Fractional Diffusion Theory and Applications Part II

Transcription

1 Fractional Diffusion Theory and Applications Part II p. 1/2 Fractional Diffusion Theory and Applications Part II 22nd Canberra International Physics Summer School 28 Bruce Henry (Trevor Langlands, Peter Straka, Susan Wearne, Claire Delides) School of Mathematics and Statistics The University of New South Wales Sydney NSW 252 Australia

2 Fractional Diffusion Theory and Applications Part II p. 2/2 Fractional Calculus One can ask what would be a differential having as its exponent a fraction. Although this seems removed from Geometry... it appears that one day these paradoxes will yield useful consequences. Gottfried Leibniz (1695) There are different ways to define fractional derivatives, all based on generalizing known results in the ordinary calculus. Excellent reference books Oldham & Spanier (1974), Miller & Ross (1993), Podlubny (1999)

3 Fractional Diffusion Theory and Applications Part II p. 3/2 Introduction to Fractional Calculus Consider ordinary derivatives of f(x) = x p, n N

4 Fractional Diffusion Theory and Applications Part II p. 3/2 Introduction to Fractional Calculus Consider ordinary derivatives of f(x) = x p, n N d n f dx n = p(p 1)...(p n 1)x p n = p! (p n)! xp n

5 Fractional Diffusion Theory and Applications Part II p. 3/2 Introduction to Fractional Calculus Consider ordinary derivatives of f(x) = x p, n N d n f dx n = p(p 1)...(p n 1)x p n = p! (p n)! xp n = Γ(p + 1)xp n Γ(p n + 1) where Γ(n + 1) = n! n N

6 Fractional Diffusion Theory and Applications Part II p. 3/2 Introduction to Fractional Calculus Consider ordinary derivatives of f(x) = x p, n N d n f dx n = p(p 1)...(p n 1)x p n = = Γ(p + 1)xp n Γ(p n + 1) But Γ(α + 1) = p! (p n)! xp n where Γ(n + 1) = n! n N e t t α dt α R +

7 Fractional Diffusion Theory and Applications Part II p. 3/2 Introduction to Fractional Calculus Consider ordinary derivatives of f(x) = x p, n N d n f dx n = p(p 1)...(p n 1)x p n = = Γ(p + 1)xp n Γ(p n + 1) But Γ(α + 1) = p! (p n)! xp n where Γ(n + 1) = n! n N e t t α dt α R + Define dα f Γ(p + 1)xp α = dxα Γ(p α + 1) α >

8 Fractional Diffusion Theory and Applications Part II p. 4/2 Riemann-Liouville Fractional Integral Consider the n fold integral (n N) d n f(x) dx n = = x 1 Γ(n) ( xn 1 x... ( x2 ( x1 ) ) f(x )dx dx 1 f(y) dy Cauchy formula (x y) n+1 )... dx

9 Fractional Diffusion Theory and Applications Part II p. 4/2 Riemann-Liouville Fractional Integral Consider the n fold integral (n N) d n f(x) dx n = = x 1 Γ(n) ( xn 1 x... ( x2 ( x1 ) ) f(x )dx dx 1 f(y) dy Cauchy formula (x y) n+1 )... dx Riemann-Liouville fractional integral D q x = d q f(x) dx q = 1 Γ(q) x f(y) dy q R+ (x y) q+1

10 Fractional Diffusion Theory and Applications Part II p. 4/2 Riemann-Liouville Fractional Integral Consider the n fold integral (n N) d n f(x) dx n = = x 1 Γ(n) ( xn 1 x... ( x2 ( x1 ) ) f(x )dx dx 1 f(y) dy Cauchy formula (x y) n+1 )... dx Riemann-Liouville fractional integral D q x = d q f(x) dx q = 1 Γ(q) x f(y) dy q R+ (x y) q+1 Improper for q < 1 but still converges for < q < 1.

11 Fractional Diffusion Theory and Applications Part II p. 4/2 Riemann-Liouville Fractional Integral Consider the n fold integral (n N) d n f(x) dx n = = x 1 Γ(n) ( xn 1 x... ( x2 ( x1 ) ) f(x )dx dx 1 f(y) dy Cauchy formula (x y) n+1 )... dx Riemann-Liouville fractional integral D q x = d q f(x) dx q = 1 Γ(q) x f(y) dy q R+ (x y) q+1 Improper for q < 1 but still converges for < q < 1. Diverges for q so it cannot define for a fractional derivative.

12 Fractional Diffusion Theory and Applications Part II p. 5/2 Memory Interpretation The R-L fractional integral w.r.t. x defines a weighted average of the function over the domain [,x] using a power law weighting function.

13 Fractional Diffusion Theory and Applications Part II p. 5/2 Memory Interpretation The R-L fractional integral w.r.t. x defines a weighted average of the function over the domain [,x] using a power law weighting function. Example f(x) = x d 1 2x dx 1 2 = 1 Γ( 1 2 ) x y (x y) 1 2 dy = 4 3 x 3 2 π d 1 2e x dx 1 2 = erf( x)e x erf( x) = 2 π x e y2 dy

14 Fractional Diffusion Theory and Applications Part II p. 6/2 Geometric Interpretation (Podlubny, 21-27) Consider g(y) = 1 Γ(q + 1) (xq (x y) q )

15 Fractional Diffusion Theory and Applications Part II p. 6/2 Geometric Interpretation (Podlubny, 21-27) Consider g(y) = 1 Γ(q + 1) (xq (x y) q ) Plot g(y) versus y for < y < x.

