fractional calculus hilfer08a

ThreefoldIntroduction to FractionalDerivatives R. Hilfer Fakultät für Mathematik und Physik Universität Stuttgart Pfaffenwaldring 27 70569 Stuttgart Germany to be published in: R.Klages et al., Anomalous Transport: Foundations and Applications , Wiley-VCH, Weinheim, 2008, Chapter 1 June 6, 2007 to be published in: R. Klages et al. (eds.), Anomalous Transport: Foundations and Applications , Wiley-VCH, Weinheim, Copyright © 2007 R. Hilfer, StuttgartV Contents 1 Threefold Introduction toFractional Derivatives 1 R. Hilfer 1.1 Historical Introduction to Fractional Derivatives 1 1.1.1 Leibniz 1 1.1.2 Euler 2 1.1.3 Paradoxa and Problems 2 1.1.4 Liouville 4 1.1.5 Fourier 5 1.1.6 Grünwald 5 1.1.7 Riemann 6 1.2 Mathematical Introduction to Fractional Derivatives 7 1.2.1 Fractional Integrals 7 1.2.1.1 Iterated Integrals 7 1.2.1.2 Riemann-Liouville Fractional Integrals 8 1.2.1.3 Weyl Fractional Integrals 9 1.2.1.4 Riesz Fractional Integrals 11 1.2.1.5 Fractional Integrals of Distributions 12 1.2.1.6 Integral Transforms 14 1.2.1.7 Fractional Integration by Parts 15 1.2.1.8 Hardy-Littlewood Theorem 15 1.2.1.9 Additivity 16 1.2.2 Fractional Derivatives 16 1.2.2.1 Riemann-Liouville Fractional Derivatives 16 1.2.2.2 General Types of Fractional Derivatives 19 1.2.2.3 Marchaud-Hadamard Fractional Derivatives 19 1.2.2.4 Weyl Fractional Derivatives 21 1.2.2.5 Riesz Fractional Derivatives 22 1.2.2.6 Grünwald-Letnikov Fractional Derivatives 22 1.2.2.7 Fractional Derivatives of Distributions 24 VIContents 1.2.2.8 Fractional Derivatives at Their Lower Limit 26 1.2.2.9 Fractional Powers of Operators 27 1.2.2.10 Pseudodifferential Operators 28 1.2.3 Eigenfunctions 29 1.3 Physical Introduction to Fractional Derivatives 31 1.3.1 Basic Questions 31 1.3.2 Fractional Space 32 1.3.3 Fractional Time 33 1.3.3.1 Basic Questions 33 1.3.3.2 Time Evolution 34 1.3.3.3 Continuity 35 1.3.3.4 Homogeneity 35 1.3.3.5 Causality 36 1.3.3.6 Fractional Time Evolution 36 1.3.3.7 Infinitesimal Generator 38 1.3.3.8 Remarks 38 1.3.4 Identification of αfrom Models 39 1.3.4.1 Bochner-Levy Fractional Diffusion 39 1.3.4.2 Montroll-Weiss Fractional Diffusion 40 1.3.4.3 Continuous Time Random Walks 42 References 44 Appendix A Tables 51 B Function Spaces 53 C Distributions 57 Index61 to be published in: R. Klages et al. (eds.), Anomalous Transport: Foundations and Applications , Wiley-VCH, Weinheim, Copyright © 2007 R. Hilfer, Stuttgart1 1 ThreefoldIntroductiontoFractionalDerivatives R. Hilfer 1.1 Historical Introduction toFractional Derivatives 1.1.1 Leibniz Already at the beginning of calculus one of its founding fath ers, namely G.W. Leibniz, investigated fractional derivatives [72, 73]. Di fferentiation, denoted as dα(α∈N), obeys Leibniz’ product rule dα(f g) =1 dαfd0g+α 1dα−1fd1g+α(α−1) 1·2dα−2fd2g+... (1.1) for integer α, and Leibniz was intrigued by the analogy with the binomial theorem pα(f+g) =1 pαfp0g+α 1pα−1fp1g+α(α−1) 1·2pα−2fp2g+... (1.2) where he uses the notation pαfinstead of fαto emphasize the formal opera- tional analogy. Moving from integer to noninteger powers α∈RLeibniz suggests that "on peut exprimer par une serie infinie une grandeur comme" dαh(with h=f g). As his first step he tests the idea of such a generalized differ ential quantity dαhagainst the rules of his calculus. In his calculus the differ ential relation dh=hdximplies d x=dh/hand d h/dx=h. One has, therefore, also d2h= hdx2and generally dαh=hdxα. Regarding dαh=hdxαwith noninteger αas a fractional differential relation subject to the rules of h is calculus, however, 21 ThreefoldIntroductiontoFractionalDerivatives leads to a paradox. Explicitly, he finds (for α=1/2) dαh dxα=dαh (dh/h)α/\e}atio\slash=h (1.3) where d x=dh/hwas used. Many decades had to pass before Leibniz’ para- dox was fully resolved. 1.1.2 Euler Derivatives of noninteger (fractional) order motivated Eu ler to introduce the Gamma function [25]. Euler knew that he needed to generalize (or interpo- late, as he calls it) the product 1 ·2·...·n=n! to noninteger values of n, and he proposed an integral n ∏ k=1k=n!=1/integraldisplay 0(−logx)ndx (1.4) for this purpose. In §27-29 of [25] he immediately applies th is formula to par- tially resolve Leibniz’ paradox, and in §28 he gives the basi c fractional deriva- tive (reproduced here in modern notation with Γ(n+1) =n!) dαxβ dxα=Γ(β+1) Γ(β−α+1)xβ−α(1.5) valid for integer and for noninteger α,β. 1.1.3 Paradoxaand Problems Generalizing eq. (1.5) to all functions that can be expanded into a power series might seem a natural step, but this "natural" definition of fr actional deriva- tives does not really resolve Leibniz’ paradox. Leibniz had implicitly assumed the rule dαeλx dxα=λαeλx(1.6) by demanding dαh=hdxαfor integer α. One might therefore take eq. (1.6) instead of eq. (1.5) as an equally "natural" starting point ( this was later done by Liouville in [76, p.3,eq.(1)]), and define fractional der ivatives as dαf dxα=∑ kckλα keλkx(1.7) 1.1 HistoricalIntroductiontoFractionalDerivatives 3 for functions representable as exponential series f(x)∼∑kckexp(λkx). Re- garding the integral (a Laplace integral) x−β=1 Γ(β)∞/integraldisplay 0e−yxyβ−1dy (1.8) as a sum of exponentials, Liouville [76, p. 7] then applied eq . (1.6) inside the integral to find dαx−β dxα=1 Γ(β)∞/integraldisplay 0e−yx(−y)αyβ−1dy=(−1)αΓ(β+α) Γ(β)xβ+α(1.9) where the last equality follows by substituting yx=zin the integral. If this equation is formally generalized to −β, disregarding existence of the integral, one finds dαxβ dxα=(−1)αΓ(−β+α) Γ(−β)xβ−α(1.10) a formula similar to, but different from eq. (1.5). Although eq. (1.10) agrees with eq. (1.5) for integer αit differs for noninteger α. More precisely, if α=1/2 and β=−1/2, then Γ(3/2) Γ(0)x−1=0/\e}atio\slash=i x√π=(−1)1/2Γ(1) Γ(1/2)x−1(1.11) revealing again an inconsistency between eq. (1.5) and eq. ( 1.10) (resp. (1.9)). Another way to see this inconsistency is to expand the expone ntial function into a power series, and to apply Euler’s rule, eq. (1.5), to i t. One finds (with obvious notation) /parenleftbiggdα dxα/parenrightbigg (1.5)exp(x) =/parenleftbiggdα dxα/parenrightbigg (1.5)∞ ∑ k=0xk k!=∞ ∑ k=0xk−α Γ(k−α+1) /\e}atio\slash=/parenleftbiggdα dxα/parenrightbigg (1.6)exp(x) =exp(x) (1.12) and this shows that Euler’s rule (1.5) is inconsistent with t he Leibniz/Liouville rule (1.6). Similarly, Liouville found inconsistencies [7 5, p.95/96] when calcu- lating the fractional derivative of exp (λx) +exp(−λx)based on the definition (1.7). A resolution of Leibniz’ paradox emerges when eq. (1.5) and ( 1.6) are com- pared for α=−1, and interpreted as integrals. Such an interpretation was 41 ThreefoldIntroductiontoFractionalDerivatives already suggested by Leibniz himself [73]. More specificall y, one has d−1ex dx−1=ex=x/integraldisplay −∞etdt/\e}atio\slash=x/integraldisplay 0etdt=ex−1=d−1 dx−1∞ ∑ k=0xk k!(1.13) showing that Euler’s fractional derivatives on the right ha nd side differs from Liouville’s and Leibniz’ idea on the left. Similarly, eq. (1 .5) corresponds to d−1xβ dx−1=xβ+1 β+1=x/integraldisplay 0yβdy. (1.14) On the other hand, eq. (1.9) corresponds to d−1x−β dx−1=x1−β 1−β=−∞/integraldisplay xy−βdy=x/integraldisplay ∞y−βdy. (1.15) This shows that Euler’s and Liouville’s definitions differ w ith respect to their limits of integration. 1.1.4 Liouville It has already been mentioned that Liouville defined fractio nal derivatives using eq. (1.7) (see [76, p.3,eq.(1)]) as dαf dxα=∑ kckλα keλkx(1.7) for functions representable as a sum of exponentials f(x)∼∑ kckexp(λkx). (1.16) Liouville seems not to have recognized the necessity of limi ts of integration. From his definition (1.7) he derives numerous integral and se ries representa- tions. In particular, he finds the fractional integral of ord erα>0 as /integraldisplayα f(x)dxα=1 (−1)αΓ(α)∞/integraldisplay 0f(x+y)yα−1dy (1.17) (see formula [A] on page 8 of [76, p.8]). Liouville then gives formula [B] for fractional differentiation on page 10 of [76] as dαf dxα=1 (−1)n−αΓ(n−α)∞/integraldisplay 0dnf(x+y) dxnyn−α−1dy (1.18) 1.1 HistoricalIntroductiontoFractionalDerivatives 5 where n−1<α<n. Liouville restricts the discussion to functions repre- sented by exponential series with λk>0 so that f(−∞) = 0. Liouville also expands the coefficients λα kin (1.7) into binomial series λα k=lim h→01 hα(1−e−hλk)α, λk>0 (1.19a) = (−1)αlim h→01 hα(1−ehλk)α, λk<0 (1.19b) and inserts the expansion into his defintion (1.7) to arrive a t formulae that contain the representation of integer order derivatives as limits of difference quotients (see [75, p.106ff]). The results may be written as dαf dxα=lim h→0/braceleftBigg 1 hα∞ ∑ m=0/bracketleftbigg (−1)m/parenleftbiggα m/parenrightbigg f(x−mh)/bracketrightbigg/bracerightBigg (1.20a) = (−1)αlim h→0/braceleftBigg 1 hα∞ ∑ m=0/bracketleftbigg (−1)m/parenleftbiggα m/parenrightbigg f(x+mh)/bracketrightbigg/bracerightBigg (1.20b) where the binomial coefficient (α m)isΓ(α−1)Γ(m−1)/Γ(α+m−1). Later, this idea was taken up by Grünwald [34], who defined fractiona l derivatives as limits of generalized difference quotients. 1.1.5 Fourier Fourier [29] suggested to define fractional derivatives by g eneralizing the for- mula for trigonometric functions, dα dxαcos(x) =cos/parenleftBig x+απ 2/parenrightBig , (1.21) from α∈Ntoα∈R. Again, this is not unique because the generalization dα dxαcos(x) = (−1)αcos/parenleftBig x−απ 2/parenrightBig (1.22) is also possible. 1.1.6 Grünwald Grünwald wanted to free the definition of fractional derivat ives from a spe- cial form of the function. He emphasized that fractional der ivatives are inte- groderivatives, and established for the first time general f ractional derivative 61 ThreefoldIntroductiontoFractionalDerivatives operators. His calculus is based on limits of difference quo tients. He studies the difference quotients [34, p.444] F[u,x,α,h]f=n ∑ k=0(−1)k/parenleftbiggα k/parenrightbiggf(x−kh) hα(1.23) with n= (x−u)/hand calls Dα[f(x)]x=x x=u=lim h→0F[u,x,α,h]f (1.24) theα-th differential quotient taken over the straight line from u to x [34, p.452]. The title of his work emphasizes the need to introduce limits of i ntegration into the concept of differentiation. His ideas were soon elaborated upon by Letnikov (see [99])and applied to differential equations by Most [89 ]. 1.1.7 Riemann Riemann, like Grünwald, attempts to define fractional diffe rentiation for gen- eral classes of functions. Riemann defines the n-th differential quotient of a function f(x)as the coeffcient of hnin the expansion of f(x+h)into inte- ger powers of h[96, p.354]. He then generalizes this definition to noninteg er powers, and demands that f(x+h) =n=∞ ∑ n=−∞cn+α(∂n+α xf)(x)hn+α(1.25) holds for n∈N,α∈R. The factor cn+αis determined such that ∂β(∂γf) = ∂β+γfholds, and found to be 1/ Γ(n+α+1). Riemann then derives the inte- gral representation [96, p.363] for negative α ∂αf=1 Γ(−α)x/integraldisplay k(x−t)−α−1f(t)dt+∞ ∑ n=1Knx−α−n Γ(−n−α+1)(1.26) where k,Knare finite constants. He then extends the result to nonnegati ve αby writing "für einen Werth von αaber, der ≥0ist, bezeichnet ∂αf dasjenige, was aus ∂α−mf (wo m >α) durch m-malige Differentiation nach x hervorgeht,..." [96, p.341]. The combination of Liouville’s and Grünwald’s pioneering work with this idea has become the definition of the Riemann-Liouv ille fractional derivatives (see Section 1.2.2.1 below). 