Foundations of Fractional Calculus

Foundations of fractional calculus

Sanja Konjik

Department of Mathematics and Informatics, University of Novi Sad, Serbia

Winter School on Non-Standard Forms of Teaching Mathematics and Physics: Experimental and Modeling Approach Novi Sad, February 6-8, 2015

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 1 / 22 derivatives integrals

f (x0 + ∆x) f (x0) f 0(x0) = lim − F (x) = f (x) dx F 0(x) = f (x) ∆x 0 ∆x ⇒ → Z

b f(x)dx Za

a b x

Calculus

Modern calculus was founded in the 17th century by Isaac Newton and Gottfried Leibniz.

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 2 / 22 Calculus

Modern calculus was founded in the 17th century by Isaac Newton and Gottfried Leibniz.

derivatives integrals

f (x0 + ∆x) f (x0) f 0(x0) = lim − F (x) = f (x) dx F 0(x) = f (x) ∆x 0 ∆x ⇒ → Z

b f(x)dx Za

a b x

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 2 / 22 Fractional Calculus was born in 1695 n fd dt n What if the order will be n = ½?

It will lead to a paradox, from which one day useful consequences will be Start drawn. JJ II J I G.F.A. de L’Hôpital G.W. Leibniz 10 / 90 (1661–1704) (1646–1716) Back Full screen Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 3 / 22 Close End Leonhard Euler (1707-1783) dα(xn) in 1738 gave a meaning to the derivative , for dxα α / N ∈

Joseph Liouville (1809-1882) in his papers in 1832-1837 gave a solid foundation to the fractional calculus, which has undergone only slight changes since then.

Georg Friedrich Bernhard Riemann (1826-1866) in a paper from 1847 which was published 29 years later (and ten years after his death) proposed a deﬁnition for fractional integration in the form that is still in use today.

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 4 / 22 4

Fig. 1 Timeline of main scientists in the area of Fractional Calculus. Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 5 / 22

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 value of books published per year ni/hi,whereni denotes the number of published objects during the period hi, i =1, , 5, is then plotted. Due to the scarcity of data the period 1966-1975 is not··· considered in the trendline calculation for the second index. Figure 2 shows the histograms for the books with author and books edited indices. The exponential trendlines reveal a good correlation factor, but dif- ferent growing rates. This discrepancy means that such indices describe only a part of the object under analysis and common sense suggests that some value in between both cases is probably closer to the “true”. These trend lines reﬂect the past and there is no guarantee that we can foresee the future as the Moore law seems to be demonstrating recently. The gamma function: ∞ Γ(z) := e−t tz−1 dt (Re z > 0) Z0 with properties

Γ(z + 1) = z Γ(z), Γ(n + 1) = n!(n N). ∈ The beta function: 1 B(z, ω) = tz−1(1 t)ω−1 dt (Re z, Re ω > 0) − Z0 with properties Γ(z)Γ(ω) B(z, ω) = . Γ(z + ω)

Special functions

Special functions, such as the gamma function, the beta function or the Mittag-Leﬄer function, have an important role in fractional calculus.

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 6 / 22 The beta function: 1 B(z, ω) = tz−1(1 t)ω−1 dt (Re z, Re ω > 0) − Z0 with properties Γ(z)Γ(ω) B(z, ω) = . Γ(z + ω)

Special functions

Special functions, such as the gamma function, the beta function or the Mittag-Leﬄer function, have an important role in fractional calculus. The gamma function: ∞ Γ(z) := e−t tz−1 dt (Re z > 0) Z0 with properties

Γ(z + 1) = z Γ(z), Γ(n + 1) = n!(n N). ∈

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 6 / 22 Special functions

Γ(z + 1) = z Γ(z), Γ(n + 1) = n!(n N). ∈ The beta function: 1 B(z, ω) = tz−1(1 t)ω−1 dt (Re z, Re ω > 0) − Z0 with properties Γ(z)Γ(ω) B(z, ω) = . Γ(z + ω)

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 6 / 22 The Mittag-Leﬄer function:

∞ zk E (z) = (Re α, Re β > 0) α,β Γ(α k + β) k=0 X with special cases: 1 E (z) = 0,1 1 z z E1,1(z) = e − E2,1(z) = cosh(√z)

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 7 / 22 The Riemann-Liouville fractional calculus

Let u L1([a, b]) and α > 0. ∈

The left Riemann-Liouville fractional integral of order α:

t α 1 α−1 aI u(t) = (t θ) u(θ) dθ. t Γ(α) − Za The right Riemann-Liouville fractional integral of order α:

b α 1 α−1 t I u(t) = (θ t) u(θ) dθ. b Γ(α) − Zt

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 8 / 22 Properties:

