R. Hilfer Mathematical and Physical Interpretations of Fractional Derivatives and Integrals Abstract: Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpre- tation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief descrip- tion of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point. Keywords: fractional derivatives and integrals, Riemann-Liouville integrals, Weyl integrals, Riesz potentials, operational calculus, functional calculus, Mikusinski calculus, Hille-Phillips calculus, Riesz-Dunford calculus, classification, phase transitions, time evolution, anomalous diffusion, continuous time random walks Mathematics Subject Classification 2010: 26A33, 34A08, 35R11 published in: Handbook of Fractional Calculus: Basic Theory, Vol. 1, Ch. 3, pp. 47-86, de Gruyter, Berlin (2019) ISBN 978-3-11-057162-2 R. Hilfer, ICP, Fakultät für Mathematik und Physik, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Deutschland DOI https://doi.org/10.1515/9783110571622 Interpretations 1 1 Prolegomena 1.1 A multitude of mathematical and physical interpretations for fractional derivatives and integrals have been developed since Gottfried Wilhelm Leibniz [123, p. 301] first noted and then asked „...on le peut exprimer per seriem infinitam, sed quid est in Geometria?“1 Derivatives D훼 and integrals I 훼 of fractional (non-integer) order arise from viewing the symbolic relations 푛 푛+1 II = I , 푛 ∈ N, (1a) 푛 푛+1 DD = D , 푛 ∈ N, (1b) 푛 푛−1 DI = I , 푛 ∈ N, (1c) for iterated integrals I and derivatives D as almost representing the additive + semigroup (N, +) of natural numbers, and extending (1) to the semigroup (R0 , +) 2 + of non-negative reals R0 := [0, ∞) ⊂ R or to the full field (R, +, · ). 1.2 Many ideas for interpreting and extending the formal relations (1) from 푛 ∈ N to 푛 ∈ Q, R or C have been proposed over the centuries. All attempts are met squarely by some basic facts of calculus. Indeed, the geometric property D2 = 0 of (exterior) derivatives [53, p.20] would restrict eq. (1b) to 푛 ≤ 2. Or the non-commutativity ID ̸= DI seems to prevent extension of eq. (1c) to 푛 = 0. Recall that the power functions 푥푛 푃 (푥) := , 푥 ≥ 0, 푛 ∈ (2) 푛 푛! N 푛 obey the rules I 푃푛 = 푃푛+1 and D 푃푛 = 푃푛−1 similar to I in (1a) and (1c). Extension to 푛 = 0 (or 푛 < 0), however, is fraught with a singularity at 푥 = 0. Many mathematical impediments arise from these basic facts. 1.3 Derivatives and integrals of fractional (non-integer) order originate from the operational analogy („analogie merveilleuse“) observed by Leibniz [122] 푛 ∑︁ (︂푛)︂ D푛(푓푔) = (D푘 푓)(D푛−푘 푔) (3) 푘 푘=0 푛 ∑︁ (︂푛)︂ P푛(푓 + 푔) = (P푘 푓)(P푛−푘 푔) (4) 푘 푘=0 between 푛-th derivative of a product (3) of two real-valued functions 푓, 푔, and 푛 푛 0 the 푛-th power of their sum (4), if powers 푃푛(푓) = 푓 = P 푓 with 푓 = 1 are 1 „... one can express that“ difference (or sum) whose exponent is a fraction* „by an infinite series, but what is it in geometry?“ *inserted from context [123, p. 300-301] + + − 2 The notations R := (0, ∞), R0 := [0, ∞), and R := (−∞, 0) will be used. 2 Hilfer written using an operational symbol P. Extension of eqs. (2)–(4) from 푛 ∈ N to 훼 ∈ C, Re 훼 > 0 (or 훼 ∈ C [162, Sec. 2.3]) requires interpolation formulae (︂푛)︂ 푛! Γ(훼 + 1) from = to , 푘 ∈ , and (5a) 푘 푘!(푛 − 푘)! Γ(푘 + 1)Γ(훼 − 푘 + 1) N 푛 1 ∞ ∏︁ ∫︁ ∫︁ from 푛! = 푘 to Γ(훼 + 1) = (− log 푥)훼 d푥 = 푥훼푒−푥d푥 (5b) 푘=1 0 0 for factorials and binomial coefficients. It was for this purpose that Euler solved the ’interpolation problem’ and introduced the Γ-function in [50, §27-29]. 