1.2 MathematicalIntroductiontoFractionalDerivatives 7 1.2 Mathematical Introduction to Fractional Derivatives The brief historical introduction has shown that fractiona l derivatives may be defined in numerous ways. A natural and frequently used appro ach starts from repeated integration and extends it to fractional inte grals. Fractional derivatives are then defined either by continuation of fract ional integrals to negative order (following Leibniz’ ideas [73]), or by integ er order derivatives of fractional integrals (as suggested by Riemann [96]). 1.2.1 Fractional Integrals 1.2.1.1Iterated Integrals Consider a locally integrable1real valued function f:G→Rwhose domain of definition G= [a,b]⊆Ris an interval with −∞≤a<b≤∞. Integrating ntimes gives the fundamental formula (In a+f)(x) =x/integraldisplay ax1/integraldisplay a...xn−1/integraldisplay af(xn)dxn...dx2dx1 =1 (n−1)!x/integraldisplay a(x−y)n−1f(y)dy (1.27) where a<x<band n∈N. This formula may be proved by induction. It reduces n-fold integration to a single convolution integral (Faltun g). The sub- script a+indicates that the integration has aas its lower limit. An analogous formula holds with lower limit xand upper limit a. In that case the subscript a−will be used. 1)A function f:G→Ris called locally integrable if it is integrable on all compact subsets K⊂G(see eq.(B.9)). 81 ThreefoldIntroductiontoFractionalDerivatives 1.2.1.2Riemann-Liouville Fractional Integrals Equation (1.27) for n-fold integration can be generalized to noninteger values ofnusing the relation (n−1)!=∏n−1 k=1k=Γ(n)where Γ(z) =1/integraldisplay 0(−logx)z−1dx (1.28) is Euler’s Γ-function defined for all z∈C. Definition 1.1 Let−∞≤a<x<b≤∞. The Riemann-Liouville fractional inte- gral of order α>0with lower limit a is defined for locally integrable functions f:[a,b]→Ras (Iα a+f)(x) =1 Γ(α)x/integraldisplay a(x−y)α−1f(y)dy (1.29a) forx>a. The Riemann-Liouville fractional integral of order α>0with upper limit bis defined as (Iα b−f)(x) =1 Γ(α)b/integraldisplay x(y−x)α−1f(y)dy (1.29b) forx<b. For α=0 (I0 a+f)(x) = ( I0 b−f)(x) = f(x) (1.30) completes the definition. The definition may be generalized t oα∈Cwith Reα>0. Formula (1.29a) appears in [96, p. 363] with a>−∞and in [76, p. 8] with a=−∞. The notation is not standardized. Leibniz, Lagrange and Li ouville used the symbol/integraltextα[22,73,76], Grünwald wrote/integraltextα[...dxα]x=xx=a, while Riemann used ∂−αx[96] and Most wrote d−αa/dx−α[89]. The notation in (1.29) is that of [52, 54, 98, 99]. Modern authors also use fα[37], Iα[97], aIαx[94], Iαx[23], aD−α x[85, 91, 102], or d−α/d(x−a)−α[92] instead of Iα a+2. The fractional integral operators Iα a+, Iα b−are commonly called Riemann- Liouville fractional integrals [94, 98, 99] although somet imes this name is re- served for the case a=0 [85]. Their domain of definition is typically chosen 2)Some authors [23, 26, 85, 91, 92, 97] employ the derivative sy mbol D also for integrals, resp. Ifor derivatives, to emphasize the similarity between fractional integration and differentiation. If th is is done, the choice of Riesz and Feller, namely I, seems superior in the sense that fractional derivatives, similar to integrals, are nonloca l operators, while integer derivatives are local operators. 1.2 MathematicalIntroductiontoFractionalDerivatives 9 asD(Iα a+) =L1([a,b])orD(Iα a+) =L1 loc([a,b])[94, 98, 99]. For the definition of Lebesgue spaces see the Appendix B. If f∈L1([a,b])then(Iα a+f)∈L1([a,b]) and(Iα a+f)(x)is finite for almost all x. Iff∈Lp([a,b])with 1 ≤p≤∞and α>1/pthen(Iα a+f)(x)is finite for all x∈[a,b]. Analogous statements hold for(Iα b−f)(x)[98]. A short table of Riemann-Liouville fractional integrals is given in Appendix A. For a more extensive list of fractional integrals see [24] . 1.2.1.3Weyl Fractional Integrals Examples (1.5) and (1.6) or (A.2) and (A.3) show that Definiti on 1.1 is well suited for fractional integration of power series, but not f or functions defined by Fourier series. In fact, if f(x)is a periodic function with period 2 π, and3 f(x)∼∞ ∑ k=−∞ckeikx(1.31) then the Riemann-Liouville fractional (Iα a+f)will in general not be periodic. For this reason an alternative definition of fractional inte grals was investi- gated by Weyl [124]. Functions on the unit circle G=R/2πZcorrespond to 2 π-periodic func- tions on the real line. Let f(x)be periodic with period 2 πand such that the integral of fover the interval [0, 2π]vanishes, so that c0=0 in eq. (1.31). Then the integral of fis itself a periodic function, and the constant of integrati on can be chosen such that the integral over [0, 2π]vanishes again. Repeating the integration ntimes one finds using (1.6) and the integral representation ck= (1/2π)/integraltext2π 0e−iksf(s)dsof Fourier coefficients ∞ ∑ k=−∞ckeikx (ik)n=1 2π2π/integraldisplay 0f(y)∞ ∑ k=−∞ k/\e}atio\slash=0eik(x−y) (ik)ndy (1.32) with c0=0. Recall the convolution formula [132, p.36] (f∗g)(t) =1 2π2π/integraldisplay 0f(t−s)g(s)ds=∞ ∑ k=−∞fkgkeikt(1.33) 3)The notation ∼indicates that the sum does not need to converge, and, if it converges, does not need to converge to f(x). 101 ThreefoldIntroductiontoFractionalDerivatives for two periodic functions f(t)∼∑∞ k=−∞fkeiktand g(t)∼∑∞ k=−∞gkeikt. Us- ing eq. (1.33) and generalizing (1.32) to noninteger nsuggests the following definition. [94, 99]. Definition 1.2 Let f∈Lp(R/2πZ), 1≤p<∞be periodic with period 2 π and such that its integral over a period vanishes. The Weyl fractional integral of order αis defined as (Iα ±f)(x) = ( Ψα ±∗f)(x) =1 2π2π/integraldisplay 0Ψα ±(x−y)f(y)dy (1.34) where Ψα ±(x) =∞ ∑ k=−∞ k/\e}atio\slash=0eikx (±ik)α(1.35) for 0<α<1. It can be shown that the series for Ψα±(x)converges and that the Weyl defi- nition coincides with the Riemann-Liouville definition [13 3] (Iα +f)(x) =1 Γ(α)x/integraldisplay −∞(x−y)α−1f(y)dy (1.36a) respectively (Iα −f)(x) =1 Γ(α)∞/integraldisplay x(y−x)α−1f(y)dy (1.36b) for 2 πperiodic functions whose integral over a period vanishes. T his is eq. (1.29) with a=−∞resp. b=∞. For this reason the Riemann-Liouville fractional integrals with limits ±∞, Iα +f=Iα (−∞)+fand Iα −f=Iα ∞−f, are often called Weyl fractional integrals [24, 85, 94, 99]. The Weyl fractional integral may be rewritten as a convoluti on (Iα ±f)(x) = ( Kα ±∗f)(x) (1.37) where the convolution product for functions on Ris defined as4 (K∗f)(x):=∞/integraldisplay −∞K(x−y)f(y)dy (1.38) 4)IfK,f∈L1(R)then(K∗f)(t)exists for almost all t∈Rand f∈L1(R). IfK∈Lp(R),f∈Lq(R)with 1 <p,q<∞and 1/p+1/q=1 then K∗f∈C0(R), the space of continuous functions vanishing at infinity. 1.2 MathematicalIntroductiontoFractionalDerivatives 11 and the convolution kernels are defined as Kα ±(x):=Θ(±x)(±x)α−1 Γ(α)(1.39) forα>0. Here Θ(x) =  1 , x>0 0 , x≤0(1.40) is the Heaviside unit step function, and xα=expαlogxwith the convention that log xis real for x>0. For α=0 the kernel K0 +(x) =K0 −(x) =δ(x) (1.41) is the Dirac δ-function defined in (C.2) in Appendix C. Note that Kα±∈L1 loc(R) forα>0. 1.2.1.4Riesz Fractional Integrals Riemann-Liouville and Weyl fractional integrals have uppe r or lower limits of integration, and are sometimes called left-sided resp. rig ht-sided integrals. A more symmetric definition was advanced in [97]. Definition1.3 Letf∈L1 loc(R)be locally integrable. The Riesz fractional integral orRiesz potential of order α>0 is defined as the linear combination [99] (Iαf)(x) =(Iα +f)(x) + ( Iα −f)(x) 2 cos(απ/2)=1 2Γ(α)cos(απ/2)∞/integraldisplay −∞f(y) |x−y|1−αdy(1.42) of right- and left-sided Weyl fractional integrals. The conjugate Riesz potential is defined by (/tildewideIαf)(x) =(Iα +f)(x)−(Iα −f)(x) 2 sin(απ/2)=1 2Γ(α)sin(απ/2)∞/integraldisplay −∞sgn(x−y)f(y) |x−y|1−αdy (1.43) Of course, α/\e}atio\slash=2k+1,k∈Zin (1.42) and α/\e}atio\slash=2k,k∈Zin (1.43). The definition is again completed with (I0f)(x) = (/tildewideI0f)(x) = f(x) (1.44) forα=0. 121 ThreefoldIntroductiontoFractionalDerivatives Riesz fractional integration may be written as a convolutio n (Iαf)(x) = ( Kα∗f)(x) (1.45a) (/tildewideIαf)(x) = (/tildewideKα∗f)(x) (1.45b) with the (one-dimensional) Riesz kernels Kα(x) =Kα+(x) +Kα−(x) 2 cos(απ/2)=|x|α−1 2 cos(απ/2)Γ(α)(1.46) forα/\e}atio\slash=2k+1,k∈Z, and /tildewideKα(x) =Kα+(x)−Kα−(x) 2 sin(απ/2)=|x|α−1sgn(x) 2 sin(απ/2)Γ(α)(1.47) forα/\e}atio\slash=2k,k∈Z. Subsequently, Feller introduced the generalized Riesz-F eller kernels [26] Kα,β(x) =|x|α−1sin[α(π/2+βsgnx)] 2 sin(απ/2)Γ(α)(1.48) with parameter β∈R. The corresponding generalized Riesz-Feller fractional integral of order αand type βis defined as (Iα,βf)(x) = ( Kα,β∗f)(x). (1.49) This formula interpolates continuously from the Weyl integ ral Iα −=Iα,−π/2 forβ=−π/2 through the Riesz integral Iα=Iα,0forβ=0 to the Weyl integral Iα +=Iα,π/2forβ=π/2. Due to their symmetry Riesz-Feller fractional integrals are readily generalized to higher dimensions. 1.2.1.5Fractional Integrals ofDistributions Fractional integration can be extended to distributions us ing the convolution formula (1.37) above. Distributions are generalized funct ions [31, 105]. They are defined as linear functionals on a space Xof conveniently chosen “test functions”. For every locally integrable function f∈L1 loc(R)there exists a distribution Ff:X→Cdefined by Ff(ϕ) =/a\}bracketle{tf,ϕ/a\}bracketri}ht=∞/integraldisplay −∞f(x)ϕ(x)dx (1.50) where ϕ∈Xis test function from a suitable space Xof test functions. By abuse of notation one often writes ffor the associated distribution Ff. Distri- butions that correspond to functions via (1.50) are called regular distributions . 1.2 MathematicalIntroductiontoFractionalDerivatives 13 Examples for regular distributions are the convolution ker nels Kα±∈L1 loc(R) defined in (1.39). They are locally integrable functions on Rwhen α>0. Dis- tributions that are not regular are sometimes called singular . An important example for a singular distribution is the Dirac δ-function. It is defined as δ:X→C /integraldisplay δ(x)ϕ(x)dx=ϕ(0) (1.51) for every test function ϕ∈X. The test function space Xis usually chosen as a subspace of C∞(R), the space of infinitely differentiable functions. A brief introduction to distributions is given in Appendix C. In order to generalize (1.37) to distributions one must defin e the convolution of two distributions. To do so one multiplies eq. (1.38) on bo th sides with a smooth test function ϕ∈C∞ c(R)of compact support. Integrating gives /a\}bracketle{tK∗f,ϕ/a\}bracketri}ht=∞/integraldisplay −∞∞/integraldisplay −∞K(x−y)f(y)ϕ(x)dydx =∞/integraldisplay −∞∞/integraldisplay −∞K(x)f(y)ϕ(x+y)dydx =/a\}bracketle{tK(x),/a\}bracketle{tf(y),ϕ(x+y)/a\}bracketri}ht/a\}bracketri}ht. (1.52) where the notation /a\}bracketle{tf(y),ϕ(x+y)/a\}bracketri}htmeans that the functional Ffis applied to the function ϕ(x+·)for fixed x. Explicitly, for fixed x Ff(ϕx) =/a\}bracketle{tf(y),ϕx(y)/a\}bracketri}ht=/a\}bracketle{tf(y),ϕ(x+y)/a\}bracketri}ht=∞/integraldisplay −∞f(y)ϕ(x+y)dx (1.53) where ϕx(·) = ϕ(x+·). Equation (1.52) can be used as a definition for the convolution of distributions provided that the right hand s ide has meaning. This is not always the case as the counterexample K=f=1 shows. In general the convolution product is not associative (see eq. (1.113)). However, associative and commutative convolution algebras exist [2 1]. Equation (1.52) is always meaningful when supp Kor supp fis compact [63]. Another case is when Kand fhave support in R+. This will be assumed in the following. Definition 1.4 Letfbe a distribution f∈C∞ 0(R)′with supp f⊂R+. Then its fractional integral is the distribution Iα 0+fdefined as /a\}bracketle{tIα 0+f,ϕ/a\}bracketri}ht=/a\}bracketle{tIα +f,ϕ/a\}bracketri}ht=/a\}bracketle{tKα +∗f,ϕ/a\}bracketri}ht (1.54) for Re α>0. It has support in R+. 