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http://www.springer.com/mathematics/analysis/book/978-3-642-30897-0

​This nine-chapter monograph introduces a rigorous investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson's type before turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular  q-Sturm–Liouville theory is also introduced; Green's function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi. Hence fractional q-calculi of the types Riemann–Liouville; Grünwald–Letnikov;  Caputo;  Erdélyi–Kober and Weyl are defined analytically. Fractional q-Leibniz rules with applications  in q-series are  also obtained with rigorous proofs of the formal  results of  Al-Salam-Verma, which remained unproved for decades. In working towards the investigation of q-fractional difference equations; families of q-Mittag-Leffler functions are defined and their properties are investigated, especially the q-Mellin–Barnes integral  and Hankel contour integral representation of  theq-Mittag-Leffler functions under consideration,  the distribution, asymptotic and reality of their zeros, establishing q-counterparts of Wiman's results. Fractional q-difference equations are studied; existence and uniqueness theorems are given and classes of Cauchy-type problems are completely solved in terms of families of q-Mittag-Leffler functions. Among many q-analogs of classical results and concepts, q-Laplace,q-Mellin and q²-Fourier transforms are studied and their applications are investigated.

Keywords: Basic Hypergeometric functions - One variable calculus - Zeros of analytics functions - q-difference equations

Related subjects: Analysis - Dynamical Systems & Differential Equations - Theoretical, Mathematical & Computational Physics

Table of contents 

• Preliminaries
• q-Difference Equations
• q-Sturm Liouville Problems
• Riemann–Liouville q-Fractional Calculi
• Other q-Fractional Calculi
• Fractional q-Leibniz Rule and Applications
• q-Mittag–Leffler Functions.- 8 Fractional q-Difference Equations
• Applications of q-Integral Transforms

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International Journal of Bifurcation and Chaos 

Volume 22, Issue 4

THEME SECTION: Fractional Dynamics and Control 

INTRODUCTION
Changpin Li, Yang Quan Chen, Blas M. Vinagre and Igor Podlubny

THEME SECTION: Fractional Dynamics and Control — Tutorials and Reviews

STABILITY OF CLOSED LOOP FRACTIONAL ORDER SYSTEMS AND DEFINITION OF DAMPING CONTOURS FOR THE DESIGN OF CONTROLLERS
Patrick Lanusse, Alain Oustaloup and Valerie Pommier-Budinger

FINITE DIFFERENCE METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS
Changpin Li and Fanhai Zeng

EXPERIENCES ON AN INTERNET LINK CHARACTERIZATION AND NETWORKED CONTROL OF A SMART WHEEL
Inés Tejado, Blas M. Vinagre, Miguel Romero, Ángel P. De Madrid and Yangquan Chen

THEME SECTION: Fractional Dynamics and Control — Papers

GENERALIZED FRACTIONAL ORDER BLOCH EQUATION WITH EXTENDED DELAY
Sachin Bhalekar, Varsha Daftardar-Gejji, Dumitru Baleanu and Richard Magin

CHAOS IN FRACTIONAL-ORDER POPULATION MODEL
Ivo Petráš

STABILITY-PRESERVING HIGH-ORDER METHODS FOR MULTITERM FRACTIONAL DIFFERENTIAL EQUATIONS
Roberto Garrappa

ON THE DNA OF ELEVEN MAMMALS
J. Tenreiro Machado, António C. Costa and Maria Dulce Quelhas

HOW TO APPROXIMATE THE FRACTIONAL DERIVATIVE OF ORDER 1 < α ≤ 2
Ercilia Sousa

ON IMPACT SCRIPTS WITH BOTH FRACTIONAL AND DRY FRICTION TYPE OF DISSIPATION
Nenad Grahovac, Miodrag Zigic and Dragan Spasic

EXISTENCE AND CONTINUATION THEOREMS OF RIEMANN–LIOUVILLE TYPE FRACTIONAL DIFFERENTIAL EQUATIONS
Chunhai Kou, Huacheng Zhou and Changpin Li

THE INCREMENTAL RATIO BASED CAUSAL FRACTIONAL CALCULUS
Manuel D. Ortigueira, Margarita Rivero and Juan J. Trujillo

A NONSTANDARD FINITE DIFFERENCE SCHEME FOR TWO-SIDED SPACE-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS
Shaher Momani, Abdullah Abu Rqayiq and Dumitru Baleanu