If α = n N then ∈ t tn tn−1 t1 n 1 n−1 aI u(t) = (t θ) u(θ) dθ = ... u(θ) dθ1 ... dθn, t (n 1)! − − Za Za Za Za n i.e., aIt u is just an n-fold integral of u. Semigroup property:

α β α+β α β α+β aIt aIt u = aIt u and t Ib t Ib u = t Ib u (α, β > 0) Consequences: α β α β β α aIt and aIt commute, i.e., aIt aIt = aIt aIt ; α µ Γ(µ + 1) µ+α aI (t a) = (t a) . t − Γ(µ + 1 + α) · −

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 9 / 22 Let u AC([a, b]) and 0 α < 1. ∈ ≤

The left Riemann-Liouville fractional derivative of order α: t α d 1−α 1 d u(θ) aD u(t) = aI u(t) = dθ. t dt t Γ(1 α) dt (t θ)α − Za − The right Riemann-Liouville fractional derivative of orderα:

b α d 1−α 1 d u(θ) t Db u(t) = t Ib u(t) = α dθ. − dt Γ(1 α) − dt t (θ t) − Z − 0 0 If α = 0 then aDt u(t) = t Db u(t) = u(t).

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 10 / 22 Euler formula:

α −µ Γ(1 µ) 1 aD (t a) = − t − Γ(1 µ α) (t a)µ+α − − − Derivation: Z t µ α µ 1 d (θ − a)− aDt (t − a)− = dθ Γ(1 − α) dt a (t − θ)α µ 1 d Z t a z− = − dz Γ(1 − α) dt 0 (t − a − z)α µ 1 d Z t a z− = − dz α z α Γ(1 − α) dt 0 (t − a) (1 − t a ) − µ µ 1 d Z 1 (t − a)− ξ− = (t − a)dξ Γ(1 − α) dt 0 (t − a)α(1 − ξ)α Z 1 1 d 1 µ α µ α = (t − a) − − ξ− (1 − ξ)− dξ Γ(1 − α) dt 0

1 − µ − α µ α = · (t − a)− − · B(1 − µ, 1 − α) Γ(1 − α)

Γ(1 − µ) µ α = · (t − a)− − Γ(1 − µ − α) Conclusions: α α−1 µ = 1 α: aDt (t a) = 0 − α − µ = 0: aDt c = 0, for any constant c R. 6 ∈ Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 11 / 22 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 AC n([a, b]) and n 1 α < n. ∈ − ≤ The left Riemann-Liouville fractional derivative of order α: d [α] d [α]+1 Dαu(t) = D{α}u(t) = I 1−{α}u(t) a t dt a t dt a t or n t α 1 d u(θ) aDt u(t) = α−n+1 dθ Γ(n α) dt a (t θ) − Z − The right Riemann-Liouville fractional derivative of order α:

[α] [α]+1 α d {α} d 1−{α} t D u(t) = t D u(t) = t I u(t) b − dt b − dt b or n b α 1 d u(θ) t Db u(t) = α−n+1 dθ Γ(n α) − dt t (θ t) − Z − Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 12 / 22 Properties: α α aDt aIt u = u α α aI aD u = u, but t t 6 n−1 α−k−1 n−k−1 α α (t a) d n−α aI aD u(t) = u(t) − aI u(t) t t − Γ(α k) dt t k=0 t=a X −

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

3 1 Dαu = Γ = √π, Dβu = 0 0 t 2 2 0 t

α β β α 1 − 3 α+β 1 − 3 0D 0D u = 0, 0D 0D u = t 2 , 0D u = t 2 t t t t −4 t −4 α β α+β α β β α 0D (0D u) = 0D u and 0D (0D u) = 0D (0D u) ⇒ t t 6 t t t 6 t t

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 13 / 22 Left Riemann-Liouville fractional operators via convolutions

Set tα−1 H(t) , α > 0 Γ(α) fα(t) :=   d N  fα+N (t), α 0, N N : N + α > 0 N + α 1 0 dt ≤ ∈ ∧ − ≤  (H is the Heaviside function) α N: ∈ t2 tn−1 f (t) = H(t), f (t) = t , f (t) = + ,... f (t) = + , 1 2 + 3 2 n (n 1)! − α N0: − ∈ 0 (n) f0(t) = δ(t), f−1(t) = δ (t),... f−n(t) = δ (t).