141 ThreefoldIntroductiontoFractionalDerivatives Iff∈C∞ 0(R)′with supp f⊂R+then also Iα 0+f∈C∞ 0(R)′with supp Iα 0+f⊂R+. 1.2.1.6Integral Transforms The Fourier transformation is defined as F{f}(k) =∞/integraldisplay −∞e−ikxf(x)dx (1.55) for functions f∈L1(R). Then F{Iα ±f}(k) = (±ik)−αF{f}(k) (1.56) holds for 0 <α<1 by virtue of the convolution theorem. The equation cannot be extended directly to α≥1 because the Fourier integral on the left hand side may not exist. Consider e.g. α=1 and f∈C∞ c(R). Then (I1 +f)(x)→const asx→∞andF/braceleftBig I1 +f/bracerightBig does not exist [94]. Equation (1.56) can be extended to all αwith Re α>0 for functions in the so called Lizorkin space [99, p.148] defined as the space of functions f∈ S(R)such that (DmF{f})(0) = 0 for allm∈N0. For the Riesz potentials one has F{Iαf}(k) =|k|−αF{f}(k) (1.57a) F/braceleftBig /tildewideIαf/bracerightBig (k) = (−i sgn k)|k|−αF{f}(k) (1.57b) for functions in Lizorkin space. The Laplace transform is defined as L{f}(u) =∞/integraldisplay 0e−uxf(x)dx (1.58) for locally integrable functions f:R+→C. Now L/braceleftbig Iα 0+f/bracerightbig(u) =u−αL{f}(u) (1.59) by the convolution theorem for Laplace transforms. The Lapl ace transform of Iα 0−fleads to a more complicated operator. 1.2 MathematicalIntroductiontoFractionalDerivatives 15 1.2.1.7Fractional Integration by Parts Iff(x)∈Lp([a,b]),g∈Lq([a,b])with 1/ p+1/q≤1+α,p,q≥1 and p/\e}atio\slash=1, q/\e}atio\slash=1 for 1/ p+1/q=1+αthen the formula b/integraldisplay af(x)(Iα a+g)(x)dx=b/integraldisplay ag(x)(Iα b−f)(x)dx (1.60) holds. The formula is known as fractional integration by par ts [99]. For f(x)∈ Lp(R),g∈Lq(R)with p>1,q>1 and 1/ p+1/q=1+αthe analogous formula ∞/integraldisplay −∞f(x)(Iα +g)(x)dx=∞/integraldisplay −∞g(x)(Iα −f)(x)dx (1.61) holds for Weyl fractional integrals. These formulae provide a second method of generalizing frac tional integra- tion to distributions. Equation (1.60) may be read as /a\}bracketle{tIα a+f,ϕ/a\}bracketri}ht=/a\}bracketle{tf, Iα b−ϕ/a\}bracketri}ht (1.62) for a distribution fand a test function ϕ. It shows that right- and left-sided fractional integrals are adjoint operators. The formula ma y be viewed as a definition of the fractional integral Iα a+fof a distribution provided that the operator Iα b−maps the test function space into itself. 1.2.1.8Hardy-Littlewood Theorem The mapping properties of convolutions can be studied with t he help of Youngs inequality. Let p,q,robey 1 ≤p,q,r≤∞and 1/ p+1/q=1+1/r. IfK∈Lp(R)and f∈Lq(R)then K∗f∈Lr(R)and Youngs inequal- ity/bardblK∗f/bardblr≤ /bardbl K/bardblp/bardblf/bardblqholds. It follows that /bardblK∗f/bardblq≤C/bardblf/bardblpif 1≤p≤q≤∞and K∈Lr(R)with 1/ r=1+ (1/q)−(1/p). The Hardy- Littlewood theorem states that these estimates remain vali d for Kα±although these kernels do not belong to any Lp(R)-space [37, 38]. The theorem was generalized to higher dimensions by Sobolev in 1938, and is a lso known as the Hardy-Littlewood-Sobolev inequality (see [37, 38, 63, 113]). Theorem 1.5 Let0<α<1,1<p<1/α,−∞≤a<b≤∞. Then Iα a+, Iα b− are bounded linear operators from Lp([a,b])to Lq([a,b])with 1/q= (1/p)−α,i.e. there exists a constant C (p,q)independent of f such that /bardblIα a+f/bardblq≤C/bardblf/bardblp. 161 ThreefoldIntroductiontoFractionalDerivatives 1.2.1.9Additivity The basic composition law for fractional integrals follows from (Kα +∗Kβ +)(x) =x/integraldisplay 0Kα +(x−y)Kβ +(y)dy=x/integraldisplay 0(x−y)α−1 Γ(α)yβ−1 Γ(β)dy =xα−1 Γ(α)xβ−1 Γ(β)1/integraldisplay 0(1−z)α−1zβ−1xdz =xα+β−1 Γ(α+β)=Kα+β +(x) (1.63) where Euler’s Beta-function Γ(α)Γ(β) Γ(α+β)=1/integraldisplay 0(1−z)α−1zβ−1xdz=B(α,β) (1.64) was used. This implies the semigroup law for exponents Iα a+Iβ a+=Iα+β a+, (1.65) also called additivity law. It holds for Riemann-Liouville , Weyl and Riesz- Feller fractional integrals of functions. 1.2.2 Fractional Derivatives 1.2.2.1Riemann-Liouville Fractional Derivatives Riemann [96, p.341] suggested to define fractional derivati ves as integer order derivatives of fractional integrals. Definition 1.6 Let−∞≤a<x<b≤∞. The Riemann-Liouville fractional derivative of order 0<α<1with lower limit a (resp. upper limit b) is defined for functions such that f∈L1([a,b])and f∗K1−α∈W1,1([a,b])as (Dα a±f)(x) =±d dx(I1−α a±f)(x) (1.66) and(D0 a±f)(x) = f(x)forα=0. For α>1 the definition is extended for functions f∈L1([a,b])with f∗Kn−α∈Wn,1([a,b])as (Dα a±f)(x) = (±1)ndn dxn(In−α a±f)(x) (1.67) 1.2 MathematicalIntroductiontoFractionalDerivatives 17 where5n= [Reα] +1 is smallest integer larger than α. Here Wk,p(G) ={f∈Lp(G): Dkf∈Lp(G)}denotes a Sobolev space defined in (B.17). For k=p=1 the space W1,1([a,b]) = AC0([a,b])coincides with the space of absolutely continuous functions. The notation for fractional derivatives is not standardize d6. Leibniz and Euler used dα[25, 72, 73] Riemann wrote ∂α x[96], Liouville preferred dα/dxα [76], Grünwald used {dαf/dxα}x=x x=aor Dα[f]x=x x=a[34], Marchaud wrote D(α) a, and Hardy-Littlewood used an index fα[37]. The notation in (1.67) follows [52, 54, 98, 99]. Modern authors also use I−α[97], I−αx[23], aDαx[85, 94, 102], dα/dxα[102, 129], dα/d(x−a)α[92] instead of Dα a+. Let f(x)be absolutely continuous on the finite interval [a,b]. Then, its derivative f′exists almost everywhere on [a,b]with f′∈L1([a,b]), and the function fcan be written as f(x) =x/integraldisplay af′(y)dy+f(a) = ( I1 a+f′)(x) +f(a) (1.68) Substituting this into Iα a+fgives (Iα a+f)(x) = ( I1 a+Iα a+f′)(x) +f(a) Γ(α+1)(x−a)α(1.69) where commutativity of I1 a+and Iα a+was used. It follows that (D Iα a+f)(x)−(Iα a+Df)(x) =f(a) Γ(α)(x−a)α−1(1.70) for 0<α<1. Above, the notations (Df)(x) =df(x) dx=f′(x) (1.71) were used for the first order derivative. This observation suggests to introduce a modified Riemann-L iouville frac- tional derivative through (/tildewideDα a+f)(x):=In−α a+f(n)(x) =1 Γ(n−α)x/integraldisplay af(n)(y) (x−y)α−n+1dy (1.72) 5)[x]is the largest integer smaller than x. 6)see footnote 2 181 ThreefoldIntroductiontoFractionalDerivatives where n= [Reα] +1. Note, that fmust be at least n-times differentiable. For- mula (1.72) is due to Liouville [76, p.10] (see eq. (1.18) abo ve), but nowadays sometimes named after Caputo [17]. The relation between (1.72) and (1.67) is given by Theorem 1.7 For f ∈ACn−1([a,b])with n = [Reα] +1the Riemann-Liouville fractional derivative (Dα a+f)(x)exists almost everywhere for Reα≥0. It can be written as (Dα a+f)(x) = (/tildewideDα a+f)(x) +n−1 ∑ k=0(x−a)k−α Γ(k−α+1)f(k)(a) (1.73) in terms of the Liouville(-Caputo) derivative defined in (1.72) . The Riemann-Liouville fractional derivative is the left in verse of Riemann- Liouville fractional integrals. More specifically, [99, p. 44] Theorem 1.8 Let f∈L1([a,b]). Then Dα a+Iα a+f(x) = f(x) (1.74) holds for all αwith Reα≥0. For the right inverses of fractional integrals one finds Theorem 1.9 Let f∈L1([a,b])andReα>0. If in addition In−α a+f∈ACn([a,b]) where n = [Reα] +1then Iα a+Dα a+f(x) = f(x)−n−1 ∑ k=0(x−a)α−k−1 Γ(α−k)/parenleftBig Dn−k−1In−α a+f/parenrightBig (a) (1.75) holds. For 0<Reα<1this becomes Iα a+Dα a+f(x) = f(x)−(I1−α a+f)(a) Γ(α)(x−a)α−1(1.76) The last theorem implies that for f∈L1([a,b])and Re α>0 with n= [Reα] +1 the equality Iα a+Dα a+f(x) = f(x) (1.77) holds only if In−α a+f∈ACn([a,b]) (1.78a) 1.2 MathematicalIntroductiontoFractionalDerivatives 19 and /parenleftBig DkIn−α a+f/parenrightBig (a) =0 (1.78b) for all k=0, 1, 2, ..., n−1. Note that the existence of g(x) = Dα a+f(x)in eq. (1.77) does not imply that f(x)can be written as (Iα a+g)(x)for some integrable function g[99]. This holds only if both conditions (1.78) are satisfied . As an example where one of them fails, consider the function f(x) = ( x−a)α−1for 0<α<1. Then Dα a+(x−a)α−1=0 exists. Now D0I1−α a+(x−a)α−1/\e}atio\slash=0 so that (1.78b) fails. There does not exist an integrable gsuch that Iα a+g= (x−a)α−1. In fact, gcorresponds to the δ-distribution δ(x−a). 1.2.2.2General Types ofFractional Derivatives Riemann-Liouville fractional derivatives have been gener alized in [52, p.433] to fractional derivatives of different types. Definition1.10 Thegeneralized Riemann-Liouville fractional derivative of o rder0< α<1and type 0≤β≤1with lower (resp. upper) limit a is defined as (Dα,β a±f)(x) =/parenleftbigg ±Iβ(1−α) a±d dx/parenleftBig I(1−β)(1−α) a± f/parenrightBig/parenrightbigg (x) (1.79) for functions such that the expression on the right hand side exists. The type βof a fractional derivative allows to interpolate continuou sly from Dα a±=Dα,0 a±to/tildewideDα a±=Dα,1 a±. A relation between fractional derivatives of the same order but different types was given in [52, p.434]. 1.2.2.3Marchaud-Hadamard Fractional Derivatives Marchaud’s approach [78] is based on Hadamards finite parts o f divergent integrals [36]. The strategy is to define fractional derivat ives as analytic con- tinuation of fractional integrals to negative orders. [see [99, p.225]] Definition 1.11 Let−∞<a<b<∞and 0 <α<1. The Marchaud fractional derivative of order αwith lower limit a is defined as (Mα a+f)(x) =f(x) Γ(1−α)(x−a)α+α Γ(1−α)x/integraldisplay af(x)−f(y) (x−y)α+1dy (1.80) 201 ThreefoldIntroductiontoFractionalDerivatives and the Marchaud fractional derivative of order αwith upper limit b is defined as (Mα b−f)(x) =f(x) Γ(1−α)(b−x)α+α Γ(1−α)b/integraldisplay xf(x)−f(y) (x−y)α+1dy (1.81) Fora=−∞(resp. b=∞) the definition is (Mα ±f)(x) =α Γ(1−α)∞/integraldisplay 0f(x)−f(x∓y) yα+1dy (1.82) The definition is completed with M0f=ffor all variants. The idea of Marchaud’s method is to extend the Riemann-Liouv ille integral from α>0 to α<0, and to define (I−α +f)(x) =1 Γ(−α)∞/integraldisplay 0y−α−1f(x−y)dy (1.83) where α>0. However, this is not possible because the integral in (1.8 3) di- verges. The idea is to subtract the divergent part of the inte gral, /integraldisplay∞ εy−α−1f(x)dy=f(x) αεα(1.84) obtained by setting f(x−y)≈f(x)fory≈0. Subtracting (1.83) from (1.84) for 0<α<1 suggests the definition (Mα +f)(x) = lim ε→0+1 Γ(−α)∞/integraldisplay εf(x)−f(x−y) yα+1dy (1.85) Formal integration by parts leads to (I1−α +f′)(x), showing that this definition contains the Riemann-Liouville definition. The definition may be extended to α>1 in two ways. The first consists in applying (1.85) to the n-th derivative dnf/dxnforn<α<n+1. The second possibility is to regard f(x−y)−f(x)as a first order difference, and to generalize to n-th order differences. The n-th order difference is (∆n yf)(x) = ( 1−Ty)nf(x) =n ∑ k=0(−1)k/parenleftbiggn k/parenrightbigg f(x−ky) (1.86) where (1f)(x) = f(x)is the identity operator and (Thf)(x) = f(x−h) (1.87) 1.2 MathematicalIntroductiontoFractionalDerivatives 21 is the translation operator. The Marchaud fractional deriv ative can then be extended to 0 <α<nthrough [94, 98] (Mα +f)(x) = lim ε→0+1 Cα,n∞/integraldisplay ε∆n yf(x) yα+1dy (1.88) where Cα,n=∞/integraldisplay 0(1−e−y)n yα+1dy (1.89) where the limit may be taken in the sense of pointwise or norm c onvergence. The Marchaud derivatives Mα ±are defined for a wider class of functions than Weyl derivatives Dα ±. As an example consider the function f(x) =const. Let fbe such that there exists a function g∈L1([a,b])with f=Iα a+g. Then the Riemann-Liouville derivative and the Marchaud der ivative coincide almost everywhere, i.e. (Mα a+f)(x) = ( Dα a+f)(x)for almost all x[99, p.228]. 