1.4 Given the operational analogy between (3) and (4), Lagrange obtained the celebrated symbolic formula [61, p. 194-195] 훼 훼 (︁ ℎ d푓 )︁ Δℎ 푓 = 푒 d푥 − 1 (6) for general 훼, where (Δℎ푓)(푥) = 푓(푥 + ℎ) − 푓(푥) is the finite difference with shift ℎ ≥ 0. Lagrange’s formula results from treating the symbol of differentiation systematically as an algebraic quantity with the provision of replacing its 푛-th powers (︂d푓 )︂푛 d푛푓 −−−−→ (7) d푥 d푥푛 with the 푛-th derivative at the end. Other authors have subsequently tried to build differential calculus on an algebraic or algorithmic basis [63, 7, 180, 31]. Recall that Leibniz’ product rule D(푓푔) = 푔(D 푓) + 푓(D 푔) (8) is the algebraic basis of D푛 and eq. (3) reduces to eq. (8) for 푛 = 1. Indeed eq. (8) is the axiomatic basis for differentiation (resp. derivation) on manifolds [33] and algebras [26, 119]. A full account of Leibniz’ influential idea to generalize eq. (1) is not possible in these pages. Many books and specialized treatises [125, 157, 168, 82, 193] contain a section on history (see also [134]). 1.5 The objective of this chapter is to collect a list of mathematical and physical interpretations in the sense of Definitions 1 and 2 below. The scope is necessarily rather restricted due to the long history of fractional calculus. Only a selection of established and widely known mathematical interpretations is aimed at here. As to the physical interpretations, there will be even less examples, because, in the opinion of this author, most physical interpretations are still tentative. Interpretations 3 2 Integrals and Derivatives 2.1 Integrals. Integrals are (weighted) sums with infinitely many terms or linear functionals on function spaces. More concretely, let (Ω, M , 휇) be a measure space with 휎-algebra M and measure (or weight) 휇, and let 푓 :Ω → R be a real-valued function on Ω. Then ∫︁ ∫︁ I 휇,Ω(푓) = 푓휇 = 푓(푥) d휇(푥) = 휇Ω(푓) ∈ R (9a) Ω Ω denotes the integral of 푓 with respect to 휇. When 휇 = 휆 is the Lebesgue measure and Ω ⊂ R is an interval, notations such as ∫︁ ∫︁ I Ω(푓) = 푓 = 푓(푥) d푥 = 휆Ω(푓) (9b) Ω Ω often suppress the dependence on the measure or weight 휇. 2.2 Let J = (푎, 푏) be an interval with −∞ ≤ 푎 < 푏 ≤ ∞ and Ω = (푎, 푥) an interval with 푎 < 푥 < 푏. For a locally integrable function 푓 : J → R the notation 푥 푥 ∫︁ ∫︁ (I 푎+ 푓)(푥) := I Ω(푓) = 푓 = 푓(푦)d푦 (10a) 푎 푎 푏 푏 ∫︁ ∫︁ (I 푓)(푥) := I (푓) = 푓 = 푓(푦)d푦 (10b) 푏− J∖Ω 푥 푥 is used to emphasize the dependence on the variable upper or lower limit 푥. 2.3 Derivatives. Let Ω ⊂ R be open and let F(Ω) be the algebra of real 3 valued functions 푓 :Ω → R with pointwise addition and pointwise multiplication . Algebraically, the derivative (or derivation) D 푓 is defined as a linear operator D : F → F on F such that the product rule (8) holds true for all 푓, 푔 ∈ F. On commutative Banach algebras every such D has range contained in the Jacobson radical of F, i.e. in the kernel of the Gelfand isomorphism [183, 194]. 2.4 Difference Quotients. Analytically, the derivative of 푓 at 푥 ∈ Ω is defined as the limit of finite difference quotients (ℎ ≥ 0) (︃ )︃ 푓(푥 ± ℎ) − 푓(푥) Tℎ − 1 (D 푓)(푥) := lim = lim ± 푓(푥) (11a) ℎ→0 ℎ ℎ→0 ℎ 3 Sets of functions such as F(Ω), or the standard Lebesgue spaces L푝(Ω) are denoted with a sans serif font. 4 Hilfer (︂Δ )︂ = lim ±ℎ 푓(푥) (11b) ℎ→0 ℎ if both limits exist, are well-defined, and equal. Here 1 is the identity (see eq. (100) in the appendix), ℎ Δ±ℎ = T± − 1 (12) are the right/left-sided difference operators appearing in eq. (6), and (︀ ℎ )︀ T± 푓 (푥) = 푓(푥 ± ℎ) (13) stands for translations of 푓 to the left (+) or right (-) by ℎ. 2.5 Iteration. The algebra of integer powers of D and I in eq. (1) is the starting point of fractional calculus. Let A : X → X be a linear operator on a linear space X with domain D(A) ⊂ X. For 푛 ∈ N the 푛-th power of an operator A푛 := A ∘ · · · ∘ A . (14) ⏟ ⏞ 푛-times is interpreted as its 푛-fold composition or iteration and defined recursively as A0 := 1, D(A0) = X (15a) A푛+1 푓 := A푛 ∘(A 푓), D(A푛+1) = {푓 ∈ D(A)| A 푓 ∈ D(A푛)} (15b) where 1 is the identity operator on X (see eq. (99), Appendix). Then the law of 푛 푚 푛+푚 푛+1 푛 exponents A A = A holds for 푛, 푚 ∈ N. This shows that A = AA and thus the domains are shrinking D(A푛+1) ⊂ D(A푛). Moreover A푛(D(A푚)) ⊂ D(A푚−푛) for 푚 ≥ 푛. 2.6 Iterated Integrals. Let J = (푎, 푏) ⊆ R with −∞ ≤ 푎 < 푏 ≤ ∞ as in 1 paragraph 2.2, and let 푓 ∈ Lloc(J) := {푓 : J → R | 푓 is integrable on all compact 1 subsets 퐾 ⊂ Ω} be locally integrable. Iterating I 푎+ : D(I 푎+) → Lloc(J) from eq.