Fractional Diffusion Equations and Anomalous Diffusion

360 Pages·2018·21.876 MB·English

by  Luiz Roberto Evangelista, Ervin Kaminski Lenzi

FRACTIONAL DIFFUSION EQUATIONS
AND ANOMALOUS DIFFUSION
Anomalousdiffusionhasbeendetectedinawidevarietyofscenarios,fromfractal
media,systemswithmemory,transportprocessesinporousmediatofluctuationsof
financial markets, tumour growth, and complex fluids. Providing a contemporary
treatment of this process, this book examines the recent literature on anomalous
diffusionandcoversarichclassofproblemsinwhichsurfaceeffectsareimportant,
offering detailed mathematical tools of usual and fractional calculus for a wide
audience of scientists and graduate students in physics, mathematics, chemistry,
and engineering. Including the basic mathematical tools needed to understand the
rules for operating with the fractional derivatives and fractional differential equa-
tions, this self-contained text presents the possibility of using fractional diffusion
equationswithanomalousdiffusionphenomenatoproposepowerfulmathematical
modelsforalargevarietyoffundamentalandpracticalproblemsinafast-growing
fieldofresearch.
luiz roberto evangelista is Professor of Theoretical Physics at the
UniversityofMaringa´ (UEM),Brazil.Hisresearchinterestslieincomplexfluids,
complex systems, and history of physics, and include mathematical physics of
liquidcrystals,diffusionproblems,andadsorption-desorptionphenomena.
ervin kaminski lenzi is Associate Professor at the University of Ponta
Grossa(UEPG),Brazil.Hisresearchinterestsareincomplexsystemsandstochas-
tic processes, and include anomalous diffusion processes, usual and fractional
diffusion equations, and modern boundary value problems with applications in
liquid-crystallinesystemsandimpedancespectroscopy.