1.2.2.4Weyl Fractional Derivatives There are two kinds of Weyl fractional derivatives for perio dic functions. The Weyl-Liouville fractional derivative is defined as [99, p.351], [94] (Dα ±f)(x) =±d dx(I1−α ±f)(x) (1.90) for 0<α<1 where the Weyl integral ±Iα ±fwas defined in (1.34). The Weyl-Marchaud fractional derivative is defined as [99, p.352], [94] (Wα ±f)(x) =1 2π2π/integraldisplay 0[f(x−y)−f(x)](D1Ψ1−α ±)(y)dy (1.91) for 0<α<1 where Ψ±(x)is defined in eq. (1.35). The Weyl derivatives are defined for periodic functions of with zero mean in Cβ(R/2πZ)where β>α. In this space (Dα ±f)(x) = ( Wα ±f)(x), i.e. the Weyl-Liouville and Weyl-Marchaud form coincide [99]. As for fractional integr als, it can be shown that the Weyl-Liouville derivative (0<α<1) (Dα +f)(x) =1 Γ(1−α)x/integraldisplay −∞f(y) (x−y)αdy (1.92) 221 ThreefoldIntroductiontoFractionalDerivatives coincides with the Riemann-Liouville derivative with lowe r limit −∞. In ad- dition one has the equivalence Dα +f=Wα +fwith the Marchaud-Hadamard fractional derivative in a suitable sense [99, p.357]. 1.2.2.5Riesz Fractional Derivatives To define the Riesz fractional derivative as integer derivat ives of Riesz poten- tials consider the Fourier transforms F/braceleftBig D I1−αf/bracerightBig (k) = ( ik)|k|α−1F{f}(k) = ( i sgn k)|k|αF{f}(k) (1.93) F/braceleftBig D/tildewidestI1−αf/bracerightBig (k) = ( ik)(−i sgn k)|k|α−1F{f}(k) =|k|αF{f}(k) (1.94) for 0<α<1. Comparing this to eq. (1.57) suggests to consider d dx(/tildewidestI1−αf)(x) =lim h→01 h/bracketleftBig (/tildewidestI1−αf)(x+h)−(/tildewidestI1−αf)(x)/bracketrightBig (1.95) as a candidate for the Riesz fractional derivative. Following [94] the strong Riesz fractional derivative of order αRαfof a function f∈Lp(R), 1≤p<∞, is defined through the limit lim h→0/vextenddouble/vextenddouble/vextenddouble/vextenddouble1 h(f∗K1−α h)−Rαf/vextenddouble/vextenddouble/vextenddouble/vextenddouble p=0 (1.96) whenever it exists. The convolution kernel defined as K1−α h=1 2Γ(1−α)sin(απ/2)/bracketleftbiggsgn(x+h) |x+h|α−sgnx |x|α/bracketrightbigg (1.97) is obtained from eq. (1.95). Indeed, this definition is equiv alent to eq. (1.94). A function f∈Lp(R)where 1 ≤p≤2 has a strong Riesz derivative of order αif and only if there exsists a function g∈Lp(R)such that |k|αF{f}(k) = F{g}(k). Then Rαf=g. 1.2.2.6Grünwald-Letnikov Fractional Derivatives The basic idea of the Grünwald approach is to generalize finit e difference quo- tients to noninteger order, and then take the limit to obtain a differential quo- tient. The first order derivative is the limit d dxf(x) = ( Df)(x) =lim h→0f(x)−f(x−h) h=lim h→0[1−T(h)] hf(x) (1.98) 1.2 MathematicalIntroductiontoFractionalDerivatives 23 of a difference quotient. In the last equality (1f)(x) = f(x)is the identity operator, and [T(h)f](x) = f(x−h) (1.99) is the translation operator. Repeated application of Tgives [T(h)nf](x) = f(x−nh) (1.100) where n∈N. The second order derivative can then be written as d2 dx2f(x) = ( D2f)(x) =lim h→0f(x)−2f(x−h) +f(x−2h) h2 =lim h→0/braceleftbigg[1−T(h)] h/bracerightbigg2 f(x), (1.101) and the n-th derivative dn dxnf(x) = ( Dnf)(x) =lim h→01 hnn ∑ k=0(−1)k/parenleftbiggn k/parenrightbigg f(x−kh) =lim h→0/braceleftbigg[1−T(h)] h/bracerightbiggn f(x) (1.102) which exhibits the similarity with the binomial formula. Th e generalization to noninteger ngives rise to fractional difference quotients defined throu gh (∆α hf)(x) =∞ ∑ k=0(−1)k/parenleftbiggα k/parenrightbigg f(x−kh) (1.103) forα>0. These are generally divergent for α<0. For example, if f(x) = 1, then N ∑ k=0(−1)k/parenleftbiggα k/parenrightbigg =1 Γ(1−α)Γ(N+1−α) Γ(N+1)(1.104) diverges as N→∞ifα<0. Fractional difference quotients were studied in [68]. Note that fractional differences obey [99] (∆α h(∆β hf))(x) = ( ∆α+β hf)(x) (1.105) Definition 1.12 The Grünwald-Letnikov fractional derivative of order α>0 is de- fined as the limit (Gα ±f)(x) = lim h→0+1 hα(∆α ±hf)(x) (1.106) of fractional difference quotients whenever the limit exis ts. The Grünwald Letnikov fractional derivative is called pointwise orstrong depending on whether the limit is taken pointwise or in the norm of a suitab le Banach space. 241 ThreefoldIntroductiontoFractionalDerivatives For a definition of Banach spaces and their norms see e.g. [128 ]. The Grünwald-Letnikov fractional derivative has been stud ied for periodic functions in Lp(R/2πZ)with 1 ≤p<∞in [94, 99]. It has the following properties. Theorem 1.13 Let f∈Lp(R/2πZ),1≤p<∞andα>0. Then the following statements are equivalent: 1.Gα +f∈Lp(R/2πZ) 2. There exists a function g ∈Lp(R/2πZ)such that (ik)αF{f(x)}(k) = F{g(x)}(k)where k ∈Z. 3. There exists a function g ∈Lp(R/2πZ)such that f (x)− F{f(x)}(0) = (Iα +g)(x)holds for almost all x. Theorem 1.14 Let f∈Lp(R/2πZ),1≤p<∞andα,β>0. Then: 1.Gα +f∈Lp(R/2πZ)implies Gβ +f∈Lp(R/2πZ)for every 0<β<α. 2.Gα +Gβ +f=Gα+β +f 3.Gα +(Iα +f) = f(x)− F{f}(0) 1.2.2.7Fractional Derivatives ofDistributions The basic idea for defining fractional differentiation of di stributions is to ex- tend the definition of fractional integration (1.54) to nega tive α. However, for Reα<0 the distribution Kα+becomes singular because xα−1is not locally integrable in this case. The extension of Kα +to Re α<0 requires regulariza- tion [31, 63, 128]. It turns out that the regularization exis ts and is essentially unique as long as (−α)/∈N0. Definition 1.15 Letfbe a distribution f∈C∞ 0(R)′with supp f⊂R+. Then thefractional derivative of order αwith lower limit 0 is the distribution Dα 0+f defined as /a\}bracketle{tDα 0+f,ϕ/a\}bracketri}ht=/a\}bracketle{tDα +f,ϕ/a\}bracketri}ht=/a\}bracketle{tK−α +∗f,ϕ/a\}bracketri}ht (1.107) where α∈Cand Kα +(x) =  Θ(x)xα−1 Γ(α), Reα>0 dN dxN/bracketleftbigg Θ(x)xα+N−1 Γ(α+N)/bracketrightbigg , Reα+N>0,N∈N(1.108) 1.2 MathematicalIntroductiontoFractionalDerivatives 25 is the kernel distribution. For α=0 one finds K0+(x) = ( d/d x)Θ(x) = δ(x) and D0 0+=1as the identity operator. For the α=−k,k∈None finds K−k +(x) =δ(k)(x) (1.109) where δ(k)is the k-th derivative of the δdistribution. The kernel distribution in (1.108) is K−α +(x) =d dx/bracketleftbigg Θ(x)x−α Γ(1−α)/bracketrightbigg =d dxK1−α +(x) (1.110) for 0<α<1. Its regularized action is /angbracketleftbig K−α +(x),ϕ(x)/angbracketrightbig=/angbracketleftbiggd dxK1−α +(x),ϕ(x)/angbracketrightbigg =−/angbracketleftBig K1−α +(x),ϕ(x)′/angbracketrightBig (1.111a) =−1 Γ(1−α)lim ε→0∞/integraldisplay εx−αϕ(x)′dx (1.111b) =−lim ε→0  ϕ(x) +C Γ(α)xα/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞ ε−∞/integraldisplay εϕ(x) +C Γ(−α)x1+αdx  (1.111c) =∞/integraldisplay 0ϕ(x)−ϕ(0) Γ(−α)x1+αdx (1.111d) where ϕ(∞)<∞was assumed in the last step and the arbitrary constant was chosen as C=−ϕ(0). This choice regularizes the divergent first term in (1.111c). If this rule is used for the distributional convol ution (K−α +∗f)(x) =1 Γ(−α)∞/integraldisplay 0f(x)−f(x−y) yα+1dy= (Mα +f)(x) (1.112) then the Marchaud-Hadamard form is recovered with 0 <α<1. It is now possible to show that the convolution of distributi ons is in general not associative. A counterexample is (1∗δ′)∗Θ=1′∗Θ=0∗Θ=0/\e}atio\slash=1=1∗δ=1∗Θ′=1∗(δ′∗Θ)(1.113) where Θis the Heaviside step function. Dα 0+fhas support in R+. The distributions in f∈C∞ 0(R)′with supp f⊂ R+form a convolution algebra [21] and one finds [31, 99] 261 ThreefoldIntroductiontoFractionalDerivatives Theorem 1.16 If f∈C∞ 0(R)′with supp f⊂R+then also Iα 0+f∈C∞ 0(R)′with Iα 0+supp f⊂R+. Moreover, for all α,β∈C Dα 0+Dβ 0+f=Dα+β 0+f (1.114) with Dα 0+f=I−α 0+f for Reα<0. For each f ∈C∞ 0(R)′with supp f⊂R+there exists a unique distribution g ∈C∞ 0(R)′with supp g⊂R+such that f =Iα 0+g. Note that Dα 0+f=Dα 0+(1f) = ( K−α +∗K0 +)∗f= (Dα 0+δ)∗f=δ(α)∗f (1.115) for all α∈C. Also, the differentiation rule Dα 0+Kβ +=Kβ−α + (1.116) holds for all α,β∈C. It contains DKβ +=Kβ−1 + (1.117) for all β∈Cas a special case. 1.2.2.8Fractional Derivatives atTheir Lower Limit All fractional derivatives defined above are nonlocal opera tors. A local frac- tional derivative operator was introduced in [40, 41, 52]. Definition 1.17 For−∞<a<∞theRiemann-Liouville fractional derivative of order 0<α<1at the lower limit a is defined by dαf dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=a=f(α)(a) = lim x→a±(Dα a±f)(x) (1.118) whenever the two limits exist and are equal. If f(α)(a)exists the function fis called fractionally differentiable at the limit a. These operators are useful for the analysis of singularitie s. They were applied in [40–42, 44, 52] to the analysis of singularities i n the theory of criti- cal phenomena and to the generalization of Ehrenfests class ification of phase transitions. There is a close relationship to the theory of r egularly varying functions [107] as evidenced by the following result [52]. Theorem 1.18 Let the function f :[0,∞[→Rbe monotonously increasing with f(x)≥0and f(0) =0, and such that (Dα,λ 0+f)(x)with 0<α<1and0≤λ≤1 is also monotonously increasing on a neighbourhood [0,δ]for small δ>0. Let 0≤ 1.2 MathematicalIntroductiontoFractionalDerivatives 27 β<λ(1−α) +α, let C ≥0be a constant and Λ(x)a slowly varying function for x→0. Then lim x→0f(x) xβΛ(x)=C (1.119) holds if and only if lim x→0(Dα,λ 0+f)(x) xβ−αΛ(x)=CΓ(β+1) Γ(β−α+1)(1.120) holds. A function fis called slowly varying at infinity if lim x→∞f(bx)/f(x) = 1 for all b>0. A function f(x)is called slowly varying at a∈Riff(1/(x−a)) is slowly varying at infinity. 1.2.2.9Fractional Powersof Operators The spectral decomposition of selfadjoint operators is a fa miliar mathematical tool from quantum mechanics [116]. Let Adenote a selfadjoint operator with domain D(A)and spectral family Eλon a Hilbert space Xwith scalar product (., .). Then (Au,v) =/integraldisplay σ(A)λd(Eλu,v) (1.121) holds for all u,v∈D(A). Here σ(A)is the spectrum of A. It is then straight- forward to define the fractional power Aαuby (Aαu,u) =/integraldisplay σ(A)λαd(Eλu,u) (1.122) on the domain D(Aα) ={u∈X:/integraldisplay σ(A)λαd(Eλu,u)<∞}. (1.123) Similarly, for any measurable function g:σ(A)→Cthe operator g(A)is defined with an integrand g(λ)in eq. (1.122). This yields an operator calculus that allows to perform calculations with functions instead of operators. Fractional powers of the Laplacian as the generator of the di ffusion semi- group were introduced by Bochner [13] and Feller [26] based o n Riesz’ frac- tional potentials. The fractional diffusion equation ∂f ∂t=−(−∆)α/2f (1.124) 281 ThreefoldIntroductiontoFractionalDerivatives was related by Feller to the Levy stable laws [74] using one di mensional frac- tional integrals I−α,βof order −αand type β[26]7. For α=2 eq. (1.124) reduces to the diffusion equation. This type of fractional d iffusion will be re- ferred to as fractional diffusion of Bochner-Levy type (see Section 1.3.4 for more discussion). Later, these ideas were extended to fractiona l powers of closed8 semigroup generators [4, 5, 69, 70]. If (−A)is the infinitesimal generator of a semigroup T(t)(see Section 1.3.3.2 for definitions of T(t)and A) on a Banach space Bthen its fractional power is defined as (−A)αf=lim ε→0+1 −Γ(−α)∞/integraldisplay εt−α−1[1−T(t)]fdt (1.125) for every f∈Bfor which the limit exists in the norm of B[93, 120, 121, 123]. This aproach is clearly inspired by the Marchaud form (1.82) . Alternatively, one may use the Grünwald approach to define fractional powers of semigroup generators [99, 122]. 