Table Of ContentFunctional and Impulsive
Differential Equations of Fractional Order
Qualitative Analysis and Applications
Functional and Impulsive
Fermented Meat Products
Differential Equations of
Health Aspects
Fractional Order
Qualitative Analysis and Applications
Edited by
Nevijo Zdolec
University of Zagreb, Faculty of Veterinary Medicine
Department of Hygiene
Technology and Food Safety Heinzelova 55
10000 Zagreb, Croatia
Ivanka M. Stamova and Gani Tr. Stamov
The University of Texas at San Antonio
Texas, USA
CRC Press
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This book is dedicated
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Preface
This book is an exposition of the most recent results related to the qualitative
analysisofavarietyoffractional-orderequations.
The mathematical investigations of derivatives of non-integer order mark
their beginning with the correspondence between Leibniz and L’Hospital in
1695(see[Leibniz1695],[Podlubny1999]).Althoughfractionalcalculushasa
more than three centuries long history, the subject of fractional differential
equations has gained considerable popularity and importance during the past
few decades, due mainly to its demonstrated applications to many real-world
phenomenastudiesinphysics,mechanics,chemistry,engineering,finance,etc.
There are several excellent books devoted to equations of fractional order.
The book written by I. Podlubny (see [Podlubny 1999]) played an outstanding
role in the development of the theory of fractional ordinary differential
equations. It is the most cited and used book which is entirely devoted to a
systematic presentation of the basic ideas and methods of fractional calculus in
the theory and applications of such equations. The monograph of Kilbas,
Srivastava and Trujillo (2006) provides uptodate developments on fractional
differential and fractional integro-differential equations involving many
differentpotentiallyusefuloperatorsoffractionalcalculus.Inadditiontothese,
the books of Abbas, Benchohra and N’Gue´re´kata (2012), Baleanu, Diethelm,
ScalasandTrujillo(2012),Caponetto,Dongola,FortunaandPetra´s˘ (2010),Das
(2011), Diethelm (2010), Hilfer (2000), Kiryakova (1994), Lakshmikantham,
Leela and Vasundhara Devi (2009), Magin (2006), Miller and Ross (1993) are
good sources for the theory of fractional operators and equations, as well as
theirnumerousapplications.
On the other hand, relatively recently functional differential equations of
fractional order have started to receive an increasing interest [Baleanu, Sadati,
Ghaderi, Ranjbar, Abdeljawad and Jarad 2010], [Banas´ and 2011],
[Benchohra, Henderson, Ntouyas and Ouahab 2008], [El-Sayed, Gaaraf and
vii
viii (cid:4) FunctionalandImpulsiveDifferentialEquationsofFractionalOrder
Hamadalla 2010], [Henderson and Ouahab 2009], [Lakshmikantham 2008].
Indeed,fractionaloperatorsareaverynaturaltooltomodelmemory-dependent
phenomena. Also, fractional calculus has been incorporated into impulsive
differential equations [Ahmad and Nieto 2011], [Ahmad and Sivasundaram
2010], [Anguraj and Maheswari 2012], [Cao and Chen 2012], [Chauhan and
Dabas 2011], [Fecˇkan, Zhou and Wang 2012]. Many interesting results on the
fundamental theory of such equations have been reported. However, due to the
lackofabookonthesetopics,manyresearchersremainunawareofthisfield.
The questions related to the qualitative theory of different classes of
differential equations are the age-old problems of a great importance. The
methods used in the qualitative investigation of their solutions, and their wide
applications have all advanced to the extent that aspects in each of these areas
havedemandedindividualattention.
Thesurveypublishedin2011byLiandZhang(see[LiandZhang2011]),is
a very good overview on the recent stability results of fractional differential
equations without impulses and delays, and the analytical methods used. It is
seen that, at the time, a few stability results rely on a restrictive modeling of
fractional differential systems: the basic hypothesis deals with
commensurability, i.e. the fractional derivative orders have to be an integer
multipleofminimalfractionalorder.Inthelastdecades,manyresearchershave
moreinterestsinthestabilityoflinearsystemsandsomemethodshaveemerged
in succession. For example, there are the frequency domain methods, Linear
Matrix Inequalities methods, and conversion methods [Li and Zhang 2011]. By
contrast,thedevelopmentofstabilityofnonlinearfractionaldifferentialsystems
evenwithoutimpulsiveperturbationswasabitslow.
During the last few years the authors’ research in the area of the qualitative
theory of different classes of functional and impulsive fractional-order
equations have undergone rapid development. A string of extensive results on
thestability,boundedness,asymptoticbehaviorandalmostperiodicityforthese
classes of equations have been obtained. The primary aim of this book is to
gatherunderonecovermanyoftheseresultswhichwillbeofprimeimportance
for researchers on the topic. It fills a void by making available a source book
whichdescribesexistingliteratureonthetopic,methodsandtheirdevelopment.
The second motivation comes from the applicable point of view, since the
qualitative properties have significant practical applications in the emerging
areas such as optimal control, biology, mechanics, medicine, bio-technologies,
electronics, economics, etc. For the applied scientists it is important to have an
introduction to the qualitative theory of fractional equations, which could help
intheirinitialstepstoadopttheresultsandmethodsintheirresearch.
Thebookconsistsoffourchapters.Itpresentsresultsfordifferentclassesof
fractional equations, including fractional functional differential equations,
fractional impulsive differential equations, fractional impulsive functional
differentialequations,whichhavenotbeencoveredbyotherbooks.Itshowsthe
Preface (cid:4) ix
manifestations of different constructive methods by demonstrating how these
effective techniques can be applied to investigate qualitative properties of the
solutions of fractional systems. Since many applications are also included, the
demonstrated techniques and models can be used in training of students in
mathematical modeling and as an instigation in the study and development of
fractional-ordermodels.
The book is addressed to a wide audience of professionals such as
mathematicians,appliedresearchersandpractitioners.
The authors are extremely grateful and very much indebted to Dr. Sandy
Norman, Chair of the Department of Mathematics at the University of Texas at
San Antonio for ensuring the opportunity for successful work on this book. In
addition, the authors have the pleasure to express their sincere gratitude to all
theirco-authors,theworkwithwhomexpandedtheirexperience.Theyarealso
thankful to all friends, colleagues and reviewers for their valuable comments
andsuggestionsduringthepreparationofthemanuscript.
I.M.StamovaandG.Tr.Stamov
Correspondingauthor:[email protected]
