Introduction into fractional calculus

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Beginings of fractional differentiation: prehistory

The idea to generalize notion dpf (x)dxp to non-integer values of p

appeared at the birth of differential calculus itself in Leibnizcorrespondence with Bernoulli who asked about meaning of thetheorem on differentiation of product of functions for non-integervalue of differentiation.

Leibniz in his letters to L’Hopital (1695) and to Wallis (1697) maderemarks of possibility to consider differentiation of order 1/2.

Euler (1738) was the one who took first step by observation thatthere is a sense to consider d

pxa

dxp for non-integer p.

Laplace (1812) proposed the differentiation of functions representablein the form

∫T (t)t−xdt.

Lacroix (1820) repeated idea of Euler and gave the formula for d1/2xa

dx1/2.

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Beginings of fractional differentiation

Fourier (1822) give the idea for the formula to define non-integerdiferentiation:

dpf (x)

dxp=

1

2π

∫λpdλ

∫f (t) cos(λ(x − t) + pπ

2) dt

Proper history starts with Abel and Liouville ...

Breaf historical outline is given in the comprehensive book onfractional integrals and derivatives:

S. G. Samko, A. A. Kilbas, and O. I. Marichev.Fractional Integrals and Derivatives.Gordon and Breach, Amsterdam, 1993.

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Fractional calculus4

Fig. 1 Timeline of main scientists in the area of Fractional Calculus.

on histograms constructed with h = 10 years bins (i.e., 1966-1975 up to 1996-2005) with exception of the last 7 years period (i.e., 2006-2012). The valueof books published per year ni/hi, where ni denotes the number of publishedobjects during the period hi, i = 1, · · · , 5, is then plotted. Due to the scarcityof data the period 1966-1975 is not considered in the trendline calculation forthe second index.

Figure 2 shows the histograms for the books with author and books editedindices. The exponential trendlines reveal a good correlation factor, but dif-ferent growing rates. This discrepancy means that such indices describe only apart of the object under analysis and common sense suggests that some valuein between both cases is probably closer to the “true”. These trend lines reflectthe past and there is no guarantee that we can foresee the future as the Moorelaw seems to be demonstrating recently.

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The Riemann-Liouville fractional calculus

Let u ∈ L1([a, b]) and α > 0.The left Riemann-Liouville fractional integral of order α:

aIαt u(t) =

1

Γ(α)

∫ ta

(t − θ)α−1u(θ) dθ.

The right Riemann-Liouville fractional integral of order α:

t Iαb u(t) =

1

Γ(α)

∫ bt

(θ − t)α−1u(θ) dθ.

(The gamma function: Γ(z) :=∫∞

0 e−t tz−1 dt, Re z > 0)

Semigroup property: aIαt aI

βt u = aI

α+βt u and t I

αb t I

βb u = t I

α+βb u

Consequences:

aIαt and aI

βt commute, i.e., aI

αt aI

βt = aI

βt aI

αt ;

aIαt (t − a)µ =

Γ(µ+ 1)

Γ(µ+ 1 + α)· (t − a)µ+α.

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Let u ∈ AC ([a, b]) and 0 ≤ α < 1.The left Riemann-Liouville fractional derivative of order α:

aDαt u(t) =

d

dtaI

1−αt u(t) =

1

Γ(1− α)d

dt

∫ ta

u(θ)

(t − θ)α dθ.

The right Riemann-Liouville fractional derivative of orderα:

tDαb u(t) =

(− d

dt

)t I

1−αb u(t) =

1

Γ(1− α)(− d

dt

)∫ bt

u(θ)

(θ − t)α dθ.

If α = 0 then aD0t u(t) = tD

0bu(t) = u(t).

when α→ 1−, aDαt u(t)→ u′(t) and tDαb u(t)→ −u′(t).Euler formula:

aDαt (t − a)−µ =

Γ(1− µ)Γ(1− µ− α)

1

(t − a)µ+α

Consequences:µ = 1− α: aDαt (t − a)α−1 = 0µ = 0: aD

αt c 6= 0, for any constant c ∈ R.