1.2.2.10 Pseudodifferential Operators The calculus of pseudodifferential operators represents a nother generalization of the operator calculus in Hilbert spaces. It has its roots i n Hadamard’s ideas [36], Riesz potentials [97], Feller’s suggestion [26] and C alderon-Zygmund singular integrals [16]. Later it was generalized and becam e a tool for treating elliptic partial differential operators with nonconstant coefficients. Definition 1.19 A (Kohn-Nirenberg) pseudodifferential operator of order α∈R σ(x, D):S(Rd)→ S(Rd)is defined as σ(x, D)f(x) =1 (2π)d/integraldisplay Rdeixkσ(x,k)F{f}(k)dk (1.126) and the function σ(x,k)is called its symbol . The symbol is in the Kohn- Nirenberg symbol class Sαif it is in C∞(R2d), and there exists a compact set K⊂Rdsuch that supp σ⊂K×Rd, and for any pair of multiindices β,γthere is a constant Cβ,γsuch that Dβ kDγ xσ(x,k)≤Cβ,γ(1+|k|)α−|β|(1.127) The Hörmander symbol class Sα ρ,δis obtained by replacing the exponent α− |β|on the right hand side with α−ρ|β|+δ|γ|where 0 ≤ρ,δ≤1. 7)Fellers motivation to introduce the type βwas this relation. 8)An operator A:B→Bon a Banach space Bis called closed if the set of pairs (x,Ax)with x∈D(A)is closed in B×B. 1.2 MathematicalIntroductiontoFractionalDerivatives 29 Pseudodifferential operators provide a unified approach to differential and integral or convolution operators that are “nearly” transl ation invariant. They have a close relation with Weyl quantization in physics [28, 116]. However, they will not be discussed further because the traditional s ymbol classes do not contain the usual fractional derivative operators. Fra ctional Riesz deriva- tives are not pseudodifferential operators in the sense abo ve. Their symbols do not fall into any of the standard Kohn-Nirenberg or Hörman der symbol classes due to lack of differentiability at the origin. 1.2.3 Eigenfunctions The eigenfunctions of Riemann-Liouville fractional deriv atives are defined as the solutions of the fractional differential equation (Dα 0+f)(x) =λf(x) (1.128) where λis the eigenvalue. They are readily identifed using eq. (A.6 ) as f(x) =x1−αEα,α(λxα) (1.129) where Eα,β=∞ ∑ k=0xk Γ(αk+β)(1.130) is the generalized Mittag-Leffler function [125,126]. More generally the eigen- value equation for fractional derivatives of order αand type βreads (Dα,β 0+f)(x) =λf(x), (1.131) and it is solved by [54, eq.124] f(x) =x(1−β)(1−α)Eα,α+β(1−α)(λxα) (1.132) where the case β=0 corresponds to (1.128). A second important special case is the equation (Dα,1 0+f)(x) =λf(x), (1.133) with Dα,1 0+=/tildewideDα 0+. In this case the eigenfunction f(x) =Eα(λxα) (1.134) 301 ThreefoldIntroductiontoFractionalDerivatives Fig. 1.1Truncatedrealpartof thegeneralizedMittag-Lefflerfunct ion −3≤Re E 0.8,0.9(z)≤3forz∈Cwith−7≤Rez≤5and−10≤ Imz≤10. The solidlineis definedby Re E 0.8,0.9(z) =0. Fig. 1.2Sameas Fig. 1.1fortheimaginarypartof E0.8,0.9(z). The solidlineis Im E 0.8,0.9(z) =0. where E α(x) = Eα,1(x)is the Mittag-Leffler function [86]. The Mittag-Leffler function plays a central role in fractional calculus. It has only recently been calculated numerically in the full complex plane [62, 108]. Figure 1.1 and 1.2 illustrate E 0.8,0.9(z)for a rectangular region in the complex plane (see [108]). The solid line in Figure 1.1 is the line Re E 0.8,0.9(z) = 0, in Figure 1.2 it is Im E 0.8,0.9(z) =0. 1.3 PhysicalIntroductiontoFractionalDerivatives 31 Note, that some authors are avoiding the operator Dα,1 0+in fractional differ- ential equations (see e.g. [7,82,84,101,111,112] or chapt ers in this volume). In their notation the eigenvalue equation (1.133) becomes (c. f. [112, eq.(22)]) d dxf(x) =λD1−α 0+f(x) (1.135) containing two derivative operators instead of one. 1.3 PhysicalIntroduction toFractional Derivatives 1.3.1 BasicQuestions An introduction to fractional derivatives would be incompl ete without an in- troduction to applications. In the past fractional calculu s has been used pre- dominantly as a convenient calculational tool [26, 76, 89]. A well known ex- ample is Riesz’ interpolation method for solving the wave eq uation [20]. In recent times, however, fractional differential equations appear as “generaliza- tions” of more or less fundamental equations of physics [3,1 2,18,23,43,46,52, 54–56,58,60,90,91,102,104,119,129]. The idea is that phy sical phenomena can be described by fractional differential equations. This pr actice raises at least two fundamental questions: 1. Are mathematical models with fractional derivatives con sistent with the fundamental laws and fundamental symmetries of nature ? 2. How can the fractional order αof differentiation be observed or how does a fractional derivative emerge from concrete models ? Both questions will be addressed here. The answer to the first question is provided by the theory of fractional time evolutions [43,47 ], the answer to the second question by anomalous subdiffusion [46, 60]. 321 ThreefoldIntroductiontoFractionalDerivatives 1.3.2 Fractional Space Fractional derivatives are nonlocal operators. Neverthel ess, numerous au- thors have proposed fractional differential equations inv olving fractional spa- tial derivatives. Particularly popular are fractional pow ers of the Laplace op- erator due to the well known work of Riesz, Feller and Bochner [13,27,97]. The nonlocality of fractional spatial derivatives raises seri ous (largely) unresolved physical problems. As an illustration of the problem with spatial fractional de rivatives consider the one dimensional potential equation for functions f∈C2(R) d2 dx2f(x) =0, x∈G (1.136) on the open interval G=]a,b[with boundary conditions f(a) = 0,f(b) = 0 with a<b. A solution of this boundary value problem is f(x) = 0 with x∈G. This trivial solution remains unchanged as long as the boun dary values f(a) = f(b) =0 remain unperturbed. All functions f∈C2(R)that vanish on [a,b]are solutions of the boundary value problem. In particular, the boundary specification f(x) =0, for x∈R\G (1.137) and the perturbed boundary specification f(x) =g(x), for x∈R\G (1.138) with g≥0 and supp g∩[a,b] =∅have the same trivial solution f=0 in G. The reason is that d2/dx2is a local operator. Consider now a fractional generalization of (1.136) that ar ises for example as the stationary limit of (Bochner-Levy) fractional diffu sion equations with a fractional Laplace operator [13]. Such a onedimensional fr actional Laplace equation reads Rαf(x) =0 (1.139) where Rαis a Riesz fractional derivative of order 0 <α<1. For the boundary specification (1.137) it has the same trivial solution f(x) = 0 for all x∈G. But this solution no longer applies for the perturbed bounda ry specification (1.138). In fact, assuming (1.138) for x∈R\Gand f(x) = 0 for x∈Gnow yields (Rαf)(x)/\e}atio\slash=0 for all x∈G. The exterior R\Gof the domain Gcannot 1.3 PhysicalIntroductiontoFractionalDerivatives 33 be isolated from the interior of Gusing classical boundary conditions. The reason is that Rαis a nonlocal operator. Locality in space is a basic and firmly established principle of physics (see e.g. [35, 115]). Of course, one could argue that relativisti c effects are negli- gible, and that fractional spatial derivatives might arise as an approximate phenomenological model describing an underlying physical reality that obeys spatial locality. However, spatial fractional derivative s imply not only action at a distance. As seen above, they imply also that the exterio r domain cannot be decoupled from the interior by conventional walls or boun dary conditions. This has far reaching consequences for theory and experimen t. In theory it invalidates all arguments based on surface to volume ratios becoming negli- gible in the large volume limit. This includes many concepts and results in thermodynamics and statistical physics that depend on the l ower dimension- ality of the boundary. Experimentally it becomes difficult t o isolate a system from its environment. Fractional diffusion would never com e to rest inside a vessel with thin rigid walls unless the equilibrium concent ration prevails also outside the vessel. A fractionally viscous fluid at rest insi de a container with thin rigid walls would have to start to move when the same fluid starts flow- ing outside the vessel. It seems therefore difficult to recon cile nonlocality in space with theory and experiment. 1.3.3 Fractional Time 1.3.3.1BasicQuestions Nonlocality in time, unlike space, does not violate basic pr inciples of physics, as long as it respects causality [43,47–49,54]. In fact, cau sal nonlocality in time is a common nonequilibrium phenomenon known as history depe ndence, hysteresis and memory. Theoretical physics postulates time translation invarian ce as a fundamental symmetry of nature. As a consequence energy conservation is fundamental, and the infinitesimal generator of time translations is a firs t order time deriva- tive. Replacing integer order time derivatives with fracti onal time derivatives raises at least three basic questions: 1. What replaces time translations as the physical time evol ution ? 2. Is the nonlocality of fractional time derivatives consis tent with the laws of nature ? 341 ThreefoldIntroductiontoFractionalDerivatives 3. Is the asymmetry of fractional time derivatives consiste nt with the laws of nature ? These questions as well as ergodicity breaking, stationari ty, long time limits and temporal coarse grainig were discussed first within ergo dic theory [47–49] and later from a general perspective in [54]. The third question requires special remarks because irreve rsibility is a long- standing and controversial subject [71]. The problem of irr eversibility may be formulated briefly in two ways. Definition1.20 (Thenormalirreversibilityproblem) Assume that time is reversible. Explain how and why time irreversible equations arise in phy sics. Definition 1.21 (The reversed irreversibility problem) Assume that time is irre- versible. Explain how and why time reversible equations ari se in physics. While the normal problem has occupied physicists and mathem aticians for more than a century, the reversed problem was apparently firs t formulated in [59]. Surprisingly, the reversed irreversibility probl em has a clear and quan- titiative solution within the theory of fractional time. Th e solution is based on the simple postulate that every time evolution of a physic al system is ir- reversible. It is not possible to repeat an experiment in the past [59]. This empiricial fact seems to reflect a fundamental law of nature t hat rivals the law of energy conservation. The mathematical concepts corresponding to irreversible t ime evolutions are operator semigroups and abstract Cauchy problems [15, 9 3]. The follow- ing brief introduction to fractional time evolutions (sect ions 1.3.3.2–1.3.3.8) is in large parts identical to the brief exposition in [59]. For more details see [54]. 1.3.3.2Time Evolution A physical time evolution {T(∆t): 0≤∆t<∞}is defined as a one-parameter family (with time parameter ∆t) of bounded linear time evolution operators T(∆t)on a Banach space B. The parameter ∆trepresents time durations. The one-parameter family fulfills the conditions [T(∆t1)T(∆t2)f](t0) = [ T(∆t1+∆t2)f](t0) (1.140) [T(0)f](t0) = f(t0) (1.141) for all ∆t1,∆t2≥0,t0∈Rand f∈B. The elements f∈Brepresent time dependent physical observables, i.e. functions on the time axis R. Note that the argument ∆t≥0 ofT(∆t)has the meaning of a time duration, while t∈R inf(t)means a time instant. Equations (1.140) and (1.141) define a s emigroup. 1.3 PhysicalIntroductiontoFractionalDerivatives 35 The inverse elements T(−∆t)are absent. This reflects the fundamental differ- ence between past and future. The linear operator A defined as Af=s-lim ∆t→0+T(∆t)f−f ∆t(1.142) with domain D(A) =/braceleftbigg f∈B: s-lim ∆t→0+T(∆t)f−f ∆texists/bracerightbigg (1.143) is called the infinitesimal generator of the semigroup. Here s-lim f=gis the strong limit and means lim /bardblf−g/bardbl=0 in the norm of Bas usual. 