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, 205 Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 / 22

Calculus Modern calculus was founded in the 7th century by Isaac Newton and Gottfried Leibniz. Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 2 / 22

Calculus Modern calculus was founded in the 7th century by Isaac Newton and Gottfried Leibniz. derivatives f f (x 0 + x) f (x 0 ) (x 0 ) = lim x 0 x integrals F (x) = f (x) dx F (x) = f (x) y b f(x)dx a a b x Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 2 / 22

Fractional Calculus was born in 695 What if the order will be n = ½? n d f n dt G.F.A. de L Hôpital (66 704) It will lead to a paradox, from which one day useful consequences will be drawn. G.W. Leibniz (646 76) Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 3 / 22

Leonhard Euler (707-783) in 738 gave a meaning to the derivative d α (x n ) dx α, for α / N Joseph Liouville (809-882) in his papers in 832-837 gave a solid foundation to the fractional calculus, which has undergone only slight changes since then. Georg Friedrich Bernhard Riemann (826-866) in a paper from 847 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, 205 4 / 22

Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 5 / 22

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, 205 6 / 22

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: with properties Γ(z) := 0 Γ(z + ) = z Γ(z), e t t z dt (Re z > 0) Γ(n + ) = n! (n N). Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 6 / 22

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: with properties The beta function: Γ(z) := 0 Γ(z + ) = z Γ(z), B(z, ω) = with properties 0 e t t z dt (Re z > 0) Γ(n + ) = n! (n N). t z ( t) ω dt (Re z, Re ω > 0) B(z, ω) = Γ(z)Γ(ω) Γ(z + ω). Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 6 / 22

The Mittag-Leffler function: E α,β (z) = k=0 z k Γ(α k + β) (Re α, Re β > 0) with special cases: E 0, (z) = z E, (z) = e z E 2, (z) = cosh( z) Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 7 / 22

The Riemann-Liouville fractional calculus Let u L ([a, b]) and α > 0. The left Riemann-Liouville fractional integral of order α: ai α t u(t) = Γ(α) t a (t θ) α u(θ) dθ. The right Riemann-Liouville fractional integral of order α: ti α b u(t) = Γ(α) b t (θ t) α u(θ) dθ. Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 8 / 22

Properties: If α = n N then ai n t u(t) = (n )! t a (t θ) n u(θ) dθ = i.e., a I n t u is just an n-fold integral of u. Semigroup property: tn tn t a a... a u(θ) dθ... dθ n, ai α t ai β t u = a I α+β t u and ti α b ti β b u = ti α+β b u (α, β > 0) Consequences: ait α and a It β commute, i.e., a It α ait β = a It β ait α ; ait α (t a) µ Γ(µ + ) = Γ(µ + + α) (t a)µ+α. Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 9 / 22

Let u AC([a, b]) and 0 α <. The left Riemann-Liouville fractional derivative of order α: ad α t u(t) = d dt a I α t u(t) = d Γ( α) dt t a u(θ) (t θ) α dθ. The right Riemann-Liouville fractional derivative of orderα: ( tdb α u(t) = d ) ti α dt b u(t) = If α = 0 then a Dt 0 u(t) = t Db 0 u(t) = u(t). ( d ) b Γ( α) dt t u(θ) (θ t) α dθ. Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 0 / 22

Euler formula: ad α t (t a) µ = Γ( µ) Γ( µ α) (t a) µ+α Derivation: ad α t (t a) µ = = = = = = = d t (θ a) µ Γ( α) dt a (t θ) α dθ d t a z µ Γ( α) dt 0 (t a z) α dz d t a z µ Γ( α) dt 0 (t a) α ( t a z dz )α d (t a) µ ξ µ Γ( α) dt 0 (t a) α (t a)dξ ( ξ) α d (t a) µ α ξ µ ( ξ) α dξ Γ( α) dt 0 µ α (t a) µ α B( µ, α) Γ( α) Γ( µ) (t a) µ α Γ( µ α) Conclusions: µ = α: a Dt α (t a) α = 0 µ = 0: a Dt α c 0, for any constant c R. Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 / 22

