Fractional Derivatives for Physicists and Engineers: Volume I Background and Theory Volume II Applications

373 Pages·2013·1.85 MB·English

by  Uchaikin V.V.

Vladimir V. Uchaikin
Fractional derivatives for
1
physicists and engineers
Volume I. Background and theory
1Springer(Berlin),HigherEducationPress(Beijing),2013

Tomyfriends
whomademylifelonger

Preface
”God made the integers; all else is the work of man”.2 For centuries, the ancients
weresatisfiedwithusingnaturalnumberscalledsimply”numbers”.Whatwecallir-
rationalnumberswasnotincludedintothisnotionbytheGreeks.Notevenrational
fractionswerecallednumbers.
Hence, numberswereconceivedasdiscontinuous, whilemagnitudeswerecon-
tinuous.Thetwonotationsappeared,therefore,entirelydistinct.Thetransferfrom
numberstomagnitudes(tolengths,forexample)wasadifficultandimportantstep.
Perhaps, the most dramatic confrontation of the notions exhibited in Zeno’s para-
doxes.Oneofthemsays,”Achillescannotovertakeatortoise.Why?Achillesmust
first reach the place from which the tortoise started. By that time, the tortoise will
have moved on a little way. Achilles must then traverse that, and still the tortoise
willbeahead.Heisalwaysnearer,yetnevermakesuptoit”.Theparadoxresolu-
tionbecamepossibleonlyafterextendingtheconcept”number”overthewholereal
axis.
Therealnumbershaveformedabasisofclassicalanalysiswhosemajorconcept
is the continuity. In frame of this conception, the set of natural and even rational
numbersisvanishing(incardinality)ascomparedwiththecontinuumofrealnum-
bers.Innumericalcalculations,weuserationalnumbersasapproximationstoreal
ones,butnamelyirrationalnumbersreflecttherealworld.Nobodycanmakearod
withrationallengthandnobodycancheckthatthelengthisrational.
Ofcourse,whenwearedealingwithasetofisolatedobjectsweusethenatural
numbersforcountingtheobjects.However,iftheobjectsarenumbersandwecon-
siderthesumofthenumbers,thesituationmaychange.Thus,thenumberninthe
expressions
n
∑
y =x +x +...+x = x
n 1 2 n j
j=1
and
2 I’mverygratefultoProf.V.KiryakovaforherremarkthatthisphraseisduetoLeopoldKro-
necker(1886).
ix
x Preface
n
z =x ·x ·...·x =∏x
n 1 2 n j
j=1
needsnocomments,itisnatural.Butifthenumbersx areidentical(andpositive),
i
theseoperationscanbeeasilycontinuedonthewholerealaxis:n={1,2,3,...}→
ν∈(−∞,∞):
yν=νx, zν=xν, ν∈R.
ThefamousEulerinventioncalledEuler’sGammafunction
∫∞
Γ(ν)= e−xxν−1dx, ν>0,
0
Γ(n+1)=n!, n=1·2·3·...
has played a crucial role in extending the concept “number of operations” on the
noninteger values. With this function, there was made a scientific breakthrough in
the differential calculus, which enriched it with differentiation and integration of
fractionalorderscalledshortlyfractionalcalculus.Thefoundationoffractionalcal-
culusisconnectedwiththenamesofRiemann,Liouille,Weyl,Gru¨nwald,Letnikov
andothers.Thoughthefirstworksinthisdirectionweremadeofabouttwocenturies
ago,theseideashadnotfoundanypracticalapplicationsforalongtime.However,
the situation changed dramatically during a couple of last decades, when about 3
thousandworkswerepublishedonthesubject.
Forbetterunderstandingofwhattheextensionofthisconceptcanbring,consider
populardifferentialequationsoftheoreticalphysicsoftheform
∂mf(x,t) ∂nf(x,t)
a +b =F, (0.1)
∂tm ∂xn
where x,t are space-time variables, a, b and F are given functions of x andt, and
m, n=0,1,2,... are integer numbers. If one of the numbers, say n, is zero, the
corresponding variable x becomes a plain parameter. Omitting it, we arrive at the
ordinarydifferentialequation:
dmf(t)
a +bf(t)=F.
dtm
Putting n=1 in Eq. (0.1) and interpreting f(t) as a velocity of a material point
performingone-dimensionalmotionunderactionoftheforceF−bf,werecognize
here the simplest version of the Newton equation. If a,b>0, then the term −bf