1.3.3.3Continuity Physical time evolution is continuous. This requirement is represented math- ematically by the assumption that s-lim ∆t→0T(∆t)f=f (1.144) holds for all f∈B, where s-lim is again the strong limit. Semigroups of operators satisfying this condition are called strongly continuous or C0- semigroups [15, 93]. Strong continuity is weaker than unifo rm continuity and has become recognized as an important continuity concept th at covers most applications [2]. 1.3.3.4Homogeneity Homogeneity of time means two different requirements: Firs tly, it requires that observations are independent of a particular instant o r position in time. Secondly, it requires arbitrary divisibility of time durations and self- consistency for the transition between time scales. Independence of physical processes from their position on t he time axis re- quires that physical experiments are reproducible if they a receteris paribus shifted in time. The first requirement, that the start of an ex periment can be shifted, is expressed mathematically as the requirement of invariance under time translations. As a consequence one demands commutativ ity of the time evolution with time translations in the form [T(τ)T(∆t)f](t0) = [ T(∆t)T(τ)f](t0) = [ T(∆t)f](t0−τ) (1.145) 361 ThreefoldIntroductiontoFractionalDerivatives for all ∆t≥0 und t0,τ∈R. Here the translation operator T(t)is defined by T(τ)f(t0) = f(t0−τ). (1.146) Note that τ∈Ris a time shift, not a duration. It can also be negative. Physi cal experiments in the past have the same outcome as in the presen t or in the future. Outcomes of past experiments can be studied in the pr esent with the help of documents (e.g. a video recording), irrespective of the fact that the experiment cannot be repeated in the past. The second requirement of homogeneity is homogeneous divis ibility. The semigroup property (1.140) implies that for ∆t>0 T(∆t)...T(∆t) =[T(∆t)]n=T(n∆t) (1.147) holds. Homogeneous divisibility of a physical time evoluti on requires that there exist rescaling factors Dnfor∆tsuch that with ∆t=∆t/Dnthe limit lim n→∞T(n∆t/Dn) =T(∆t) (1.148) exists und defines a time evolution T(∆t). The limit n→∞corresponds to two simultaneous limits n→∞,∆t→0, and it corresponds to the passage from a microscopic time scale ∆tto a macroscopic time scale ∆t. 1.3.3.5Causality Causality of the physical time evolution requires that the v alues of the image function g(t) = ( T(∆t)f)(t)depend only upon values f(s)of the original function with time instants s<t. 1.3.3.6Fractional Time Evolution The requirement (1.145) of homogeneity implies that the ope rators T(∆t)are convolution operators [114,128]. Let Tbe a bounded linear operator on L1(R) that commutes with time translations, i.e. that fulfills eq. (1.145). Then there exists a finite Borel measure µsuch that (T f)(s) = ( µ∗f)(s) =/integraldisplay f(s−x)µ(dx) (1.149) holds [128], [114, p.26]. Applying this theorem to physical time evolution op- erators T(∆t)yields a convolution semigroup µ∆tof measures T(∆t)f(t) = (µ∆t∗f)(t) µ∆t1∗µ∆t2=µ∆t1+∆t2(1.150) 1.3 PhysicalIntroductiontoFractionalDerivatives 37 with ∆t1,∆t2≥0. For ∆t=0 the measure µ0is the Dirac-measure concen- trated at 0. The requirement of causality implies that the support supp µ∆t⊂R+= [0,∞)of the semigroup is contained in the positive half axis. The convolution semigroups with support in the positive hal f axis[0,∞)can be characterized completely by Bernstein functions [10]. A n arbitrarily often differentiable function b:(0,∞)→Rwith continuous extension to [0,∞)is called Bernstein function if for all x∈(0,∞) b(x)≥0 (1.151) (−1)ndnb(x) dxn≤0 (1.152) holds for all n∈N. Bernstein functions are positive, monotonously increasi ng and concave. The characterization is given by the following theorem [10, p.68]. There exists a one-to-one mapping between the convolution semigr oups{µt:t≥0} with support on [0,∞)and the set of Bernstein functions b:(0,∞)→R[10]. This mapping is given by /integraldisplay∞ 0e−uxµ∆t(dx) =e−∆tb(u)(1.153) with ∆t>0 and u>0. The requirement of homogeneous divisibility further restr icts the set of ad- missible Bernstein functions. It leaves only those measure sµthat can appear as limits lim n→∞,∆t→0µ∆t∗...∗µ∆t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright nfactors=lim n→∞µn∆t/Dn=µ∆t(1.154) Such limit measures µexist if and only if b(x) = xαwith 0 <α≤1 and Dn∼n1/αholds [11, 32, 54]. The remaining measures define the class of fractional time ev olutions Tα(∆t)that depend only on one parameter, the fractional order α. These remaining fractional measures have a density and they can be written as [43, 47–49, 54] Tα(∆t)f(t0) =∞/integraldisplay 0f(t0−s)hα/parenleftBigs ∆t/parenrightBigds ∆t(1.155) 381 ThreefoldIntroductiontoFractionalDerivatives where ∆t≥0 and 0 <α≤1. The density functions hα(x)are the one- sided stable probability densities [43,47–49,54]. They ha ve a Mellin transform [45, 103, 131] M{hα(x)}(s) =∞/integraldisplay 0xs−1hα(x)dx=1 αΓ((1−s)/α) Γ(1−s)(1.156) allowing to identify hα(x) =1 αxH10 11/parenleftBigg 1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(0, 1) (0, 1/ α)/parenrightBigg (1.157) in terms of H-functions [30, 45, 95, 103]. 1.3.3.7Infinitesimal Generator The infinitesimal generators of the fractional semigroups Tα(∆t) Aαf(t) =−(Mα +f)(t) =−1 Γ(−α)/integraldisplay∞ 0f(t−s)−f(t) sα+1ds (1.158) are fractional time derivatives of Marchaud-Hadamard type [51,98]. This fun- damental and general result provides the basis for generali zing physical equa- tions of motion by replacing the integer order time derivati ve with a fractional time derivative as the generator of time evolution [43, 54]. For α=1 one finds h1(x) = δ(x−1)from eq. (1.158), and the frac- tional semigroup Tα=1(∆t)reduces to the conventional translation semigroup T1(∆t)f(t0) = f(t0−∆t). The special case α=1 occurs more frequently in the limit (1.154) than the cases α<1 in the sense that it has a larger domain of at- traction. The fact that the semigroup T1(∆t)can often be extended to a group on all of Rprovides an explanation for the seemingly fundamental reve rsibil- ity of mechanical laws and equations. This solves the "rever sed irreversibility problem". 1.3.3.8Remarks Homogeneous divisibility formalizes the fact that a verbal statement in the present tense presupposes always a certain time scale for th e duration of an instant. In this sense the present should not be thought of as a point, but as a short time interval [48, 54, 59]. 1.3 PhysicalIntroductiontoFractionalDerivatives 39 Fractional time evolutions seem to be related to the subject ive human ex- perience of time. In physics the time duration is measured by comparison with a periodic reference (clock) process. Contrary to this , the subjective hu- man experience of time amounts to the comparison with an hour glass, i.e. with a nonperiodic reference. It seems that a time duration i s experienced as “long” if it is comparable to the time interval that has pas sed since birth. This phenomenon seems to be reflected in fractional stationa ry states defined as solutions of the stationarity condition Tα(∆t)f(t) = f(t). Fractional sta- tionarity requires a generalization of concepts such as “st ationarity” or “equi- librium”. This outlook could be of interest for nonequilibr ium and biological systems [43, 47–49, 54]. Finally, also the special case α→0 challenges philosophical remarks [59]. In the limit α→0 the time evolution operator degenerates into the identity . This could be expressed verbally by saying that for α=0 “becoming” and “being” coincide. In this sense the paradoxical limit α→0 is reminiscent of the eternity concept known from philosophy. 1.3.4 Identification of αfrom Models Consider now the second basic question of Section 1.3.1: How can the frac- tional order αbe observed in experiment or identified from concrete models . To the best knowledge of this author there exist two examples where this is possible. Both are related to diffusion processes. There do es not seem to exist an example of a rigorous identification of αfrom Hamiltonian models, although it has been suggested that such a relation might exi st (see [129]). 1.3.4.1Bochner-Levy Fractional Diffusion The term fractional diffusion can refer either to diffusion with a fractional Laplace operator or to diffusion equations with a fractiona l time derivative. Fractional diffusion (or Fokker-Planck) equations with a f ractional Laplacian may be called Bochner-Levy diffusion . The identification of the fractional or- derαin Bochner-Levy diffusion equations has been known for more than five decades [13, 14, 26]. For a lucid account see also [27]. The fr actional order α in this case is the index of the underlying stable process [13 , 27]. With few ex- ceptions [77] these developments in the nation of mathemati cs did, for many years, not find much attention or application in the nation of physics although eminent mathematical physicists such as Mark Kac were thoro ughly famil- 401 ThreefoldIntroductiontoFractionalDerivatives iar with Bochner-Levy diffusion [65]9. A possible reason might be the unre- solved problem of locality discussed above. Bochner himsel f writes “Whether this (equation) might have physical interpretation, is not known to us” [13, p.370]. 1.3.4.2Montroll-Weiss Fractional Diffusion Diffusion equations with a fractional time derivative will be called Montroll- Weiss diffusion although fractional time derivatives do not appear in the original paper [87] and the connection was not discovered un til 30 years later [46, 60]. As shown in Section 1.3.3, the locality probl em does not arise. Montroll-Weiss diffusion is expected to be consistent with all fundamental laws of physics. The fact that the relation between Montroll -Weiss theory and fractional time derivatives was first established in [46, 60 ] seems to be widely unknown at present, perhaps because this fact is never menti oned in widely read reviews [82] and popular introductions to the subject [ 112]10. There exist several versions of diffusion equations with fr actional time derivatives, and they differ physically or mathematically from each other [54, 82, 104, 127, 130]. Of interest here will be the fraction al diffusion equa- tion for f:Rd×R+→R Dα,1 0+f(r,t) =C∆f(r,t) (1.159) with a fractional time derivative of order αand type 1. The Laplace operator is ∆and the fractional diffusion constant is C. The function f(r,t)is assumed to obey the initial condition f(r, 0+) = f0δ(r). Equation (1.159) was introduced in integral form in [104], but the connection with [87] was no t given. An alternative to eq. (1.159), introduced in [53, 54], is Dα,0 0+f(r,t) =C∆f(r,t) (1.160) with a Riemann-Liouville fractional time derivative Dα 0+of type 0. This equa- tion does not describe diffusion of Montroll-Weiss type [53 ]. It has therefore been called “inconsistent” in [81, p.3566]. As emphasized i n [53] the choice of Dα 0+in (1.159) is physically and mathematically consistent, bu t corresponds to a modified initial condition, namely I1−α 0+f(r, 0+) = f0δ(r). Similarly, frac- tional diffusion equations with time derivative Dα,β 0+of order αand type βhave been investigated in [54]. For α=1 they all reduce to the diffusion equation. 9)Also, Herrmann Weyl, who pioneered fractional as well as fun c- tional calculus and worked on the foundations of physics, se ems not to have applied fractional derivatives to problems in physi cs. 10)Note that, contrary to [112, p.51], fractional derivatives are never mentioned in [6]. 