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Definition of R-L fractional derivatives can be extended for any α ≥ 1:write α = [α] + {α}, where [α] denotes the integer part and {α},0 ≤ {α} < 1, the fractional part of α. Let u ∈ ACn([a, b]) andn − 1 ≤ α < n.

The left Riemann-Liouville fractional derivative of order α:

aDαt u(t) =

( ddt

)[α]aD{α}t u(t) =

( ddt

)[α]+1aI

1−{α}t u(t)

or

aDαt u(t) =

1

Γ(n − α)( ddt

)n ∫ ta

u(θ)

(t − θ)α−n+1 dθ

The right Riemann-Liouville fractional derivative of order α:

tDαb u(t) =

(− d

dt

)[α]tD{α}b u(t) =

(− d

dt

)[α]+1t I

1−{α}b u(t)

or

tDαb u(t) =

1

Γ(n − α)(− d

dt

)n ∫ bt

u(θ)

(θ − t)α−n+1 dθ

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Properties:

aDαt aI

αt u = u

aIαt aD

αt u 6= u, but

aIαt aD

αt u(t) = u(t)−

n−1∑k=0

(t − a)α−k−1Γ(α− k)

dn−k−1

dt

∣∣∣∣t=a

(aI

n−αt u(t)

)Contrary to fractional integration, Riemann-Liouville fractionalderivatives do not obey either the semigroup property or thecommutative law: E.g. u(t) = t1/2, a = 0, α = 1/2 and β = 3/2.

0Dαt u = Γ

(3

2

)=

1

2

√π, 0D

βt u = 0

0Dαt

(0D

βt u)

= 0, 0Dβt

(0D

αt u)

= −14t−

32 , 0D

α+βt u = −

1

4t−

32

⇒ 0Dαt (0Dβt u) 6= 0Dα+βt u and 0Dαt (0Dβt u) 6= 0Dβt (0Dαt u)

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Left Riemann-Liouville fractional operators via convolutions

In the distributional setting: (S′+ - space of Schwartz’ distributions supported in [0,∞))

fα(t) :=

H(t)

tα−1

Γ(α), α > 0( d

dt

)Nfα+N(t), α ≤ 0,N ∈ N : N + α > 0

(∈ S ′+)

Then:

α ∈ N : f1(t) = H(t), f2(t) = t+, . . . fn(t) =tn−1+

(n−1)!

−α ∈ N0 : f0(t) = δ(t), f−1(t) = δ′(t), . . . f−n(t) = δ(n)(t)

The convolution operator fα∗ is the left Riemann-Liouville operator ofdifferentiation for α < 0integration for α ≥ 0

For 0 < α < 1: 0Dαt u(t) = f−α ∗ u(t)

L[0Dαt u(t)](s) = sαLu(s) = sαũ(s) (s ∈ C)F [0Dαt u(t)](ω) = (iω)αFu(ω) = (iω)αû(ω) (ω ∈ R)

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Fractional identities

Linearity:

aDαt (µu + νv) = µ · aDαt u + ν · aDαt v

The Leibnitz rule does not hold in general:

aDαt (u · v) 6= u · aDαt v + aDαt u · v

For analytic functions u and v :

aDαt (u · v) =

∞∑i=0

(α

i

)(aD

α−it u) · v (i),

(α

i

)=

(−1)i−1αΓ(i − α)Γ(1− α)Γ(i + 1)

For a real analytic function u:

aDαt u =

∞∑i=0

(α

i

)(t − a)i−α

Γ(i + 1− α)u(i)(t)

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Integration by parts formulae

Let 0 < α < 1.

For f ∈ Lp(a, b) and g ∈ Lq(a, b), p ≥ 1, q ≥ 1, 1p + 1q ≤ 1 + α, wehave ∫ b

aaIαt f (x)g(x)dx =

∫ ba

f (x)t Iαb g(x)dx .

For f ∈ t Ib(Lp(a, b)) and g ∈ Lq(a, b), p ≥ 1, q ≥ 1, 1p + 1q ≤ 1 + α,we have ∫ b

aaD

αt f (x)g(x)dx =

∫ ba

f (x)tDαb g(x)dx .

