2.2 Mathematical Introduction to Fractional Derivatives

[22.2.1] Consider a locally integrable¹ (This is a footnote:) ¹ A function f:G→R is called locally integrable if it is integrable on all compact subsets K⊂G (see eq.(B.9)). real valued function f:G→R whose domain of definition G=a,b⊆R is an interval with -∞≤a<b≤∞. [22.2.2] Integrating

[page 23, §0]    n times gives the fundamental formula

where a<x<b and n∈N. [23.0.1] This formula may be proved by induction. [23.0.2] It reduces n-fold integration to a single convolution integral (Faltung). [23.0.3] The subscript a+ indicates that the integration has a as its lower limit. [23.0.4] An analogous formula holds with lower limit x and upper limit a. [23.0.5] In that case the subscript a- will be used.

[23.1.1] Equation (2.27) for n-fold integration can be generalized to noninteger values of n using the relation n-1!=∏k=1n-1k=Γ⁢n where

is Euler’s Γ-function defined for all z∈C.

[page 24, §1]    [24.1.1] Formula (2.29a) appears in [96, p.363] with a>-∞ and in [76, p.8] with a=-∞. [24.1.2] The notation is not standardized. [24.1.3] Leibniz, Lagrange and Liouville used the symbol ∫α [73, 22, 76], Grünwald wrote ∫α…⁢d⁢xαx=ax=x, while Riemann used ∂x-α [96] and Most wrote da-α/d⁢x-α [89]. [24.1.4] The notation in (2.29) is that of [99, 98, 54, 52]. [24.1.5] Modern authors also use fα [37], Iα [97], Ixαa [94], Ixα [23], Dx-αa [85, 102, 91], or d-α/d⁢x-a-α [92] instead of Ia+α² (This is a footnote:) ² Some authors [97, 26, 92, 23, 85, 91] employ the derivative symbol D also for integrals, resp. I for derivatives, to emphasize the similarity between fractional integration and differentiation. If this is done, the choice of Riesz and Feller, namely I, seems superior in the sense that fractional derivatives, similar to integrals, are nonlocal operators, while integer derivatives are local operators..

[24.2.1] The fractional integral operators Ia+α,Ib-α are commonly called Riemann-Liouville fractional integrals [99, 98, 94] although sometimes this name is reserved for the case a=0 [85]. [24.2.2] Their domain of definition is typically chosen as D⁢Ia+α=L1⁢a,b or D⁢Ia+α=Lloc1⁢a,b [99, 98, 94]. [24.2.3] For the definition of Lebesgue spaces see the Appendix B. [24.2.4] If f∈L1⁢a,b then Ia+α⁢f∈L1⁢a,b and Ia+α⁢f⁢x is finite for almost all x. [24.2.5] If f∈Lp⁢a,b with 1≤p≤∞ and α>1/p then Ia+α⁢f⁢x is finite for all x∈a,b. [24.2.6] Analogous statements hold for Ib-α⁢f⁢x [98].

[24.3.1] A short table of Riemann-Liouville fractional integrals is given in Appendix A. [24.3.2] For a more extensive list of fractional integrals see [24].

[24.4.1] Examples (2.5) and (2.6) or (A.3) and (A.5) show that Definition 2.1 is well suited for fractional integration of power series, but not for functions defined by Fourier series. [24.4.2] In fact, if f⁢x is a periodic function with period 2⁢π, and³ (This is a footnote:) ³The notation ∼ indicates that the sum does not need to converge, and, if it converges, does not need to converge to f⁢x.

then the Riemann-Liouville fractional Ia+α⁢f will in general not be periodic. [24.4.3] For this reason an alternative definition of fractional integrals was investigated by Weyl [124].

[24.5.1] Functions on the unit circle G=R/2⁢π⁢Z correspond to 2⁢π-periodic functions on the real line. [24.5.2] Let f⁢x be periodic with period 2⁢π and such that the integral of f over the interval 0,2⁢π vanishes, so that c0=0 in eq. (2.31). [24.5.3] Then the integral of f is itself a periodic function, and the constant of integration can be chosen such that the integral over 0,2⁢π vanishes again. [24.5.4] Repeating the integration n times one finds using (2.6) and the integral representation

[page 25, §0]    ck=1/2⁢π⁢∫02⁢πe-i⁢k⁢s⁢f⁢s⁢d⁢s of Fourier coefficients

with c0=0. [25.0.1] Recall the convolution formula [132, p.36]

for two periodic functions f⁢t∼∑k=-∞∞fk⁢ei⁢k⁢t and g⁢t∼∑k=-∞∞gk⁢ei⁢k⁢t. [25.0.2] Using eq. (2.33) and generalizing (2.32) to noninteger n suggests the following definition. [99, 94].