1.3 PhysicalIntroductiontoFractionalDerivatives 41 Before discussing how αarises from an underlying continuous time random walk it is of interest to give an overall comparison of ordina ry diffusion with α=1 and fractional diffusion of the form (1.159) with α/\e}atio\slash=1. This is con- veniently done using the following table published in [46]. The first column gives the results for α=1, the second for 0 <α<1 and the third for the limit α→0. The first row compares the infinitesimal generators of time evolution Aα. The second row gives the fundamental solution f(k,u)in Fourier-Laplace space. The third row gives f(k,t)and the fourth f(r,t). In the fifth and sixth row the asymptotic behaviour is collected for r2/tα→0 and r2/tα→∞. α=1 0 <α<1 α→0 Aαd dt/tildewideDα 0+ →1 f(k,u)f0 u+Ck2f0uα−1 uα+Ck2→f0 u(1+Ck2) f(k,t) f0e−Ctk2f0Eα/parenleftBig −Ctk2/parenrightBig →f0 1+Ck2 f(r,t)f0e−r2/4Ct (4πCt)−d/2f0 (r2π)d/2Hd α/parenleftbiggr2 4Ctα/parenrightbiggf0|r|1−d 2/radicalbig C(2π)dKd−2 d/parenleftbigg|r|√ C/parenrightbigg r2 tα→0 t−d/2 |r|2−d tα|r|(2/d)−(d/2) r2 tα→∞ exp/bracketleftbigg −r2 4Ct/bracketrightbigg exp −cα/parenleftbiggr2 4Ctα/parenrightbigg 1 2−α exp/parenleftbigg −|r|√ C/parenrightbigg In the table E α,β(x)denotes the generalized Mittag-Leffler function from eq. (1.130), Kν(x)is the modified Bessel function [1], d>2,cα= (2−α)αα/(2−α) and the shorthand Hd α(x)=H20 12/parenleftBigg x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1,α) (d/2, 1),(1, 1)/parenrightBigg (1.161) was used for the H-function H20 12. For information on H-functions see [30, 54, 79, 95]. 421 ThreefoldIntroductiontoFractionalDerivatives The results in the table show that the normal diffusion ( α=1) is slowed down for 0 <α<1 and comes to a complete halt for α→0. For more discussion of the solution see [46]. 1.3.4.3Continuous Time Random Walks The fractional diffusion equation (1.159) can be related ri gorously to the mi- croscopic model of Montroll-Weiss continuous time random w alks (CTRW’s) [64,87] in the same way as ordinary diffusion is related to ra ndom walks [27]. The fractional order αcan be identified and has a physical meaning related to waiting times in the Montroll-Weiss model. The relation bet ween fractional time derivatives and CTRW’s was first exposed in [46, 60]. The relation was established in two steps. First, it was shown in [60] that Mon troll-Weiss con- tinuous time random walks with a Mittag-Leffler waiting time density are rig- orously equivalent to a fractional master equation. Then, i n [46] this under- lying random walk model was connected to the fractional diff usion equation (1.159) in the usual asymptotic sense [109] of long times and large distances11. For additional results see also [50, 53, 54, 57] The basic integral equation for separable continuous time r andom walks describes a random walker in continuous time without correl ation between its spatial and temporal behaviour. It reads [39, 64, 87, 88, 118] f(r,t) =δr,0Φ(t) +t/integraldisplay 0ψ(t−t′)∑ r′λ(r−r′)f(r′,t′)dt′(1.162) where f(r,t)denotes the probability density to find the walker at positio n r∈Rdafter time tif it started from r=0at time t=0. The function λ(r)is the probability for a displacement by rin each step, and ψ(t)gives the probability density of waiting time intervals between steps. The transi tion probabilities obey ∑rλ(r) =1, and Φ(t) =1−/integraltextt 0ψ(t′)dt′is the survival probability at the initial site. The fractional master equation introduced in [60] with init al condition f(r, 0) =δr,0reads Dα,1 0+f(r,t) =∑ r′w(r−r′)f(r′,t) (1.163) with fractional transition rates w(r)obeying ∑rw(r) =0. Note, that eq. (1.162) contains a free function ψ(t)that has no counterpart in eq. (1.163). The rig- 11)This is emphasized in eqs. (1.8) and (2.1) in [46] that are, of course, asymptotic. 1.3 PhysicalIntroductiontoFractionalDerivatives 43 orous relation between eq. (1.162) and eq. (1.163), first est ablished in [60], is given by the relation λ(k) =1+ταw(k) (1.164) for the Fourier transformed transition rates w(r)and probabilities λ(r), and the choice ψ(t) =tα−1 ταEα,α/parenleftbigg −/parenleftbiggt τ/parenrightbiggα/parenrightbigg (1.165) for the waiting time density, where τ>0 is a characteristic time constant. With E α,α(0) =1 it follows that ψ(t)∼tα−1(1.166) fort→0. From E α,α(x)∼x−2forx→∞one finds ψ(t)∼t−α−1(1.167) fort→∞. For α=1 the waiting time density becomes the exponential distribution, and for α→0 it approaches 1/ t. It had been observed already in the early 1970’s that continu ous time ran- dom walks are equivalent to generalized master equations [9 , 66]. Similarly, the Fourier-Laplace formula f(k,u) =uα−1/(uα+Ck2) (1.168) for the solution of CTRW’s with algbraic tails of the form (1. 167) was well known (see [117, eq.(21),p.402] [110, eq.(23),p.505] [67, eq.(29),p.3083]). Com- parison with row 2 of the table makes the connection between t he fractional diffusion equation (1.159) and the CTRW-equation (1.162) e vident. However, this connection with fractional calculus was not made befor e the appearance of [46, 60]. In particular, there is no mention of fractional derivatives or frac- tional calculus in [6]. 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Ein Kalkül für gebroch- ene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren. Teil II: Gruppenerzeuger. Compositio Math. , 22:104, 1970. 122 U. Westphal. An approach to fractional powers of operators via fractional differ- ences. Proc. London Math. Soc. , 29:557, 1974. 123 U. Westphal. Fractional powers of in- finitesimal generators of semigroups. In R. Hilfer, editor, Applications of Fractional Calculus in Physics , page 131, Singapore, 2000. World Scientific. 124 H. Weyl. Bemerkungen zum Begriff des Differentialquotienten gebrochener Ord- nung. Vierteljahresschrift der Naturforsch. Ges. Zürich , 62:296, 1917. 125 A. Wiman. Über den Fundamentalsatz in der Theorie der Funktionen Eα(x).Acta Math. , 29:191, 1905. 126 A. Wiman. Über die Nullstellen der Funktionen Eα(x).Acta Math. , 29:217, 1905. 127 W. Wyss. The fractional diffusion equa- tion. J. Math. Phys. , 27:2782, 1986. 128 K. Yosida. Functional Analysis . Springer, Berlin, 1965. 129 G. Zaslavsky. Fractional kinetics of hamil- tonian chaotic systems. In R. Hilfer, ed- itor, Applications of Fractional Calculus in Physics , page 202, Singapore, 2000. World Scientific. 50References 130 G. Zaslavsky. Chaos, fractional kinet- ics, and anomalous transport. Phys.Rep. , 371:461, 2002. 131 V .M. Zolotarev. Mellin-Stieltjes trans- forms in probability theory. Theor. Prob. Appl. , II:433, 1957.132 A. Zygmund. Trigonometric Series , vol- ume I. Cambridge University Press, Cambridge, 1959. 133 A. Zygmund. Trigonometric Series , vol- ume II. Cambridge University Press, Cambridge, 1959. to be published in: R. Klages et al. (eds.), Anomalous Transport: Foundations and Applications , Wiley-VCH, Weinheim, Copyright © 2007 R. Hilfer, Stuttgart51 A Tables Letα∈C,x>a f(x) ( Iα a+f)(x) f(λx) λ−α(Iα λa+f)(λx),λ>0 (A.1) (x−a)β Γ(β+1) Γ(α+β+1)(x−a)α+β(A.2) Reβ>0 eλxeλa(x−a)αE1,α+1(λ(x−a)) (A.3) λ∈R (x−a)β−1eλx Γ(β)eλa Γ(α+β)(x−a)α+β−11F1(β;α+β;λ(x−a)) Reβ>0 (A.4) (x−a)β−1log(x−a)Γ(β)(x−a)α+β−1 Γ(α+β)[ψ(β−ψ(α+β) +log(x−a)] (A.5) (x−a)β−1Eγ,β((x−a)γ) ( x−a)α+β−1Eγ,α+β((x−a)γ) (A.6) Reβ>0, Re γ>0 to be published in: R. Klages et al. (eds.), Anomalous Transport: Foundations and Applications , Wiley-VCH, Weinheim, Copyright © 2007 R. Hilfer, Stuttgart53 B FunctionSpaces The set Gdenotes an interval, a domain in Rdor a measure space (G,A,µ)[8] depending on the context. Kstands for RorC.γ= (γ1, ...,γd)∈Nd 0is a multiindex and |γ|=∑d i=1γi. For the definition of Hilbert and Banach spaces the reader may consult e.g. [128]. The following notation is used for various spaces of continuous functions: C0(G):={f:G→K|fis continuous } (B.1) Ck(G):={f∈C0(G)|fisk-times continuously differentiable } (B.2) Ck 0(G):={f∈Ck(G)|fvanishes at the boundary ∂G} (B.3) Ck b(G):={f∈Ck(G)|fis bounded } (B.4) Ck c(G):={f∈Ck(G)|fhas compact support } (B.5) Ck ub(G):={f∈Ck(G)|fis bounded and uniformly continuous } (B.6) ACk([a,b]):={f∈Ck([a,b])|f(k)is absolutely continuous } (B.7) For compact Gthe norm on these spaces is /bardblf/bardbl∞:=sup x∈G|f(x)|. (B.8) The Lebesgue spaces over (G,A,µ)are defined as Lp loc(G,µ):={f:G→K|fpis integrable on every compact K⊂G}(B.9) Lp(G,µ):={f:G→K|fpis integrable } (B.10) with norm /bardblf/bardblp:= /integraldisplay G|f(s)|pdµ(s) 1/p . (B.11) 54B FunctionSpaces Forp=∞ L∞(G,µ):={f:G→K|fis measurable and /bardblf/bardbl∞<∞} (B.12) where /bardblf/bardbl∞:=sup{|z|:z∈fess(G)} (B.13) and fess(G):={z∈C:µ({x∈G:|f(x)−z|<ε})/\e}atio\slash=0 for all ε>0} (B.14) is the essential range of f . The Hölder spaces Cα(G)with 0 <α<1 are defined as Cα(G):={f:G→K|∃c≥0 s.t.|f(x)−f(y)| ≤c|x−y|α,∀x,y∈G}(B.15) with norm /bardblf/bardblα:=/bardblf/bardbl∞+cα (B.16) where cαis the smallest constant cin (B.15). For α>1 the Hölder space Cα(G) contains only the constant functions and therefore αis chosen as 0 <α<1. The spaces Ck,α(G),k∈N, consist of those functions f∈Ck(G)whose partial derivatives of order kall belong to Cα(G). The Sobolev spaces are defined by Wk,p(G) =/braceleftBigg f∈Lp(G):fisk-times differentiable in the sense of distributions and Dγf∈ Lp(G)for all γ∈Nd 0with|γ| ≤k/bracerightBigg (B.17) where the derivative Dγ=∂γ1 1...∂γd dwith multiindex γ= (γ1, ...,γd)∈Nd 0is understood in the sense of distributions. A distribution fis in Wk,p(G)if and only if for each γ∈Nd 0with|γ| ≤kthere exists fγ∈Lp(G)such that /integraldisplay Gφfγdx= (−1)|γ|/integraldisplay G(Dγφ)fdx (B.18) for all test functions φ. In the special case d=1 one has f∈Wk,p(G)if and only if f∈Ck−1(G),f(k−1)∈AC(G), and f(j)∈Lp(G)forj=0, 1, ..., k. The Sobolev spaces are equipped with the norm /bardblf/bardblWk,p(G)=∑ |γ|≤m/bardblDγf/bardblp (B.19) 55 (see [2]). A function is called rapidly decreasing if it is infinitely many times differentiable, i.e. f∈C∞(Rd)and lim |x|→∞|x|nDγf(x) =0 (B.20) for all n∈Nand γ∈Nd. The test function space S(Rd):={f∈C∞(Rd)|fis rapidly decreasing } (B.21) is called Schwartz space . to be published in: R. Klages et al. (eds.), Anomalous Transport: Foundations and Applications , Wiley-VCH, Weinheim, Copyright © 2007 R. Hilfer, Stuttgart57 C Distributions Distributions are generalized functions [31]. They were in vented to overcome the differentiability requirements for functions in analy sis and mathematical physics [63, 105]. Distribution theory has also a physical o rigin. A physical observable fcan never be measured at a point x∈Rdbecause every measure- ment apparatus averages over a small volume around x[115]. This “smearing out” can be modelled as an integration with smooth “test func tions” having compact support. LetXdenote the space of admissible test functions. Commonly use d test function spaces are C∞(Rd), the space of infinitely often differentiable func- tions, C∞c(Rd), the space of smooth functions with compact support (see (B. 5)), C∞ 0(Rd), the space of smooth functions vanishing at infinity (see (B. 3)), or the so called Schwartz space S(Rd)of smooth functions decreasing rapidly at in- finity (see (B.21)). A distribution F:X→Kis a linear and continuous mapping that maps ϕ∈Xto a real ( K=R) or complex ( K=C) number1. There exists a canonical correspondence between functions and distribu tions. More pre- cisely, for every locally integrable function f∈L1 loc(Rd)there exists a distri- bution Ff=/a\}bracketle{tf, ./a\}bracketri}ht(often also denoted with the same symbol f) defined by Ff(ϕ) =/a\}bracketle{tf,ϕ/a\}bracketri}ht=/integraldisplay Rdf(x)ϕ(x)dx (C.1) for every test function ϕ∈X. Distributions that can be written in this way are called regular distributions . Distributions that are not regular are sometimes called singular . The mapping f→ /a\}bracketle{t f, ./a\}bracketri}htthat assigns to a locally integrable fits associated distribution is injective and continuous. T he set of distribu- tions is again a vector space, namely the dual space of the vec tor space of test functions, and it is denoted as X′where Xis the test function space. 