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Caputo fractional derivatives

Let u ∈ ACn([a, b]) and n − 1 ≤ α < n.The left Caputo fractional derivative of order α:

caD

αt u(t) =

1

Γ(n − α)

∫ ta

u(n)(θ)

(t − θ)α−n+1 dθ

The right Caputo fractional derivative of order α:

ctD

αb u =

1

Γ(n − α)

∫ bt

(−u)(n)(θ)(θ − t)α−n+1 dθ

In general, R-L and Caputo fractional derivatives do not coincide:

aDαt u =

caD

αt u +

n−1∑k=0

(t − a)k−αΓ(k − α + 1)u

(k)(a+).

If u(k)(a+) = 0, (k = 0, 1, . . . , n − 1) or u ∈ S ′+ then aDαt u = caDαt u.12 / 17

Symmetrized fractional derivatives: generalisation of thefirst order

Let u ∈ AC ([a, b]), and 0 < α < 1.

Ca Eβb u(x) =

1

2(Ca D

βx u(x)− Cx Dβb )u(x) =

1

Γ(1− β)

∫ ba

u′(θ)

|t − θ|β dθ.

For a = −∞ and b =∞

Eβx u(x) =1

2

1

Γ(1− β) |x |−β ∗ u′(x).

E0x u(x) = 0 and Eβx u(x)→ u′(x), as β → 1.F(Eβx u(x)

)= i sin(βπ2 ) sgn(ξ)|ξ|β Fu(ξ)

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Symmetrized fractional derivatives: generalisation of thesecond order derivative

of order α ∈ (1, 2)

Eαx u =1

2

(Dα+ + D

α−)u =

1

2Γ (2− α)1

|x |α−1∗ d

2

dx2u (x) .

of order α ∈ (2, 3)

Eαx u =1

2

(Dα+ + D

α−)u =

1

2Γ (3− α)sgn x

|x |α−2∗ d

3

dx3u (x) .

In both casesF(Eαx w) = |ξ|α cos(

απ

2)ŵ

so for the definition we can take

Eαx w = F−1(|ξ|α cos(απ2 )

).

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Distributed order fractional derivative

For φ ∈ E ′(R) and ϕ ∈ S(R) one defines distributed order fractionalderivative

∫suppφ φ(α)D

αu(t) dα as follows:〈∫suppφ

φ(α)Dαu(t) dα,ϕ(t)〉

:=〈φ(α), 〈Dαu(t), ϕ(t)〉

〉,

γ → 〈Dγu(t), ϕ(t)〉 : R→ R is smooth.

L(∫

suppφφ(α)Dαu(t) dα

)(s) = ũ(s)〈φ(α), sα〉 (Re s > 0)

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Books: Theory...

Oldham, K.B., Spanier, J.

The Fractional Calculus. Academic Press

New York, 1974.

S. G. Samko, A. A. Kilbas, and O. I. Marichev.

Fractional Integrals and Derivatives.

Gordon and Breach, Amsterdam, 1993.

Podlubny, I.

Fractional Differential Equations,

Vol. 198 of Mathematics in Science and Engineering. Academic Press, SanDiego, 1999.

K. Diethelm.

The Analysis of Fractional Differential Equations.

Springer, Berlin (2010).

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Books: ... and Applications

Miller, K.S., Ross, B.

An Introduction to the Fractional Integrals and Derivatives - Theory andApplications

John Willey and Sons, Inc., New York, 1993.

Gorenflo, R., Mainardi, F.

Fractional calculus: Integral and differential equations of fractional order

In Mainardi F. Carpinteri, A., Eds., Fractals and Fractional Calculus inContinuum Mechanics, Vol. 378 of CISM Courses and Lectures, pages223-276. Springer-Verlag, Wien and New York, 1997.

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.

Theory and Applications of Fractional Di fferential Equations

Elsevier, Amsterdam, 2006.

T. M. Atanackovicc, S. Pilipovicc, B. Stankovicc, D.Zorica

Fractional Calculus with Applications in Mechanics. Vol 1 and Vol 2

ISTE Ltd and John Wiley and Sons, Inc, 2014.17 / 17