The convolution operator fα is the left Riemann-Liouville operator of diﬀerentiation for α < 0 ∗ integration for α 0 ≥ Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 14 / 22 Heaviside function:

0, t < 0 H(t) = 1 , t = 0  2  1 t > 0 

Dirac delta distribution generalized function that vanishes everywhere except at the origin, with integral 1 over R

Convolution

+∞ f u(t) = f (t θ)u(θ) dθ ∗ − Z−∞ Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 15 / 22 8Willbeinsertedbytheeditor

Taylor’s Formula/ Serie The remainder

m 1 α+j − RL Da+ f(x0) α+j R (x)= f(x)= (x x0) m Γ (α + j +1) − RL α+m RL α+m (80) j=0 I D f(x) ! a+ a+ +Rm(x), α > 0 m Γ (α) cj (x0) (j+1)α 1 f(x)= (x x0) − Γ ((j +1)α) − (m+1)x j=0 ( RLDα ) f(ξ) ! R (x)= a+ . m Γ ((m+1)α+1) (81) +Rm(x), where α [0, 1] and (x a)(m+1)α, ξ [a, x] ∈ − ∈ 1 α RL α j c (x)=(x x ) − [ D f] (x) j − 0 a C α Da+f(a) α f(x)=f(a)+ Γ (α+1) (x a) CDα CDα f(a) − (82) + a+ a+ (x a)2α + ..., Γ (2α+1) − m 1 − αk 1 f(x)= ak x + Rm(x), Rm(x)= . Γ (αm+1) k=0 (83) ! x αm 1 (αk) (α ) . (x z) − D f(z)dz D k f(0) 0 − where x>0andak = Γ (α +1) k "

In (83), α0 =0andtheαk,(k =1, ..., m)areanincreasingsequenceofreal (αk) RL 1 (αk αk 1)RL 1+αk 1 numbers such that 0 < αk αk 1,andD = I0+− − − D0+ − .Formore information see e.g. [1–7]. − − Analytical expressions of some fractional derivatives 4AnalyticalExpressionsofSomeFractionalDerivatives

RL α f(x),x>a Da+f(x) α k(x a)− k Γ (1− α) (53) − β Γ (β+1) β α (x a) , (β) > 1 Γ (β+1 α) (x a) − (54) − ℜ − λa − α− λx e (x a)− E1,1 α(λ(x a)) = e , λ =0 − λa − − (55) ̸ e Ex a( α, λ) λ − − λ (a p) α a x (x p) ,a p>0 Γ (1± α) (x a)− 2F1 1, λ, 1 α; a−p (56) ± ± − − − − ± β λ Γ (β+1) λ # β α $ (x a) (x p) , Γ (β+1 α) (a p) (x a) − − ± − ± − · a x (57) (β) > 1 a p>0 2F1 β +1, λ; β α; a−p ℜ − ∧ ± · − − ± β λ Γ (β+1) # λ β α $ (x a) (p x) , Γ (β+1 α) (p a) (x a) − − − − − − · x a (58) p>x>a (β) > 1 2F1 β +1, λ; β α; p−a ∧ℜ − λa · − − − Γ (β+1)e # β α $ β λx Γ (β+1 α) (x a) − (x a) e , (β) > 1 − − · (59) − ℜ − 1F1(β +1, β +1 α; λ(x a)) α (x a)− · − − − .[ F (1, 1 α,iλ(x a)) sin(λ(x a)) 2iΓ (1 α) 1 1 − − − (60) − − F (1, 1 α, iλ(x a))] 1 1 − − −

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 16 / 22 Caputo fractional derivatives

Let u AC n([a, b]) and n 1 α < n. ∈ − ≤

The left Caputo fractional derivative of order α:

1 t u(n)(θ) c Dαu(t) = dθ a t Γ(n α) (t θ)α−n+1 − Za − The right Caputo fractional derivative of order α:

1 b ( u)(n)(θ) c Dαu = − dθ t b Γ(n α) (θ t)α−n+1 − Zt −

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 17 / 22 Properties: In general, R-L and Caputo fractional derivatives do not coincide:

[α] [α] α d 1−{α} 1−{α} d c α aD u = aI u = aI u = D u, t dt t 6 t dt a t but n−1 k−α α c α (t a) (k) + aD u = D u + − u (a ), t a t Γ(k α + 1) k=0 X − and n−1 k−α α c α (b t) (k) − t D u = D u + − u (b ). b t b Γ(k α + 1) k=0 X − (k) + α c α If u (a ) = 0 (k = 0, 1,..., n 1) then aDt u = aDt u; (k) − − α c α If u (b ) = 0 (k = 0, 1,..., n 1) then t Db u = t Db u. c α c α − aDt k = t D k = 0, k R. b ∈