[25.2.1] It can be shown that the series for Ψ±α⁢x converges and that the Weyl definition coincides with the Riemann-Liouville definition [133]

for 2⁢π periodic functions whose integral over a period vanishes. [25.2.2] This is eq. (2.29) with a=-∞ resp. b=∞. [25.2.3] For this reason the Riemann-Liouville

[page 26, §0]    fractional integrals with limits ±∞, I+α⁢f=I-∞+α⁢f and I-α⁢f=I∞-α⁢f, are often called Weyl fractional integrals [24, 99, 85, 94].

[26.1.1] The Weyl fractional integral may be rewritten as a convolution

where the convolution product for functions on R is defined as⁴ (This is a footnote:) ⁴ If K,f∈L1⁢R then K*f⁢t exists for almost all t∈R and f∈L1⁢R. [26.1.2] If K∈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.

and the convolution kernels are defined as

for α>0. [26.1.3] Here

is the Heaviside unit step function, and xα=exp⁡α⁢log⁡x with the convention that log⁡x is real for x>0. [26.1.4] For α=0 the kernel

is the Dirac δ-function defined in (C.2) in Appendix C. [26.1.5] Note that K±α∈Lloc1⁢R for α>0.

[26.2.1] Riemann-Liouville and Weyl fractional integrals have upper or lower limits of integration, and are sometimes called left-sided resp. right-sided integrals. [26.2.2] A more symmetric definition was advanced in [97].

[27.1.1] Riesz fractional integration may be written as a convolution

with the (one-dimensional) Riesz kernels

for α≠2⁢k+1,k∈Z, and

for α≠2⁢k,k∈Z. [27.1.2] Subsequently, Feller introduced the generalized Riesz-Feller kernels [26]

with parameter β∈R. [27.1.3] The corresponding generalized Riesz-Feller fractional integral of order α and type β is defined as

[27.1.4] This formula interpolates continuously from the Weyl integral I-α=Iα,-π/2 for β=-π/2 through the Riesz integral Iα=Iα,0 for β=0 to the Weyl integral I+α=Iα,π/2 for β=π/2. [27.1.5] Due to their symmetry Riesz-Feller fractional integrals are readily generalized to higher dimensions.

[page 28, §1]

[28.1.1] Fractional integration can be extended to distributions using the convolution formula (2.37) above. [28.1.2] Distributions are generalized functions [105, 31]. [28.1.3] They are defined as linear functionals on a space X of conveniently chosen ‘‘test functions’’. [28.1.4] For every locally integrable function f∈Lloc1⁢R there exists a distribution Ff:X→C defined by

where φ∈X is test function from a suitable space X of test functions. [28.1.5] By abuse of notation one often writes f for the associated distribution Ff. [28.1.6] Distributions that correspond to functions via (2.50) are called regular distributions. [28.1.7] Examples for regular distributions are the convolution kernels K±α∈Lloc1⁢R defined in (2.39). [28.1.8] They are locally integrable functions on R when α>0. [28.1.9] Distributions that are not regular are sometimes called singular. [28.1.10] An important example for a singular distribution is the Dirac δ-function. [28.1.11] It is defined as δ:X→C

for every test function φ∈X. [28.1.12] The test function space X is usually chosen as a subspace of C∞⁢R, the space of infinitely differentiable functions. [28.1.13] A brief introduction to distributions is given in Appendix C.