FRACTIONAL DIFFUSION
EQUATIONS AND ANOMALOUS
DIFFUSION
LUIZ ROBERTO EVANGELISTA
UniversityofMaringa´(UEM)
ERVIN KAMINSKI LENZI
UniversityofPontaGrossa(UEPG)
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107143555
DOI:10.1017/9781316534649
©LuizRobertoEvangelistaandErvinKaminskiLenzi2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2018
PrintedintheUnitedKingdombyClays,StIvesplc
LibraryofCongressCataloging-in-PublicationData
Names:Evangelista,L.R.,author.|Lenzi,ErvinKaminski,1975-author.
Title:Fractionaldiffusionequationsandanomalousdiffusion/LuizRoberto
Evangelista,UniversidadeEstadualdeMaringa´,Brazil;ErvinKaminski
Lenzi,UniversidadeEstadualdePontaGrossa.
Description:Cambridge:CambridgeUniversityPress,[2018]
Identifiers:LCCN2017035420|ISBN9781107143555(hardback:alk.paper)
Subjects:LCSH:Differentialequations,Partial.|Fractionalcalculus.|Diffusion.
Classification:LCCQA377.E94482017|DDC515/.83–dc23LCrecord
availableathttps://lccn.loc.gov/2017035420
ISBN978-1-107-14355-5Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
ToLucasEvangelistaGomes,
whohasjustarrivedamongus
L.R.E.
ToAliceandPedro,withlove
E.K.L.
Setuse’or,lettore,acrederlento
cio`ch’iodiro`,nonsara`maraviglia,
che´ioche’lvidi,apenailmiconsento.
[Ifthouart,Reader,slownowtobelieve
WhatIshallsay,itwillnomarvelbe,
ForIwhosawithardlycanadmitit.]
(Dante,InfernoXXV,46–48)
Contents
Preface pageix
1 MathematicalPreliminaries 1
1.1 IntegralTransforms 1
1.2 SpecialFunctionsofFractionalCalculus 17
1.3 IntegralTransformsofSpecialFunctions 43
2 ASurveyofFractionalCalculus 46
2.1 TheOriginsofFractionalCalculus 46
2.2 TheGru¨nwald–LetnikovOperator 57
2.3 TheCaputoOperator 61
2.4 TheRiesz–WeylOperator 62
2.5 IntegralTransformsofFractionalOperators 63
2.6 AGeneralisedFourierTransform 68
3 FromNormaltoAnomalousDiffusion 71
3.1 HistoricalPerspectivesonDiffusionProblems 71
3.2 Continuous-TimeRandomWalk 90
3.3 DiffusionEquation 95
4 FractionalDiffusionEquations 101
4.1 FractionalTimeDerivative:SimpleSituations 101
4.2 FractionalSpatialDerivative:SimpleSituations 111
4.3 SorptionandDesorptionProcesses 114
4.4 ReactionTerms 124
4.5 ReactionandCTRWFormalism 134
5 FractionalDiffusionEquations 139
5.1 1Dand2DCases:DifferentDiffusiveRegimes 139
5.2 3DCase:ExternalForceandReactionTerm 145
vii
viii Contents
5.3 ReactiononaSolidSurface:AnomalousMassTransfer 151
5.4 HeterogeneousMediaandTransportthroughaMembrane 158
6 FractionalNonlinearDiffusionEquations 169
6.1 NonlinearDiffusionEquations 170
6.2 NonlinearDiffusionEquations:IntermittentMotion 173
6.3 FractionalSpatialDerivatives 182
6.4 d-DimensionalFractionalDiffusionEquations 188
7 AnomalousDiffusion 200
7.1 TheAdsorption–DesorptionProcessinAnisotropicMedia 200
7.2 FractionalDiffusionEquationsinAnisotropicMedia 209
7.3 TheCombModel 220
8 FractionalSchro¨dingerEquations 234
8.1 TheSchro¨dingerEquationandAnomalousBehaviour 234
8.2 Time-DependentSolutions 242
8.3 CTRWandtheFractionalSchro¨dingerEquation 249
8.4 MemoryandNonlocalEffects 254
8.5 NonlocalEffectsontheEnergySpectra 264
9 AnomalousDiffusionandImpedanceSpectroscopy 271
9.1 ImpedanceSpectroscopy:Preliminaries 271
9.2 ThePNPTimeFractionalModel 280
9.3 AnomalousDiffusionandMemoryEffects 286
9.4 AnomalousInterfacialConditions 292
10 ThePoisson–Nernst–PlanckAnomalousModels 306
10.1 PNPAModelsandEquivalentCircuits 306
10.2 PNPAModels:AFramework 313
References 323
Index 341
Preface
TheveryirregularstateofmotionobservedbyRobertBrownforsmallpollengrains
suspended in water initiated one of the the most fascinating fields of science. The
importance of such discovery – the so-called diffusion process – is immeasurable;
it has been found in many contexts and is widespread in nature. A characteristic
featureofthisrandommotionisthelineargrowthwithtimeexhibitedbythemean
square displacement, which is typical of a Markovian process. In contrast with
this situation, a large class of systems and processes present a diffusion behaviour
characterisedbyanonlineartimedependenceofthesamequantity,thusconstituting
whatiscalledanomalousdiffusionbehaviour.
Thelastdecadeshavewitnessedanincreasedinterestintheanomalousdiffusion
processes that seem to be indeed present in a variety of experimental scenarios
in physics, chemistry, biology, and several other branches of engineering; it is a
rapidly growing field of research, attracting the attention of the scientific commu-
nity.Thishappensfromthetheoreticalside–duetothenewmathematicalproblems
evoked–butalsofromthepointofviewofexperimentalorpracticalapplications.
Itisnoteworthythatthenumberofstudiesreportingexperimentalproblemsdealing
with anomalous diffusion has strongly increased – this attests to the ubiquity of a
phenomenoninitiallyconsideredarareevent.
The power of the mathematical tools based on fractional calculus, on the other
hand, has also attracted the attention of the community working with pure and
appliedmathematics.Theassociationofthesetechniqueswiththediffusionalprob-
lem represents in practice a new field of research. It was shown in several ways
that fractional calculus, if it is not unique, is nevertheless a suitable or even the
naturalmathematicalframeworktousetofacethehighcomplexityrepresentedby
anomalous diffusion phenomena. One powerful way of using these mathematical
tools to analyse diffusion processes leads naturally to the necessity to search for
solutionsoffractionallinearandnonlineardiffusionequations.
ix