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 18 / 22 Fractional identities

Linearity: α α α aD (µu + νv) = µ aD u + ν aD v t · t · t The Leibnitz rule does not hold in general: α α α aD (u v) = u aD v + aD u v t · 6 · t t · For analytic functions u and v: ∞ i−1 α α α−i (i) α ( 1) αΓ(i α) aD (u v) = (aD u) v , = − − t · i t · i Γ(1 α)Γ(i + 1) i=0 X − For a real analytic function u: ∞ i−α α α (t a) (i) aD u = − u (t) t i Γ(i + 1 α) i=0 X − Fractional integration by parts: b b α α f (t)aDt g dt = g(t)t Db f dt Za Za Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 19 / 22 Laplace and Fourier transform

Fourier transform: +∞ −iωx u(ω) =u ˆ(ω) = e u(x) dx (ω R) F ∈ Z−∞ Laplace transform:

+∞ u(s) =u ˜(s) = e−st u(t) dt (Re s > 0) L Z0 Properties:

n−1 [Dnu](ω) = (iω)n u(ω); [Dnu](s) = sn u(s) sn−k−1Dk u(a) F F L L − k=0 X n−1 [Dαu](ω) = (iω)α u(ω); [Dαu](s) = sα u(s) sn−k−1Dk I n−αu(a) F F L L − k=0 X Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 20 / 22 Will be inserted by the editor 11

α α απi sgn(κ)/2 where ( iκ) = κ e∓ . ∓ | |

5.2 Some Laplace transforms

L f(t) (s) f(t) α β { } k α k!s − d Eα,β ( at ) αk+β 1 ± (sα a)k+1 t − d( atα)k (92) 1∓ λt ± sα λ eα (93) − α 1 n n!s − αn ∂ α (sα λ)n+1 t ∂λ Eα (λt ) (94) −n! ∂ n λz (sα λ)n+1 ∂λ! e"α (95) α −β s − β 1 α sα a !t − "Eα,β( at ) (96) α∓1 ± s − α sα a Eα ( at ) (97) 1∓ α 1± α sα a t − Eα,α( at ) (98) 1 ∓β ± s − β 1 s a t − E1,β( at)=Et(β 1, a) (99) ∓ ± − ± β 1 1 β 1 t − β t − E (0) = E (β 1, 0) = (100) s 1,β t − Γ (β) 1 1 (101) √s √πt 1 t s√s 2 π (102) n n 1 1 # 2 t − 2 sn√s ,(n =1, 2, ) 1 3 5 (2n 1)√π (103) s ··· ·1 · ···at − 3 e (1 + 2at) (104) (s a) 2 √πt − √s a √s b 1 ebt eat (105) 2√πt3 − − − 2− 1 1 aea terfc a√t (106) √s+a √πt ! " − 2 √s 1 + aea terf a√t (107) s a2 √πt ! " − √s 1 2a a2t a√t τ 2 2 e− ! "e dτ (108) s+a √πt − √π 0 1 1 a2t √ √s(s a2) a e erf a t $ (109) − 1 2 a2t a√t τ 2 √s(s+a2) a√π e− ! 0 " e dτ (110) a2t b2 a2 e b a$erf a√t − 2 (111) (s a2) √s+b b t − − ( ) be erfc b√t −2 % ! "& Sanja Konjik (Uni Novi1 Sad) Foundations of fractionalea terfc calculusa√t Novi Sad, Feb 6-8,(112) 2015 21 / 22 √s(√s+a) ! " 1 1 e aterf √b a√t (113) (s+a)√s+b √b a −! " − − b2 a2 a2t b √ b2t √ √s(s a−2)(√s+b) e a erf a !t 1 + "e erfc b t (114) n − (1 s)− n! − √ n+ 1 H2n t (115) s 2 (2n)!%√πt ! " & ! " (1 s)n n! − √ n+ 3 H2!n+1 " t (116) s 2 (2n+1)!√π √s+2a √s at − √s ae− [I1 (at)+I!0 (at")] (117) 1 1 (a+b)t a b e 2 I t (118) √s+a√s+b − 0 −2 k 1 Γ (k) 2 1 t − 2 (a+b)t a b k k ,(k 0) √π ! e− " I 1 − t (119) (s+a) (s+b) a b k 2 2 ≥ 1 − − 1 2 (a+b)t a b a b 1 3 te− ' ( I0 −2 t + I1 −2! t " (120) (s+a) 2 (s+b) 2 % ! " ! "& References

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Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 2015 22 / 22