can be interpreted as a friction force, and we meet the relaxation problem. When
a denotes mass and f means the coordinate of a particle, we again see the New-
ton equation describing one-dimensional motion along the x-axis. This time, the
term−bf meanstheelasticforceandtheequationdescribestheharmonicoscillator
drivenbytheforceF (assumingb=const).
Preface xi
Table0.1
m,n 1D-equations 3D-equations Phys.sense Math.type
dv dv
1,0 |a| +bv=F |a| +bv=F Damped –
dt dt
motion
d2x d2r
2,0 |a| +bx=F |a| +br=F Oscillation –
dt2 dt2
∂f ∂(bf) ∂f
1,1 |a| + =F |a| +∇(bf)=F Continuity –
∂t ∂x ∂t
∂f ∂2f ∂f
1,2 |a| −|b| =F |a| −|b|∇2f =F Diffusion Parabolic
∂t ∂x2 ∂t
∂2f ∂2f ∂2f
2,2 |a| −|b| =F |a| −|b|∇2f =F Waves Hyperbolic
∂t2 ∂x2 ∂t2
∂2f
0,2 af+b =F af+b∇2f =F Staticfields Elliptic
∂x2
µ,ν No
( )
non- a∂µf +b ∂2 ν/2f=F a∂µf +b△ν/2f =F ? classified
∂tµ ∂x2 ∂tµ
integers t>0, −∞<x<∞ yet
Fig.0.1 Continuousmanifoldoffractionalpartialequations.
xii Preface
Choosing m = n = 1, we obtain a one-dimensional continuity equation. This
is the simplest of partial differential equations of mathematical physics. The other
popularversionsofequationsofmathematicalphysicsarerepresentedinTable0.1.
Theyarewell-knownanddonotneedanycomments.Inthelastline,youcanseethe
equationscontainingtime-andspace-derivativesoffractionalorders.Theseopera-
tionsaresignificantlylessfamiliartothemajorityofphysicistsandengineers.One
could pay no serious attention to such exotic mathematical construction, but only
oneglanceatFig.0.1mayshakethescepticism:weseethatthesetofwell-known
andwell-investigateddifferentialequationsofmathematicalphysicsisrepresented
byonlyafewpointson(µ,ν)diagram,whilethecontinuoussetofallotherpoints
ofthewhole(µ,ν)-planeisaterraincognita,whichcannotbutattractattentionof
graduateandpostgraduatestudents,promisingscientistsandyoungengineers.This
is the readership the book is addressed in the first place. Nevertheless, I hope that
itcanattractattentionofmoreexperiencedresearches,bothphysicistsandmathe-
maticians, as comparatively new tools for investigation and modelling of complex
naturalprocesses.
Iwas55yearsoldwhenIwentonalong-yeartripoverthis“fractional”landand
foundmanytreasuresthere:aboutthreethousandarticlesusingthetoolsforsolving
different problems in physics and engineering have been published during the last
twodecades.Theyinclude
• inversemechanicalproblems,
• stochastickineticsanddynamicalchaos,
• motioninviscousfluid,
• heatflowspreading,
• electrochemistryofelectrodes,
• percolationthroughporousmedia,
• rheologyofviscoelasticmaterials,
• electricalandradioengineering,
• plasmaphysics,
• quantumopticsandnanophysics,
• astrophysicsandcosmology,
• biophysicsandmedicine.
Fortunately,ourgroupmanagedtoparticipateindevelopingfractionalapproach
to description of anomalous (dispersive) transport in disordered semiconductors,
non-Debye relaxation in solid dielectrics, penetration of light beam through a tur-
bulentmedium,transportofresonanceradiationinplasma,blinkingfluorescenceof
quantumdots,subrecoillasercoolingofatoms,penetrationandaccelerationofcos-
mic ray in the Galaxy, large-scale statistical cosmography and solving some other
problems.Theseinvestigationsallowedustobecomeawareofdeeplinksbetween
fractionalcalculus,non-GaussianLe`vy-stablestatisticsandstochasticfractals.The
presence of a time-fractional derivative in the equation is interpreted as a special
propertyoftheprocessunderconsiderationcalledthememory,theafter-effect,or,if
wehandlewithastochasticprocess,thenon-Markovianproperty.Fractionalderiva-
tives with respect to coordinates reflect a medium with inhomogeneities of some
Preface xiii
specialkindcalledself-similarinhomogeneitiesorfractals.Wemeetsuchstructures
inturbulentflows,plasma,andinterstellarmedia.
As many my colleagues, I’m convinced that the fractional derivative or, as it is