Definition of R-L fractional derivatives can be extended for any α : write α = [α] + {α}, where [α] denotes the integer part and {α}, 0 {α} <, the fractional part of α. Let u AC n ([a, b]) and n α < n. The left Riemann-Liouville fractional derivative of order α: ( d ) [α] ( adt α u(t) = ad {α} d ) [α]+ t u(t) = ai {α} t u(t) dt dt or ad α t u(t) = d ) n t Γ(n α)( dt a u(θ) dθ (t θ) α n+ The right Riemann-Liouville fractional derivative of order α: ( tdb α u(t) = d ) [α] ( td {α} dt b u(t) = d ) [α]+ ti {α} dt b u(t) or tdb α u(t) = ( d ) n b u(θ) dθ Γ(n α) dt t (θ t) α n+ Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 2 / 22

Properties: ad α t ai α t u = u ai α t ad α t u u, but n ait α adt α u(t) = u(t) k=0 (t a) α k Γ(α k) d n k dt t=a ( ai n α t ) u(t) Contrary to fractional integration, Riemann-Liouville fractional derivatives do not obey either the semigroup property or the commutative law: E.g. u(t) = t /2, a = 0, α = /2 and β = 3/2. ( ) 3 0Dt α u = Γ = π, 0Dt β u = 0 2 2 0D α t ( ) ( ) 0Dt β u = 0, 0Dt β 0Dt α u = 4 t 3 2, 0Dt α+β u = 4 t 3 2 0 D α t ( 0 D β t u) 0 D α+β t u and 0D α t ( 0 D β t u) 0 D β t ( 0 D α t u) Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 3 / 22

Left Riemann-Liouville fractional operators via convolutions Set H(t) tα f α (t) := Γ(α), α > 0 ( d ) Nfα+N (t), α 0, N N : N + α > 0 N + α 0 dt (H is the Heaviside function) α N: f (t) = H(t), f 2 (t) = t +, f 3 (t) = t2 + 2,... f n(t) = tn + (n )!, α N 0 : f 0 (t) = δ(t), f (t) = δ (t),... f n (t) = δ (n) (t). The convolution operator f α 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, 205 4 / 22

Heaviside function: 0, t < 0 H(t) = 2, t = 0 t > 0 Dirac delta distribution generalized function that vanishes everywhere except at the origin, with integral over R Convolution f u(t) = + f (t θ)u(θ) dθ Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 5 / 22

information see e.g. [ 7]. Analytical expressions of some fractional derivatives 4AnalyticalExpressionsofSomeFractionalDerivatives k f(x),x>a (x a) β, R(β) > e λx, λ 0 (x ± p) λ, a± p>0 (x a) β (x ± p) λ, R(β) > a ± p>0 (x a) β (p x) λ, p>x>a R(β) > (x a) β e λx, R(β) > sin(λ(x a)) RL D α a+ f(x) k(x a) α Γ ( α) (53) Γ (β+) Γ (β+ α) (x a)β α (54) e λa (x a) α E, α (λ(x a)) = e λa (55) E x a ( α, ( λ) ) (a±p) λ Γ ( α) (x a) α 2F, λ, α; a x a±p (56) Γ (β+) Γ (β+ α) (a ± p)λ (x a) β α ) (57) 2F (β +, λ; β α; a x a±p Γ (β+) Γ (β+ α) (p a)λ (x a) β α ) 2F (β +, λ; β α; x a p a (58) Γ (β+)e λa Γ (β+ α) (x a)β α F (β +, β + α; λ(x a)) (59) (x a) α 2iΓ ( α).[ F (, α,iλ(x a)) F (, α, iλ(x a))] (60) Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 6 / 22

Caputo fractional derivatives Let u AC n ([a, b]) and n α < n. The left Caputo fractional derivative of order α: c ad α t u(t) = t Γ(n α) a u (n) (θ) dθ (t θ) α n+ The right Caputo fractional derivative of order α: c t Db α u = b ( u) (n) (θ) dθ Γ(n α) t (θ t) α n+ Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 7 / 22