[28.2.1] In order to generalize (2.37) to distributions one must define the convolution of two distributions. [28.2.2] To do so one multiplies eq. (2.38) on both sides with a smooth test function φ∈Cc∞⁢R of compact support. [28.2.3] Integrating gives

where the notation f⁢y,φ⁢x+y means that the functional Ff is applied to the function φ(x+⋅) for fixed x. [28.2.4] Explicitly, for fixed x

[page 29, §0]    where φx(⋅)=φ(x+⋅). [29.0.1] Equation (2.52) can be used as a definition for the convolution of distributions provided that the right hand side has meaning. [29.0.2] This is not always the case as the counterexample K=f=1 shows. [29.0.3] In general the convolution product is not associative (see eq. (2.113)). [29.0.4] However, associative and commutative convolution algebras exist [21]. [29.0.5] Equation (2.52) is always meaningful when supp⁢K or supp⁢f is compact [63]. [29.0.6] Another case is when K and f have support in R+. [29.0.7] This will be assumed in the following.

[29.2.1] If f∈C0∞⁢R′ with supp⁢f⊂R+ then also I0+α⁢f∈C0∞⁢R′ with supp⁢I0+α⁢f⊂R+.

[29.3.1] The Fourier transformation is defined as

for functions f∈L1⁢R. [29.3.2] Then

holds for 0<α<1 by virtue of the convolution theorem. [29.3.3] The equation cannot be extended directly to α≥1 because the Fourier integral on the left hand side may not exist. [29.3.4] Consider e.g. α=1 and f∈Cc∞⁢R. [29.3.5] Then I+1⁢f⁢x→const as x→∞ and F⁢I+1⁢f does not exist [94]. [29.3.6] Equation (2.56) can be extended to all α with 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 Dm⁢F⁢f⁢0=0 for all m∈N0.

[29.4.1] For the Riesz potentials one has

for functions in Lizorkin space.

[page 30, §1]    [30.1.1] The Laplace transform is defined as

for locally integrable functions f:R+→C. [30.1.2] Now

by the convolution theorem for Laplace transforms. [30.1.3] The Laplace transform of I0-α⁢f leads to a more complicated operator.

[30.2.1] If f⁢x∈Lp⁢a,b,g∈Lq⁢a,b with 1/p+1/q≤1+α, p,q≥1 and p≠1, q≠1 for 1/p+1/q=1+α then the formula

holds. [30.2.2] The formula is known as fractional integration by parts [99]. [30.2.3] For f⁢x∈Lp⁢R,g∈Lq⁢R with p>1,q>1 and 1/p+1/q=1+α the analogous formula

holds for Weyl fractional integrals.

[30.3.1] These formulae provide a second method of generalizing fractional integration to distributions. [30.3.2] Equation (2.60) may be read as

for a distribution f and a test function φ. [30.3.3] It shows that right- and left-sided fractional integrals are adjoint operators. [30.3.4] The formula may be viewed as a definition of the fractional integral Ia+α⁢f of a distribution provided that the operator Ib-α maps the test function space into itself.

[30.4.1] The mapping properties of convolutions can be studied with the help of Youngs inequality. Let p,q,r obey 1≤p,q,r≤∞ and 1/p+1/q=1+1/r. [30.4.2] If K∈Lp⁢R and f∈Lq⁢R then K*f∈Lr⁢R and Youngs inequality K*fr≤Kp⁢fq holds. [30.4.3] It follows that K*fq≤C⁢fp if

[page 31, §0]    1≤p≤q≤∞ and K∈Lr⁢R with 1/r=1+1/q-1/p. [31.0.1] The Hardy-Littlewood theorem states that these estimates remain valid for K±α although these kernels do not belong to any Lp⁢R-space [37, 38]. [31.0.2] The theorem was generalized to higher dimensions by Sobolev in 1938, and is also known as the Hardy-Littlewood-Sobolev inequality (see [37, 38, 113, 63]).

[31.2.1] The basic composition law for fractional integrals follows from

where Euler’s Beta-function

was used. [31.2.2] This implies the semigroup law for exponents

also called additivity law. [31.2.3] It holds for Riemann-Liouville, Weyl and Riesz-Feller fractional integrals of functions.

[31.3.1] Riemann [96, p.341] suggested to define fractional derivatives as integer order derivatives of fractional integrals.

[32.1.1] Here Wk,p⁢G=f∈Lp⁢G:Dk⁢f∈Lp⁢G denotes a Sobolev space defined in (B.17). [32.1.2] For k=p=1 the space W1,1⁢a,b=AC0⁢a,b coincides with the space of absolutely continuous functions.