A FRACTIONAL ORDER UNIVERSAL HIGH GAIN ADAPTIVE STABILIZERR
Yan Li and Yangquan Chen

FRACTIONAL ORDER SPECTRA AND THEIR APPLICATION IN THE THEORY OF VISCOELASTICITY
Kai-Xin Hu and Ke-Qin Zhu

STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING
F. Hu, W. Q. Zhu and L. C. Chen

ON STABILITY OF COMMENSURATE FRACTIONAL ORDER SYSTEMS
Jocelyn Sabatier and Christophe Farges  

FINITE DIFFERENCE SCHEMES FOR VARIABLE-ORDER TIME FRACTIONAL DIFFUSION EQUATION
Hongguang Sun, Wen Chen, Changpin Li and Yangquan Chen

EXISTENCE RESULTS FOR FRACTIONAL BOUNDARY VALUE PROBLEM VIA CRITICAL POINT THEORY
Feng Jiao and Yong Zhou

PATTERN FORMATION IN FRACTIONAL REACTION–DIFFUSION SYSTEMS WITH MULTIPLE HOMOGENEOUS STATES
Bohdan Datsko, Yury Luchko and Vasyl Gafiychuk

CHAOS IN DIFFUSIONLESS LORENZ SYSTEM WITH A FRACTIONAL ORDER AND ITS CONTROL
Yong Xu, Rencai Gu, Huiqing Zhang and Dongxi Li

DEALING WITH FRACTIONAL DYNAMICS OF IP NETWORK DELAYS
Inés Tejado, S. Hassan Hosseinnia, Blas M. Vinagre, Xiaona Song and Yangquan Chen

MEAN EXIT TIME AND ESCAPE PROBABILITY FOR A TUMOR GROWTH SYSTEM UNDER NON-GAUSSIAN NOISE
Jian Ren, Chujin Li, Ting Gao, Xingye Kan and Jinqiao Duan

Tutorials and Reviews

KNEADINGS, SYMBOLIC DYNAMICS AND PAINTING LORENZ CHAOS
Roberto Barrio, Andrey Shilnikov and Leonid Shilnikov

Papers

CHAOS OF THE PROPAGATING PULSE WAVE IN A RING OF SIX-COUPLED BISTABLE OSCILLATORS
Kyohei Kamiyama and Tetsuro Endo

DYNAMICS OF A PREY-DEPENDENT DIGESTIVE MODEL WITH STATE-DEPENDENT IMPULSIVE CONTROL
Linning Qian, Qishao Lu, Jiaru Bai and Zhaosheng Feng

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS
Albert C. J. Luo and Jianzhe Huang

SCALE-FREE LUBY TRANSFORM CODES
Yuli Zhao, Francis C. M. Lau, Zhiliang Zhu and Hai Yu

MULTIPLE SOLUTIONS OF A GENERALIZED SINGULAR PERTURBED BRATU PROBLEM
Taoufik Bakri, Yuri A. Kuznetsov, Ferdinand Verhulst and Eusebius Doedel

A SPECTRAL RADIUS ESTIMATE AND ENTROPY OF HYPERCUBES
William Geller, Bruce Kitchens, Michał Misiurewicz and Michał Rams

THE FIRST "LOST" INTERNATIONAL CONFERENCE ON NONLINEAR OSCILLATIONS (I.C.N.O.)
Jean-Marc Ginoux

NEURONS ARE POISED NEAR THE EDGE OF CHAOS
Leon Chua, Valery Sbitnev and Hyongsuk Kim

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Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation
C.S. Feng, S.L. Chen

An approximate compact analytical expression for the Blasius velocity profile
Ö. Savaş

Homotopy analysis method with a non-homogeneous term in the auxiliary linear operator
Anant Kant Shukla, T.R. Ramamohan, S. Srinivas

Numerical solution of hybrid fuzzy differential equations using improved predictor–corrector method
Hyunsoo Kim, Rathinasamy Sakthivel

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Classical Papers
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Integration and differentiation to a variable fractional order

Stefan G. Samko and Bertram Ross

Publication information: S. G. Samko and B. Ross. Integration and differentiation to a variable fractional order, Integr. Trans. Spec. Funct. 1, 277–300, 1993. http://www.tandfonline.com/doi/abs/10.1080/10652469308819027

Abstract. Integration and differentiation of functions to a variable order (f^n (x)) is studied in two ways: 1) using the Riemann-Liouville definition, 2) using Fourier transforms. Some properties and the inversion formula are obtained.          