oftencalled,thefractionaldifferintegralgivenbytheexpression
∫x
1 dn
f(ν)(x)= (x−ξ)n−ν−1f(ξ)dξ
a Γ(n−ν)dxn
0
with n−1≤ν<n if ν≥0 and n=0 if ν<0, is much more than merely a se-
quence of differential and special integral operators3. Important properties of Na-
ture underlie this mathematical concept. Starting to write this book, I decided to
begin it with discussion of these properties, enveloping, in my opinion, heredity,
non-locality, self-similarity and stochasticity. This is why the first three chapters
of the book united into the first part ”Background” contain description of various
naturalphenomenademonstratingsuchproperties.
Chapter1containsamodernexpositionoftheVolterraheredityconceptwhose
maintoolistheintegraloperation
∫x
f(x)7→ K(x,ξ)f(ξ)dξ
0
with the kernel K(x,ξ) interpreted as a memory in case x and ξ are time, or as
non-locality in case the variables are spatial coordinates. The variety of physical
processesincludingmechanical,molecular,hydrodynamical,thermodynamicalphe-
nomenademonstratinghereditarypropertiesaredescribedinthischapter.
InChapter2wereviewphysicalprocessescharacterizedbypower-typememory
functions and basic mechanisms generated this property. This list may shake the
opinion that the exponential function is the queen-function of theoretical physics
andshowthatwithoutpowerfunctionslikewithoutmaidsofhonourthequeencourt
wouldbemoretiresome.
Chapter3opensawidepanoramaofstochasticprocesseswhichshowsthatprob-
abilisticlongtailsofpowertyperesultsfromself-similarityoftheprocessesandthe
latterisconnectedwithexistenceoflimitdistributions,namelywiththeLevy-stable
laws.ThereaderwillgointothefractionalBrownianmotioncontinuous-timeran-
domwalkprocesses,fractionalPoissonprocessandwalkingonfractals.
Thesecondpartofthebook,”Theory”,containstheelementsoffractionalcalcu-
lustheorywithreviewofvariousfractionalequations,theiranalyticalandnumerical
solutions.
Chapter4servesasamathematicalintroductiontofractionalcalculuscontaining
basic definitions of fractional operators, their properties and rules of applications.
The reader can recognize many of them as corresponding generalization of well-
3Herearetworemarkablequotations:”ThefractionalcalculusisthecalculusoftheXXIcentury”
(K.Nishimoto,1989)and”WemaysaythatNatureworkswithfractionaltimederivatives”(S.
Westerlund,1991).
xiv Preface
knownanalogousfrominteger-ordercalculussuchastheLeibnitzruleortheTaylor
formula,
Chapter5showshowequationswithfractionalderivativesaresolved.Thereader
will meet description of some analytical methods of solution and many examples
of their applications to ordinary and partial fractional equations. Of course, this
reviewcannotpretendonastrictandexhaustiveexposition,butitwillbeusefulfor
physicistsandengineersasafirstacquaintance.
Chapter6containsanintroductiontonumericalmethodsoffractionalequation
solution. Starting with the fractional difference operators and based on this con-
cept the Gru¨nwald-Letnikov definition of fractional derivatives, the reader is ac-
quaintedwiththefinite-differencemethodsofcomputingfractionalintegrals,frac-
tionalderivatives,andfractionalequationsofvariouskinds.Thelastsectionofthis
chapterisdevotedtosomeaspectsofMonteCarlotechniques.
Thethirdpart,openingthesecondvolumeofthebook,exposesawidefieldof
applications of fractional calculus in modern physics including mechanics, hydro-
dynamics,viscoelasticity,thermodynamics,electrodynamics,plasmaphysics,quan-
tumphysics,andcosmicrayphysics.
Finally,thefourth,lastpartofthebookcontainsvariousauxiliarymaterial(spe-
cial functions, notation of fractional derivatives, main formulas of fractional cal-
culus, tables and graphs of some functions, playing a special role in solution of
fractionalequations).
Ulyanovsk(Russia),March2012 VladimirUchaikin