Foundations of fractional calculus

Transcript

1 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

2 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, / 22

3 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, / 22

4 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, / 22

5 Leonhard Euler ( ) in 738 gave a meaning to the derivative d α (x n ) dx α, for α / N Joseph Liouville ( ) in his papers in gave a solid foundation to the fractional calculus, which has undergone only slight changes since then. Georg Friedrich Bernhard Riemann ( ) 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, / 22

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

7 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, / 22

8 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, / 22

9 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, / 22

10 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, / 22

11 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, / 22

12 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, / 22

13 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, / 22

14 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

15 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, / 22

16 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 = D α t ( ) ( ) 0Dt β u = 0, 0Dt β 0Dt α u = 4 t 3 2, 0Dt α+β u = 4 t 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, / 22

17 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, / 22

18 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, / 22

19 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, / 22

20 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, / 22

21 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, / 22

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, / 22

23 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, / 22

24 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, / 22 a 2 t

25 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 Springer-Verlag, Wien and New York, 997. Podlubny, I. Fractional Differential Equations, volume 98 of Mathematics in Science and Engineering. Academic Press, San Diego, :255203(2pp), 200. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 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, / 22