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Editorial Fractional and Time-Scales Differential Equations Dumitru Baleanu, 1 Ali H. Bhrawy, 2 Delfim F. M. Torres, 3 and Soheil Salahshour 4 1 Department of Mathematics and Computer Sciences, Cankaya University, TR-06810 Yenimahalle, Ankara, Turkey 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal 4 Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran Correspondence should be addressed to Dumitru Baleanu; [email protected] Received 30 December 2013; Accepted 30 December 2013; Published 9 February 2014 Copyright © 2014 Dumitru Baleanu et al. is 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. e theory and applications of fractional differential equa- tions (FDEs) are gaining more relevance since they are used extensively in the modeling of various processes in physics, chemistry, engineering, and other areas of science. As it is known, the behavior of dynamics of most complex systems of the real world phenomena has memory. erefore, the modeling of dynamics of these types of systems by FDEs has more advantages than the classical ones, in which such effects are neglected. On the other hand, the time-scales formalism unifies the theories of difference and differential equations. Accordingly, the time-scales analysis constitutes a good tool to study both discrete and continuous systems. Advanced analytical and numerical techniques and com- putational methods are of important interest for classical, fractional, fuzzy fractional, and time-scales differential equa- tions. e papers of this special issue contain some new algo- rithms and techniques designed to investigate classical, frac- tional, fractal, fuzzy fractional, and time-scales differential equations of general interest. New insights of existence and uniqueness theorems of some differential equations were also presented. In the following we summarize briefly the content of the special issue. e second Noether theorem, the local observability of systems, and the fractional Cauchy problem within Riemann-Liouville fractional delta derivative on time scales were reported and the solutions of fractional discrete systems with sequential h-differences were obtained. Besides, the oscillation criteria for fourth-order nonlinear dynamic equations on time scales were depicted. Local fractional series expansion method for wave and diffusion equations and some mappings for special functions on Cantor sets were presented. Application of fuzzy fractional kinetic equations to modeling of the acid hydrolysis reaction, the solutions of linear fractional differential equations with uncertainty, and an operational matrix based on Legendre polynomials for solving fuzzy fractional-order differential equations were also investigated. A class of fractional-order differential models of biological systems with memory to model the interaction of immune system with tumor cells and with HIV infection of CD4 + T-cells was reported. Fractional- order total variation image restoration based on primal-dual algorithm and the fractional dynamics of genetic algorithms using hexagonal space tessellation were pointed out. e positive solution using bifurcation techniques for boundary value problems of fractional differential equations was a contribution of our special issue. e existence results for a class of fractional differential equations with boundary value conditions and with delay, a modified generalized Laguerre spectral methods for fractional differential equations on the half line, a Jacobi collocation method for solving nonlin- ear Burgers-type equations, and new wavelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth kinds were shown. e investigation of the nonlinear frac- tional Jaulent-Miodek and Whitham-Broer-Kaup equations within Sumudu transform, the Bernstein operational matri- ces applied for solving the fractional quadratic Riccati dif- ferential equations, and the approximate solutions of Fisher’s type equations with variable coefficients are topics covered by Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 365250, 2 pages http://dx.doi.org/10.1155/2014/365250

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Page 1: Editorial Fractional and Time-Scales Differential Equationsdownloads.hindawi.com/journals/aaa/2014/365250.pdfEditorial Fractional and Time-Scales Differential Equations DumitruBaleanu,

EditorialFractional and Time-Scales Differential Equations

Dumitru Baleanu,1 Ali H. Bhrawy,2 Delfim F. M. Torres,3 and Soheil Salahshour4

1 Department of Mathematics and Computer Sciences, Cankaya University, TR-06810 Yenimahalle, Ankara, Turkey2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia3 Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal4 Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran

Correspondence should be addressed to Dumitru Baleanu; [email protected]

Received 30 December 2013; Accepted 30 December 2013; Published 9 February 2014

Copyright © 2014 Dumitru Baleanu 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 theory and applications of fractional differential equa-tions (FDEs) are gaining more relevance since they are usedextensively in the modeling of various processes in physics,chemistry, engineering, and other areas of science. As it isknown, the behavior of dynamics of most complex systemsof the real world phenomena has memory. Therefore, themodeling of dynamics of these types of systems by FDEs hasmore advantages than the classical ones, in which such effectsare neglected. On the other hand, the time-scales formalismunifies the theories of difference and differential equations.Accordingly, the time-scales analysis constitutes a good toolto study both discrete and continuous systems.

Advanced analytical and numerical techniques and com-putational methods are of important interest for classical,fractional, fuzzy fractional, and time-scales differential equa-tions.

The papers of this special issue contain some new algo-rithms and techniques designed to investigate classical, frac-tional, fractal, fuzzy fractional, and time-scales differentialequations of general interest. New insights of existence anduniqueness theorems of some differential equations were alsopresented.

In the following we summarize briefly the content ofthe special issue. The second Noether theorem, the localobservability of systems, and the fractional Cauchy problemwithin Riemann-Liouville fractional delta derivative on timescales were reported and the solutions of fractional discretesystems with sequential h-differences were obtained. Besides,the oscillation criteria for fourth-order nonlinear dynamicequations on time scaleswere depicted. Local fractional series

expansion method for wave and diffusion equations andsome mappings for special functions on Cantor sets werepresented. Application of fuzzy fractional kinetic equationsto modeling of the acid hydrolysis reaction, the solutionsof linear fractional differential equations with uncertainty,and an operational matrix based on Legendre polynomialsfor solving fuzzy fractional-order differential equations werealso investigated. A class of fractional-order differentialmodels of biological systems with memory to model theinteraction of immune system with tumor cells and withHIV infection of CD4+ T-cells was reported. Fractional-order total variation image restoration based on primal-dualalgorithm and the fractional dynamics of genetic algorithmsusing hexagonal space tessellation were pointed out. Thepositive solution using bifurcation techniques for boundaryvalue problems of fractional differential equations was acontribution of our special issue. The existence results for aclass of fractional differential equations with boundary valueconditions and with delay, a modified generalized Laguerrespectral methods for fractional differential equations on thehalf line, a Jacobi collocation method for solving nonlin-ear Burgers-type equations, and new wavelets collocationmethod for solving second-order multipoint boundary valueproblems using Chebyshev polynomials of third and fourthkinds were shown. The investigation of the nonlinear frac-tional Jaulent-Miodek and Whitham-Broer-Kaup equationswithin Sumudu transform, the Bernstein operational matri-ces applied for solving the fractional quadratic Riccati dif-ferential equations, and the approximate solutions of Fisher’stype equations with variable coefficients are topics covered by

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 365250, 2 pageshttp://dx.doi.org/10.1155/2014/365250

Page 2: Editorial Fractional and Time-Scales Differential Equationsdownloads.hindawi.com/journals/aaa/2014/365250.pdfEditorial Fractional and Time-Scales Differential Equations DumitruBaleanu,

2 Abstract and Applied Analysis

our special issue. Another part of contributionswas focussingon two efficient generalized Laguerre spectral algorithms forsome fractional initial value problems, the approximation ofeigenvalues of Sturm-Liouville problems by using Hermiteinterpolation and the presentation of the numerical solutionsof fractional Fokker-Planck equations by using the iterativeLaplace transform method.

Dumitru BaleanuAli H. Bhrawy

Delfim F. M. TorresSoheil Salahshour

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