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Variable Order and Distributed Order Fractional Operators

Carl F. Lorenzo and Tom T. Hartley

Publication information: Carl F. Lorenzo and Tom T. Hartley: Variable Order and Distributed Order Fractional Operators. Nonlinear Dynamics 29: 57–98, 2002. http://www.springerlink.com/content/dlcdmm5p4lmjdg2g/

Abstract. Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed order fractional operators. Definitions based on the Riemann–Liouville definition are introduced and the behavior of the new operators is studied. Several time domain definitions that assign different arguments to the order q in the Riemann–Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms, and memory characteristics of the defining kernels.

A measure (m₂) for memory retentiveness of the order history is introduced. A generalized linear argument for the order q allows the concept of ‘tailored’ variable order fractional operators whose m₂ memory may be chosen for a particular application. Memory retentiveness (m₂) and order dynamic behavior are investigated and applications are shown.

The concept of distributed order operators where the order of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed order are defined and their various applications are outlined.

Other papers about variable-order fractional operators:

[1] C. F. M. Coimbra. Mechanics with variable-order differential operators. Ann. Phys. (Leipzig), 2003, 12(11–12):692-703.

[2] A. V. Chechkin, R. Gorenflo, I. M. Sokolov. Fractional diffusion in inhomogeneous media. J. Physics A: Math. General., 2005, 38: L679-L684.

[3] H. G. Sun, W. Chen, Y. Q. Chen. Variable-order fractional differential operators in anomalous diffusion modeling. Physica A, 2009, 388: 4586-4592.

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Prof. Virginia Kiryakova

Institute of Mathematics and Informatics (IMI) – Bulgarian Academy of Sciences (BAS), Bulgaria
Email: virginia@diogenes.bg

Scientific Degrees: M. Sc. (1975); Ph.D. (1987); Dr.Sc. (2010);
Specializations: Belarus (1990), UK (1992), Japan (1997);
Research interests and topics: fractional calculus, special functions and integral transforms

Managing Editor of the specialized journal “Fractional Calculus and Applied Analysis” (FCAA), www.math.bas.bg/~fcaa (all Back Vols 1998-2010, by IMI-BAS),
and currently at http://www.springerlink.com/content/1311-0454 (since 2011, Vol. 14 – by Springer and Versita), new website http://versita.com/fcaa/ .

Author of the monograph: Virginia Kiryakova, “Generalized Fractional Calculus and Applications”, Longman (Harlow, UK) & John Wiley (N. York, USA), 1994.
Author of more than 100 scientific articles on the mentioned research topics. These works are cited by other foreign authors more than 500 times.

Awards: Academic Prize for Mathematical Sciences of Bulgarian Academy of Sciences (1996); Badge of Honour of the Town of Sofia (1994); 3rd Prize, Bronze Medal at 11th Internat. Mathematical Olympiad (1969, Bucurest).
Visiting Professor in: Great Britain, Japan, USA, Kuwait, Holland, Lebanon, Tunisia, Spain, Portugal, Russia, Belarus, Poland, Slovak R., Hungary, Italy, Serbia, Macedonia, Turkey, UAE, etc.

Editorship: – Editor-in-Chief of “International Journal of Applied Mathematics” (Acad. Publications, Sofia); – Associate Editor of “Mathematica Balkanica” (journal of Math. Association South-Eastern Europe); – Member of Editotial Boards of 8 international mathematical journals; – Editor of Proc. Vols of Internat. Conferences: “Complex Analysis and Operational Calculus’ 91”; “Transform Methods and Special Functions’ 1994, 1996, 1999, 2003”; “Geometric Function Theory and Applications’ 2010”, etc.

