer-review process.Theissue contains eight research papers. The issue of robuststability for fractional order Hopfield neural networks withparameter uncertainties is rigorously investigated. Based onthe fractional order Lyapunov direct method, the sufficientcondition of the existence, uniqueness, and globally robuststability of the equilibrium point is presented. Moreover, thesufficient condition of the robust synchronization betweensuch neural systems with the same parameter uncertain-ties is proposed owing to the robust stability analysis ofits synchronization error system. In addition, for different

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 849395, 2 pageshttp://dx.doi.org/10.1155/2014/849395

2 Mathematical Problems in Engineering

parameter uncertainties, the quasi-synchronization betweenthe classes of neural networks is investigated with linearcontrol. And the quasi-synchronization error bound canbe controlled by choosing the suitable control parameters.Moreover, robust synchronization andquasi-synchronizationbetween the classes of neural networks are discussed. Theauthors have discussed the robust stability and synchroniza-tion for the fractional order delayed neural networks (FDNN)with parameter uncertainties.

Some selected wavelet methods have been investigatedin order to analyze their accuracy and efficiency with regardto solving linear and nonlinear fractional differential equa-tions. The authors have discussed the challenges faced byresearchers in this field and emphasize the importance ofinterdisciplinary effort for advancing the study on variouswavelets in order to solve differential equations of arbitraryorder. Several wavelet methods such as Haar wavelet method,cubic B-spline wavelet method, Legendre wavelet method,Legendre multiwavelet method, and Chebyshev waveletmethodhave been examined for solving fractional differentialequations. The Legendre multiwavelet method along withGalerkin method can be applied for providing approximatesolutions for initial value problems of fractional nonlinearpartial differential equations. Using these wavelet methodsthe fractional differential equations have been reduced to asystem of algebraic equations and this system can be easilysolved by any usual methods.

The distributed coordination of fractional multiagentsystemswith external disturbances is also discussed.The stateobserver of fractional dynamical system is presented, and anadaptive pinning controller is designed for a little part ofagents in multiagent systems without disturbances. Based ondisturbance observers, the controllers are composited withthe pinning controller and the state observer. By applyingthe stability theory of fractional order dynamical systems,the distributed coordination of fractional multiagent systemswith external disturbances can be reached asymptotically.

Two integral operators involving Appell’s functions orHorn’s function in the kernel are considered. Composition ofsuch functions with generalized Bessel functions of the firstkind is expressed in terms of generalized Wright functionand generalized hypergeometric series. Many special cases,including cosine and sine function, are also discussed bythe authors and some profound consequences have beendrawn.

The existence of solutions for a nonlinear boundaryvalue problem of impulsive fractional differential equations isstudied with 𝑝-Laplacian operator.The research of boundaryvalue problems for 𝑝-Laplacian equations of fractional orderhas just begun in recent years.

There are very few results about chaos synchronization ofthe fractional order time-delay chaotic systems available inthe literature. Here chaos synchronization of different frac-tional order time-delay chaotic systems is also considered.Based on the Laplace transform theory, the conditions forachieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active controltechnique. To verify the effectiveness and feasibility of thedeveloped method, numerical simulations are provided by

the authors. The authors imposed an improved version ofAdams-Bashforth-Moulton method.

Another paper contains the analytical solutions of thefractional order partial differential equations with time andspace fractional derivatives which have been solved bymeans of the homotopy decomposition method (HDM). Toshow the accuracy and efficiency of this method, three testexamples have been considered. The evaluations show thatthe homotopy decompositionmethod is extremely successfuland suitable. The authors claimed that HDM is an influentialand professional tool for FPDEs. He also proclaimed thatHDM is superior to the ADM, VIM, HAM, and HPM.

The hydrodynamic dispersion equation was general-ized using the concept of variational order derivative. Thedevelopment of partial differential equations such as theadvection-dispersion equation (ADE) begins with assump-tions about the random behavior of a single particle. Whenassumptions underlying the ADE are relaxed, a fractionalADE (FADE) can arise, with a noninteger-order derivativeon time or space terms. Fractional ADEs are nonlocal;they describe transport affected by hydraulic conditionsat a distance. Space fractional ADEs arise when velocityvariations are heavy tailed and describe particle motionthat accounts for variation in the flow field over the entiresystem. The modified equation was numerically solved viathe Crank-Nicholson scheme. The stability and convergenceof the scheme in this case were presented. The numericalsimulations show that the modified equation is more reliablein predicting the movement of pollution in the deformableaquifers than the constant fractional and integer derivatives.

At present, the use of fractional order partial differentialequation in real-physical systems is commonly encounteredin the fields of science and engineering. The efficient com-putational tools are required for analytical and numericalapproximations of such physical models. The present issuehas addressed recent trends and developments regarding theanalytical and numerical methods that may be used in thefractional order dynamical systems. Eventually, it may beexpected that the present special issue would certainly helpfulto explore the researchers with their new arising fractionalorder problems and elevate the efficiency and accuracy of thesolution methods for those problems in use nowadays.

Santanu Saha RayAbdon Atangana

S. C. Oukouomi NoutchieMuhammet Kurulay

Necdet BildikAdem Kilicman

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Editorial Fractional Calculus and Its Applications in