1)For vector valued distributions see [106] 58C Distributions Important examples for singular distributions are the Dira cδ-function and its derivatives. They are defined by the rules /integraldisplay δ(x)ϕ(x)dx=ϕ(0) (C.2) /integraldisplay δ(n)(x)ϕ(x) = (−1)ndnϕ dxn/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=0(C.3) for every test function ϕ∈Xand n∈N. Clearly, δ(x)is not a function, because if it were a function, then/integraltext δ(x)ϕ(x)dx=0 would have to hold. Another example for a singular distribution is the finite par t or principal value P{1/x}of 1/ x. It is defined by /angbracketleftbigg P/braceleftbigg1 x/bracerightbigg ,ϕ/angbracketrightbigg =lim ε→0+/integraldisplay |x|≥εϕ(x) xdx (C.4) forϕ∈C∞ c(R). It is a singular distribution on R, but regular on R\ {0}where it coincides with the function 1/ x. Equation (C.2) illustrates how distributions circumvent t he limitations of differentiation for ordinary functions. The basic idea is t he formula for partial integration /integraldisplay G∂if(x)ϕ(x)dx=−/integraldisplay Gf(x)∂iϕ(x)dx (C.5) valid for f∈C1 c(G),ϕ∈C1(G),i=1, ..., dand G⊂Rdan open set. The formula is proved by extending fϕas 0 to all of Rdand using Leibniz’ product rule. Rewriting the formula as /a\}bracketle{t∂if,ϕ/a\}bracketri}ht=−/a\}bracketle{tf,∂iϕ/a\}bracketri}ht (C.6) suggests to view ∂ifagain as a linear continuous mapping (integral) on a space Xof test functions ϕ∈X. Then the formula is a rule for differenti- ating fgiven that ϕis differentiable. Distributions on the test function space S(Rd)are called tempered distribu- tions . The space of tempered distributions is the dual space S(Rd)′. Tempered distributions generalize locally integrable functions gr owing at most polyno- mially for |x| → ∞. All distributions with compact support are tempered. Square integrable functions are tempered distributions. T he derivative of a tempered distribution is again a tempered distribution. S(Rd)is dense in Lp(Rd)for all 1 ≤p<∞but not in L∞(Rd). The Fourier transform and its inverse are continous maps of the Schwartz space onto itself . A distribution 59 fbelongs to S(Rd)′if and only if it is the derivative of a continuous func- tion with slow growth, i.e. it is of the form f=Dγ[(1+|x|2)k/2g(x)]where k∈N,γ∈Ndand gis a bounded continuous function on Rd. Note that the exponential function is not a tempered distribution. A distribution f∈ S(Rd)′is said to have compact support if there exists a compact subset K⊂Rdsuch that /a\}bracketle{tf,ϕ/a\}bracketri}ht=0 for all test functions with supp ϕ∩K=∅. The Dirac δ-function is an example. Other examples are Radon measures on a compact set K. They can be described as linear func- tionals on C0(K). If the set Kis sufficiently regular (e.g. if it is the closure of a region with piecewise smooth boundary) then every distrib ution with com- pact support in Kcan be written in the form f=∑ |γ|≤NDγfγ (C.7) where γ= (γ1, ...,γd),γj≥0 is a multiindex, |γ|=∑γiand fγare continu- ous functions of compact support. Here N≥0 and the partial derivatives in Dγare distributional derivatives defined above. A special cas e are distribu- tions with support in a single point taken as {0}. Any such distributions can be written in the form f=∑ |γ|≤NcγDγδ (C.8) where δis the Dirac δ-function and cγare constants. The multiplication of a distribution fwith a smooth function gis defined by the formula /a\}bracketle{tg f,ϕ/a\}bracketri}ht=/a\}bracketle{tf,gϕ/a\}bracketri}htwhere g∈C∞(G). A combination of multi- plication by a smooth function and differentiation allows t o define differential operators A=∑ |γ|≤maγ(x)Dγ(C.9) with smooth aγ(x)∈C∞(G). They are well defined for all distributions in C∞ c(G)′. A distribution is called homogeneous of degree α∈Cif f(λx) =λαf(x) (C.10) for all λ>0. Here λα=exp(αlogλ)is the standard definition. The Dirac δ- distribution is homogeneous of degree −d. For regular distributions the defi- nition coincides with homogeneity of functions f∈L1 loc(Rd). The convolution 60C Distributions kernels Kα±from eq. (1.39) are homogeneous of degree α−1. Homogeneous distributions remain homogeneous under differentiation. A homogeneous lo- cally integrable function gonRd\ {0}of degree αcan be extended to homo- geneous distributions fon all of Rd. The degree of homogeneity of fmust again be α. As long as α/\e}atio\slash=−d,−d−1,−d−2, ... the integral /a\}bracketle{tgβ,ϕ/a\}bracketri}ht=/integraldisplay g/parenleftbiggx |x|/parenrightbigg |x|βddx (C.11) which converges absolutely for Re β>−dcan be used to define f=gαby analytic continuation from the region Re β>−dto the point α. For α= −d,−d−1, ..., however, this is not always possible. An example is th e function 1/|x|onR\ {0}. It cannot be extended to a homogeneous distribution of degree −1 on all of R. For f∈L1 loc(G1)and g∈L1 loc(G2)their tensor product is the function (f⊗ g)(x,y) = f(x)g(y)defined on G1×G2. The function f⊗ggives a functional /a\}bracketle{tf⊗g,ϕ(x,y)/a\}bracketri}ht=/a\}bracketle{tf(x),/a\}bracketle{tg(y),ϕ(x,y)/a\}bracketri}ht/a\}bracketri}ht (C.12) forϕ∈C∞ c(G1×G2). For two distributions this formula defines the their tensor product. An example is a measure µ(x)⊗δ(y)concentrated on the surface y=0 in G1⊗G2where µ(x)is a measure on G1. The convolution of distribution defined in the main text (see eq. (1.52) can then be defined by the formula /a\}bracketle{tf∗g,ϕ/a\}bracketri}ht=/a\}bracketle{t(f⊗g)(x,y),ϕ(x+y)/a\}bracketri}ht (C.13) whenever one of the distributions forghas compact support. to be published in: R. Klages et al. (eds.), Anomalous Transport: Foundations and Applications , Wiley-VCH, Weinheim, Copyright © 2007 R. Hilfer, Stuttgart61 Index H-function, 38, 41 p-integrable function, 53 additivity, 16 adjoint operators, 15 analysis of singularities, 26 anomalous subdiffusion, 31 Banach space, 24, 28, 34, 53 Bernstein function, 37 Bessel function, 41 binomial formula, 1, 23 Bochner-Levy diffusion, 27, 28, 39 Borel measure, 36 Cauchy problem, 34 causality, 36 classification of phase transitions, 26 conjugate Riesz potential, 11 continuity in time, 35 continuous time random walk, 42 convolution – distributions, 13 – functions, 10 – integral, 7 – kernel, 11, 13, 22, 60 – operators, 11, 13, 29, 36 – periodic functions, 9 – product, 10, 13 – semigroup, 36 critical phenomena, 26 CTRW, 42, 43 difference quotient, 5, 6, 22, 23 diffusion, 27, 31–33, 39–43 – anomalous, 31 – fractional, 27, 32, 33, 39–43 Dirac δfunction, 11, 13, 58, 59 Dirac-measure, 37 distribution(s), 57 – convolution, 13, 60 – differentiation, 58– Dirac δ, 13, 58 – direct product, 60 – homogeneous, 59 – regular, 13, 57 – singular, 13, 57 – tempered, 58 – with compact support, 59 Ehrenfest classification, 26 eigenfunction(s) – fractional derivatives, 29 ergodicity breaking, 34 essential range, 54 Euler, 2, 4 Euler Beta function, 16 exponential series, 3, 5 Fokker-Planck equation, 39 Fourier, 5 Fourier series, 9 Fourier transformation, 14, 22 fractional derivative – at lower limit, 26 – definition – – generalized Riemann-Liouville, 19 – – Grünwald-Letnikov, 23 – – lower limit, 16 – – Marchaud-Hadamard, 19 – – of distributions, 24 – – Riemann-Liouville, 16 – – Riesz, 22 – – upper limit, 16 – – Weyl-Liouville, 21 – – Weyl-Marchaud, 21 – eigenfunction, 29 – generalized Riemann-Liouville, 19 – Grünwald-Letnikov, 23 – history, 1 – – Euler, 2 – – Fourier, 5 – – Grünwald, 6 – – Leibniz, 2 62Index – – Liouville, 2, 4 – – paradoxa, 2 – – Riemann, 6 – interpolation, 19 – Liouville(-Caputo), 19 – local, 26 – Marchaud-Hadamard, 19, 25, 38 – notation, 17 – of distributions, 24 – order, 16, 19, 21–24 – Riemann-Liouville, 16, 19 – Riesz, 22 – symbols, 17 – type, 19 – Weyl-Liouville, 21 – Weyl-Marchaud, 21 fractional difference quotient, 23 fractional differential equation, 29, 31 fractional diffusion, 39, 40, 42 – Bochner-Levy, 27, 28, 39 – Montroll-Weiss, 40 fractional integral, 7 – by parts, 15 – common symbols, 8 – composition law, 16 – definition – – lower limit, 8 – – of distributions, 13 – – Riemann-Liouville, 8 – – Riesz, 11 – – Riesz-Feller, 12 – – upper limit, 8 – – Weyl, 10 – domain of definition, 8 – interpolation, 12 – inverse, 18 – notation, 8 – of distributions, 12 – Riemann, 6 – Riemann-Liouville, 8 – Riesz, 11, 12 – Riesz-Feller, 12 – Weyl, 9, 10 fractional integration – by parts, 15 fractional master equation, 42 fractional powers, 27 fractional stationarity, 39 fractional time, 33 fractional time evolution, 36 function(s) –p-integrable, 53 – Bernstein, 37 – Bessel, 41 – Beta, 16 – continuous differentiable, 53– essential range, 54 – Gamma, 2, 8 – generalized, 57 – Hölder continuous, 54 – locally integrable, 7, 53 – Mittag-Leffler, 29 – periodic, 9, 10, 21 – regularly varying, 26 – slowly varying, 27 – smooth, 53 Gamma function, 2, 8 Grünwald, 5 Grünwald-Letnikov derivative, 22 Hardy-Littlewood theorem, 15, 16 Heaviside step function, 11 Hilbert space, 27, 53 homogeneity of time, 35 homogenous divisibilty, 36–38 Hölder space, 54 Hörmander symbol class, 28 identity operator, 20, 23 infinitesimal generator, 28, 38 integral transforms, 14 integration – by parts, 15 – fractional, 8, 9, 11, 12 – – Riemann-Liouville, 8 – – Riesz, 11 – – Weyl, 9 – iterated, 7 integroderivatives, 5 irreversibility problem – normal, 34 – reversed, 34 iterated integrals, 7 Kohn-Nirenberg pseudodifferential opera- tor, 28 Laplace transformation, 14 Laplacian, 27, 39 Lebesgue space, 53 Leibniz, 1, 2 Leibniz’ paradox, 2 Leibniz’ product rule, 1 Liouville, 4 Lizorkin space, 14 local fractional derivative, 26 locality, 33, 40 locally integrable function, 53 Marchaud fractional derivative, 19 master equation, 42 Index63 – fractional, 42 Mellin transformation, 38 Mittag-Leffler function, 29 Montroll-Weiss diffusion, 40 nonlocality, 32–34 – space, 32 – time, 33 operator(s) – adjoint, 15 – closed, 28 – convolution, 11, 13, 29, 36 – fractional powers, 27 – identity, 20, 23 – Kohn-Nirenberg, 28 – Kohn-Nirenberg symbol, 28 – Laplace, 27, 39 – pseudodifferential, 28 – semigroup of, 34 – spectral decomposition, 27 – symbol, 28 – translation, 20, 23 principal value, 58 pseudodifferential operators, 28 random walk – continuous time, 42 rapidly decreasing function, 55 regularly varying function, 26 Riemann, 6 Riemann-Liouville derivatives, 16 Riemann-Liouville fractional integrals, 8 Riesz fractional derivatives, 22 Riesz fractional integrals, 11 Riesz kernel, 12 Riesz potential, 11 – conjugate, 11 Riesz-Feller kernel, 12Schwartz space, 55, 58 semigroup, 16, 27, 28, 34–38 – generator, 35 – of convolutions, 36 singularities, 26 slowly varying function, 27 Sobolev space, 54 space – Banach, 24, 28, 34, 53 – continuous functions, 53 – Hilbert, 27, 53 – Hölder, 54 – Lebesgue, 53 – Lizorkin, 14 – Schwartz, 55 – Sobolev, 54 – test functions, 57 stationarity, 34, 39 subdiffusion, 31 test functions, 57 time – continuity, 35 – evolution, 34 – fractional, 33 – homogeneity, 35 – homogenous divisibilty, 36–38 – infinitesimal generator, 38 – translation, 33, 35, 36 time evolution, 34 translation invariance, 33, 35, 36 translation operator, 20, 23 types of fractional derivative, 19 unit step function, 11 Weyl fractional integrals, 9 Youngs inequality, 15