Properties: In general, R-L and Caputo fractional derivatives do not coincide: but and ( d adt α u = dt ) [α] ai {α} t ( u a I {α} d ) [α]u t = c dt a Dt α u, n adt α u = c adt α (t a) k α u + Γ(k α + ) u(k) (a + ), k=0 n tdb α u = c t Db α u + (b t) k α Γ(k α + ) u(k) (b ). k=0 If u (k) (a + ) = 0 (k = 0,,..., n ) then a D α t u = c ad α t u; If u (k) (b ) = 0 (k = 0,,..., n ) then t D α b u = c t D α b u. c adt α k = c t Db α k = 0, k R. Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 8 / 22

Fractional identities Linearity: adt α (µu + νv) = µ adt α u + ν adt α v The Leibnitz rule does not hold in general: i=0 ad α t (u v) u ad α t v + a D α t u v For analytic functions u and v: ( ) α adt α (u v) = ( a Dt α i u) v (i), i For a real analytic function u: adt α u = i=0 ( α i Fractional integration by parts: b a f (t) a D α t g dt = ) (t a) i α Γ(i + α) u(i) (t) b a ( ) α = ( )i αγ(i α) i Γ( α)γ(i + ) g(t) t D α b f dt Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 9 / 22

Laplace and Fourier transform Fourier transform: Fu(ω) = û(ω) = Laplace transform: + e iωx u(x) dx (ω R) Properties: Lu(s) = ũ(s) = + 0 e st u(t) dt (Re s > 0) F[D n u](ω) = (iω) n Fu(ω); F[D α u](ω) = (iω) α Fu(ω); n L[D n u](s) = s n Lu(s) s n k D k u(a) k=0 n L[D α u](s) = s α Lu(s) s n k D k I n α u(a) k=0 Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 20 / 22

L {f(t)} (s) f(t) k!s α β (s α a) k+ t αk+β d k Eα,β(±at α ) s α λ e λt n!s α d(±at α ) k (92) α (93) (s α λ) t ( αn n λ) n+ Eα (λt α ) (94) ( n! ) n (s α λ) n+ λ e λz α (95) s α β s α a t β E α,β (±at α ) (96) s α s α a E α (±at α ) (97) s α a t α E α,α (±at α ) (98) s β s a t β E,β (±at) =E t (β, ±a) (99) s β s s s 2 t β E,β (0) = E t (β, 0) = tβ Γ (β) (00) πt (0) t π (02) s n s,(n =, 2, ) 2 n t n 2 s (s a) 3 2 πt e at ( + 2at) (04) ( s a s b 2 πt e bt e at) (05) 3 s+a s s a 2 s s+a 2 πt 2a s(s a2 ) 2 s(s+a2 ) a t a t π e a2 3 5 (2n ) (03) π πt ae a2t erfc ( a t ) (06) πt + ae a2t erf ( a t ) (07) π e a2 t a t e τ 2 dτ (08) 0 a ea2t erf ( a t ) (09) e τ 2 dτ (0) 0 b 2 a 2 e [ a2 t b a erf ( a t )] (s a 2 )( s+b) be b2t erfc ( b t ) () Sanja Konjik (Uni Novi Sad) Foundations of fractional ( calculus ) Novi Sad, Feb 6-8, 205 2 / 22 a 2 t

References Samko, S.G., Kilbas, A.A., Marichev, O.I. Fractional Integrals and Derivatives - Theory and Applications. Gordon and Breach Science Publishers, Amsterdam, 993. Gorenflo, R., Mainardi, F. Fractional calculus: Integral and differential equations of fractional order. In Mainardi F. Carpinteri, A., editor, Fractals and Fractional Calculus in Continuum Mechanics,, volume 378 of CISM Courses and Lectures, pages 223 276. Springer-Verlag, Wien and New York, 997. Podlubny, I. Fractional Differential Equations, volume 98 of Mathematics in Science and Engineering. Academic Press, San Diego, 999. 43:255203(2pp), 200. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006. Miller, K.S., Ross, B. An Introduction to the Fractional Integrals and Derivatives - Theory and Applications. John Willey & Sons, Inc., New York, 993. Oldham, K.B., Spanier, J. The Fractional Calculus. Academic Press, New York, 974. Atanacković, T. M., Pilipović, S., Stanković, B., Zorica, D. Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles. Wiley-ISTE, London, 204. Sanja Konjik (Uni Novi Sad) Foundations of fractional calculus Novi Sad, Feb 6-8, 205 22 / 22