Organizer of International Conferences in Bulgaria: “Transform Methods and Special Functions” (1994, 1996, 1999, 2003, 2011), “Geometric Function Theory and Applications” (2010) and member of Org. Committees of “Complex Analysis and Applications” (1985, 1987, 1991); “Intern. MASSЕE Congress” (2003); “Second Intern. Conf. Appl. Math.” (2005).
Member of International Program Committees abroad: Belarus (AMADE; 1995, 1999, 2003, 2005, 2009, 2011), Lebanon (RTST; 1999, 2002, 2005), France (FDA’ 2004), Holland (ENOC’ 2005), Portugal (FDA’ 2006; Symp. Fract. Signals Systems’ 2009), Turkey (FDA’ 2008), Russia (Intern.Russian – Abkhazian Symposia; 2009, 2010), Spain (FDA’ 2010), USA (FDTA’ 2011), UAE (4th Intern. Conference on Math. Sci. ICM-2012); China (FDA’ 2012).

Selected Papers:

Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A
Kiryakova Virginia
APPLIED MATHEMATICS AND COMPUTATION Volume: 218 Issue: 3 Special Issue: SI Pages: 883-892 DOI: 10.1016/j.amc.2011.01.076 Published: OCT 1 2011

Recent history of fractional calculus
Machado J. Tenreiro; Kiryakova Virginia; Mainardi Francesco
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 16 Issue: 3 Pages: 1140-1153 DOI: 10.1016/j.cnsns.2010.05.027, 2011

The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus
Kiryakova Virginia
COMPUTERS & MATHEMATICS WITH APPLICATIONS Volume: 59 Issue: 5 Special Issue: SI Pages: 1885-1895 DOI: 10.1016/j.camwa.2009.08.025, 2010

The special functions of fractional calculus as generalized fractional calculus operators of some basic functions
Kiryakova Virginia
Source: COMPUTERS & MATHEMATICS WITH APPLICATIONS Volume: 59 Issue: 3 Pages: 1128-1141 DOI: 10.1016/j.camwa.2009.05.014, 2010

Fungal resistance of Triticum durum - T. monococcum ssp aegilopoides amphiploid
Plamenov D.; Belchev I.; Kiryakova V.; et al.
JOURNAL OF PLANT DISEASES AND PROTECTION Volume: 116 Issue: 2 Pages: 60-62, 2009

Influence of some agronomy factors on spike components after a rare incidence of fusarium head blight epiphytoty of winter wheat ii. Effect of post-harvest residue incorporation
Milev G.; Tonev T. K.; Kiryakova V.
BULGARIAN JOURNAL OF AGRICULTURAL SCIENCE Volume: 14 Issue: 4 Pages: 410-416, 2008

Influence of some agronomy factors on spike components after a rare incidence of fusarium head blight epiphytoty of winter wheat - I. Effect of long-term crop rotation, mineral fertilization and sowing term
Tonev T. K.; Kiryakova V.; Milev G.
BULGARIAN JOURNAL OF AGRICULTURAL SCIENCE Volume: 14 Issue: 3 Pages: 321-328, 2008

Distribution and characterization of Aegilops and Triticum species from the Bulgarian Black Sea coast
Spetsov Penko; Plamenov Dragomir; Kiryakova Vanya
CENTRAL EUROPEAN JOURNAL OF BIOLOGY Volume: 1 Issue: 3 Pages: 399-411 DOI: 10.2478/s11535-006-0027-1, 2006

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Jobs

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2 PhD Positions, Computational Methods, Swansea University

(From NA Digest, V. 12, # 24)

PhD Research Studentships: Computational Methods for Flow in Porous Media

Civil and Computational Engineering Centre, School of Engineering, Swansea University

Applications are invited for two 3-year research studentships to develop numerical methods for modelling flow in porous media. The projects will focus on development of computational methods for approximation of the flow equations in subsurface reservoirs with complex geologies. The projects will lead to the degree of PhD. Further details are given in

http://www.swan.ac.uk/staff/academic/engineering/edwardsmichael/

The studentships will be based at Swansea University, in a lively and vibrant research environment, with a strength in computational methods and modelling.

Candidates should hold (or expect to hold) a good honours degree (minimum 2.1) in a relevant discipline (including Mathematics, Computational/Numerical Methods, Physics or Engineering) together with a masters degree in computational (numerical) methods and techniques for solving partial differential equations.

Applicants should apply by sending a full CV outlining their research interests and names and addresses of at least two academic referees to Prof M G Edwards, Civil and Computational Engineering Centre, School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP or by email to M.G.Edwards@swansea.ac.uk.

These industrially funded studentships are open to suitably qualified applicants world wide and cover full fees and a stipend of up to 17000 per year depending on qualifications and experience.

Closing date: Apply as soon as possible.