Fractional Differential Equations
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Hindawi Publishing Corporation International Journal of Differential Equations Volume 2010, Article ID 215856, 2 pages doi:10.1155/2010/215856 Editorial Fractional Differential Equations Fawang Liu, 1 Mark M. Meerschaert, 2 Shaher Momani, 3 Nikolai N. Leonenko, 4 Wen Chen, 5 and Om P. Agrawal 6 1 School of Mathematical Sciences, Queensland University of Technology, P.O. Box 2434, Brisbane, Qeensland 4001, Australia 2 Department of Statistics and Probability, Michigan State University, A416 Wells Hall, East Lansing, MI 48823, USA 3 Department of Mathematics, Mu’tah University, P.O. Box 7, Mu’tah 61710, Jordan 4 School of Mathematics, Cardiff University, Cardiff CF2 4YH, UK 5 Department of Engineering Mechanics, Hohai University, Xikang Road Number 1, Nanjing, Jiangsu 210098, China 6 Department of Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, USA Correspondence should be addressed to Fawang Liu, [email protected] Received 13 April 2010; Accepted 13 April 2010 Copyright q 2010 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. It is my pleasure to present this special issue of the International Journal of Differential Equations dedicated to Fractional Differential Equations FDEs. Fractional differential equations are generalizations of ordinary differential equations to an arbitrary noninteger order. Fractional differential equations have attracted consider- able interest because of their ability to model complex phenomena. These equations capture nonlocal relations in space and time with power-law memory kernels. Due to the extensive applications of FDEs in engineering and science, research in this area has grown significantly all around the world. This special issue of Fractional Differential Equations consists of one review article paper 1 and 12 original articles covering various aspects of FDEs and their applications by the prominent researchers in the field. The review article surveys the properties of a transcendental function of the Wright type, nowadays known as the M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes. The second paper presents a computationally effective numerical method for the time-space fractional Fokker-Planck equations with a nonlinear source term, together with a stability and convergence analysis of the numerical method.
Transcript of Fractional Differential Equations
Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2010, Article ID 215856, 2 pagesdoi:10.1155/2010/215856
EditorialFractional Differential Equations
Fawang Liu,1 Mark M. Meerschaert,2 Shaher Momani,3Nikolai N. Leonenko,4 Wen Chen,5 and Om P. Agrawal6
1 School of Mathematical Sciences, Queensland University of Technology, P.O. Box 2434,Brisbane, Qeensland 4001, Australia
2 Department of Statistics and Probability, Michigan State University, A416 Wells Hall,East Lansing, MI 48823, USA
3 Department of Mathematics, Mu’tah University, P.O. Box 7, Mu’tah 61710, Jordan4 School of Mathematics, Cardiff University, Cardiff CF2 4YH, UK5 Department of Engineering Mechanics, Hohai University, Xikang Road Number 1,Nanjing, Jiangsu 210098, China
6 Department of Mechanical Engineering and Energy Processes, Southern Illinois University,Carbondale, IL 62901, USA
Correspondence should be addressed to Fawang Liu, [email protected]
Received 13 April 2010; Accepted 13 April 2010
Copyright q 2010 Fawang Liu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
It is my pleasure to present this special issue of the International Journal of DifferentialEquations dedicated to Fractional Differential Equations (FDEs).
Fractional differential equations are generalizations of ordinary differential equationsto an arbitrary (noninteger) order. Fractional differential equations have attracted consider-able interest because of their ability to model complex phenomena. These equations capturenonlocal relations in space and time with power-law memory kernels. Due to the extensiveapplications of FDEs in engineering and science, research in this area has grown significantlyall around the world.
This special issue of Fractional Differential Equations consists of one review article(paper 1) and 12 original articles covering various aspects of FDEs and their applications bythe prominent researchers in the field.
The review article surveys the properties of a transcendental function of the Wrighttype, nowadays known as the M-Wright function, entering as a probability densityin a relevant class of self-similar stochastic processes. The second paper presents acomputationally effective numerical method for the time-space fractional Fokker-Planckequations with a nonlinear source term, together with a stability and convergence analysisof the numerical method.
2 International Journal of Differential Equations
The third paper uses He’s variational iterationmethod for solving the fractional Riccatidifferential equation. The fourth paper proposes a fractional iteration method and has beenapplied to study the Hirota-Satsuma coupled KdV of time-fractional-order equation.
The fifth paper develops a formulation for fractional time-optimal control problems.The sixth paper derives conditions for the speed of spread of solutions of fractionalscalar reaction-diffusion equations to be finite or infinite. The seventh paper studies theRiesz fractional derivatives for the Korteweg-de Vries solitons and proves that these Rieszpotentials and their Hilbert transforms are linearly independent solutions of a Sturm-Liouville problem.
The eighth paper gives the analytical solution of the linear fractionally dampedoscillator equation by means of Laplace transform. The ninth paper proves the existenceand uniqueness of a positive solution to the nonzero boundary value problem for a coupledsystem of fractional differential equations. The tenth paper gives sufficient conditions onthe existence of a periodic solution for a class of compound singular fractional differentialsystems with delay, involving Nishimoto fractional derivative. The eleventh paper discussesthe existence and uniqueness of the solutions of the nonhomogeneous linear differentialequations of arbitrary positive real order by using the fractional B-Splines wavelets andthe Mittag-Leffler function. The twelfth paper proves the existence of solutions of thefractional Langevin equation with two different fractional orders in a Banach space usingthe contraction mapping principle and Krasnoselskii’s fixed point theorem. The final paperaddresses the selection and meaning of variable-order operators for dynamic modelling.
Thus, this special issue provides a wide spectrum of current research in the area ofFDEs, and I hope that experts in this and related fields would find it useful.
Finally, I would like to acknowledge the guest editors Mark M. Meerschaert, ShaherMomani, Nikolai N. Leonenko, We Chen, and Om P. Agrawal for their assistance inproducing this special edition within the specified timeframe.
Fawang LiuMark M. Meerschaert
Shaher MomaniNikolai N. Leonenko
Wen ChenOm P. Agrawal
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