[32.2.1] The notation for fractional derivatives is not standardized⁶ (This is a footnote:) ⁶see footnote 2.1. [32.2.2] Leibniz and Euler used dα [73, 72, 25] Riemann wrote ∂xα [96], Liouville preferred dα/d⁢xα [76], Grünwald used dα⁢f/d⁢xαx=ax=x or Dα⁢fx=ax=x [34], Marchaud wrote Daα, and Hardy-Littlewood used an index fα [37]. [32.2.3] The notation in (2.67) follows [99, 98, 54, 52]. Modern authors also use I-α [97], Ix-α [23], Dxαa [94, 85, 102], dα/d⁢xα [129, 102], dα/d⁢x-aα [92] instead of Da+α.

[32.3.1] Let f⁢x be absolutely continuous on the finite interval a,b. [32.3.2] Then, its derivative f′ exists almost everywhere on a,b with f′∈L1⁢a,b, and the function f can be written as

Substituting this into Ia+α⁢f gives

where commutativity of Ia+1 and Ia+α was used. [32.3.3] It follows that

for 0<α<1. [32.3.4] Above, the notations

were used for the first order derivative.

[page 33, §1]    [33.1.1] This observation suggests to introduce a modified Riemann-Liouville fractional derivative through

where n=Re⁢α+1. [33.1.2] Note, that f must be at least n-times differentiable. [33.1.3] Formula (2.72) is due to Liouville [76, p.10] (see eq. (2.18) above), but nowadays sometimes named after Caputo [17].

[33.2.1] The relation between (2.72) and (2.67) is given by

[33.3.1] The Riemann-Liouville fractional derivative is the left inverse of Riemann-Liouville fractional integrals. [33.3.2] More specifically, [99, p.44]

[33.4.1] For the right inverses of fractional integrals one finds

[33.5.1] The last theorem implies that for f∈L1⁢a,b and Re⁢α>0 with n=Re⁢α+1 the equality

[page 34, §0]    holds only if

for all k=0,1,2,…,n-1. [34.0.1] Note that the existence of g⁢x=Da+α⁢f⁢x in eq. (2.77) does not imply that f⁢x can be written as Ia+α⁢g⁢x for some integrable function g [99]. [34.0.2] This holds only if both conditions (2.78) are satisfied. [34.0.3] As an example where one of them fails, consider the function f⁢x=x-aα-1 for 0<α<1. [34.0.4] Then Da+α⁢x-aα-1=0 exists. [34.0.5] Now D0⁢Ia+1-α⁢x-aα-1≠0 so that (2.78b) fails. [34.0.6] There does not exist an integrable g such that Ia+α⁢g=x-aα-1. [34.0.7] In fact, g corresponds to the δ-distribution δ⁢x-a.

[34.1.1] Riemann-Liouville fractional derivatives have been generalized in [52, p.433] to fractional derivatives of different types.

[34.3.1] The type β of a fractional derivative allows to interpolate continuously from Da±α=Da±α,0 to D~a±α=Da±α,1. [34.3.2] A relation between fractional derivatives of the same order but different types was given in [52, p.434].

[34.4.1] Marchaud’s approach [78] is based on Hadamards finite parts of divergent integrals [36]. [34.4.2] The strategy is to define fractional derivatives as analytic continuation of fractional integrals to negative orders. [see [99, p.225]]

[35.1.1] The idea of Marchaud’s method is to extend the Riemann-Liouville integral from α>0 to α<0, and to define

where α>0. [35.1.2] However, this is not possible because the integral in (2.83) diverges. [35.1.3] The idea is to subtract the divergent part of the integral,

obtained by setting f⁢x-y≈f⁢x for y≈0. [35.1.4] Subtracting (2.83) from (2.84) for 0<α<1 suggests the definition

[35.1.5] Formal integration by parts leads to I+1-α⁢f′⁢x, showing that this definition contains the Riemann-Liouville definition.

[35.2.1] The definition may be extended to α>1 in two ways. [35.2.2] The first consists in applying (2.85) to the n-th derivative dn⁢f/d⁢xn for n<α<n+1. [35.2.3] The second possibility is to regard f⁢x-y-f⁢x as a first order difference, and to generalize to n-th order differences. [35.2.4] The n-th order difference is

where 1⁢f⁢x=f⁢x is the identity operator and

[page 36, §0]    is the translation operator. [36.0.1] The Marchaud fractional derivative can then be extended to 0<α<n through [94, 98]