16 Fractional Diffusion Theory and Applications Part II p. 6/2 Geometric Interpretation (Podlubny, 21-27) Consider g(y) = 1 Γ(q + 1) (xq (x y) q ) Plot g(y) versus y for < y < x. For each y along this curve construct a fence with a height f(y).

17 Fractional Diffusion Theory and Applications Part II p. 6/2 Geometric Interpretation (Podlubny, 21-27) Consider g(y) = 1 Γ(q + 1) (xq (x y) q ) Plot g(y) versus y for < y < x. For each y along this curve construct a fence with a height f(y). The standard integral x f(y)dy is the area of the projection of this fence onto the (y,f(y)) plane

18 Fractional Diffusion Theory and Applications Part II p. 6/2 Geometric Interpretation (Podlubny, 21-27) Consider g(y) = 1 Γ(q + 1) (xq (x y) q ) Plot g(y) versus y for < y < x. For each y along this curve construct a fence with a height f(y). The standard integral x f(y)dy is the area of the projection of this fence onto the (y,f(y)) plane The fractional integral D q x f is the area of the projection of the fence onto the (g(y),f(y)) plane.

19 Fractional Diffusion Theory and Applications Part II p. 7/2 Riemann-Liouville Fractional Derivative The ordinary derivative of a fractional integral ( ) Dxf(x) q = dq f(x) dx q = dn d (n q) f(x) dx n dx (n q) q R +, n = q +1

20 Fractional Diffusion Theory and Applications Part II p. 7/2 Riemann-Liouville Fractional Derivative The ordinary derivative of a fractional integral ( ) Dxf(x) q = dq f(x) dx q = dn d (n q) f(x) dx n dx (n q) q R +, n = q +1 Examples d 1 2 x p dx 1 2 = d dx! d 1 2 x p dx 1 2 = d dx 1 Γ( 1 2 ) Z x y p (x y) 1 2 dy! = Γ(p + 1)xp 1 2 Γ(p )

21 Fractional Diffusion Theory and Applications Part II p. 7/2 Riemann-Liouville Fractional Derivative The ordinary derivative of a fractional integral ( ) Dxf(x) q = dq f(x) dx q = dn d (n q) f(x) dx n dx (n q) q R +, n = q +1 Examples d 1 2 x p dx 1 2 = d dx! d 1 2 x p dx 1 2 = d dx 1 Γ( 1 2 ) Z x y p (x y) 1 2 dy! = Γ(p + 1)xp 1 2 Γ(p ) 1 2 D 1 2 x e x = x Γ( 1 2 ) 1 F 1 (1, 1 2,x)

22 Fractional Diffusion Theory and Applications Part II p. 7/2 Riemann-Liouville Fractional Derivative The ordinary derivative of a fractional integral ( ) Dxf(x) q = dq f(x) dx q = dn d (n q) f(x) dx n dx (n q) q R +, n = q +1 Examples d 1 2 x p dx 1 2 = d dx! d 1 2 x p dx 1 2 = d dx 1 Γ( 1 2 ) Z x y p (x y) 1 2 dy! = Γ(p + 1)xp 1 2 Γ(p ) 1 2 D 1 2 x e x = x Γ( 1 2 ) 1 F 1 (1, 1 2,x) D α x (constant) = x α Γ(1 α) (constant)

23 Fractional Diffusion Theory and Applications Part II p. 8/2 Tautochrone Problem Find the shape of wire x(y) such that the time of descent τ of a frictionless bead falling under gravity is a constant independent of the starting point.

24 Fractional Diffusion Theory and Applications Part II p. 8/2 Tautochrone Problem Find the shape of wire x(y) such that the time of descent τ of a frictionless bead falling under gravity is a constant independent of the starting point. 1 2 v2 = g(h y) v = 2g(h y) conservation of energy

25 Fractional Diffusion Theory and Applications Part II p. 8/2 Tautochrone Problem Find the shape of wire x(y) such that the time of descent τ of a frictionless bead falling under gravity is a constant independent of the starting point. 1 2 v2 = g(h y) v = 2g(h y) conservation of energy ( ) ( ) 2 v = ds 1 + dx dy dy dt = arc length parameterization dt

26 Fractional Diffusion Theory and Applications Part II p. 8/2 Tautochrone Problem Find the shape of wire x(y) such that the time of descent τ of a frictionless bead falling under gravity is a constant independent of the starting point. 1 2 v2 = g(h y) v = 2g(h y) conservation of energy ( ) ( ) 2 v = ds 1 + dx dy dy dt = arc length parameterization dt τ 2g dt = h ) 2 ( 1 + dx dy dy = h y h f(y) (h y) 1 2 dy

27 Fractional Diffusion Theory and Applications Part II p. 9/2 Abel s Solution (1823) 2gτ = h f(y) (h y) 1 2 dy

28 Fractional Diffusion Theory and Applications Part II p. 9/2 Abel s Solution (1823) 2gτ = h f(y) (h y) 1 2 dy = π D 1 2 h f(h)

29 Fractional Diffusion Theory and Applications Part II p. 9/2 Abel s Solution (1823) 2gτ = h f(y) (h y) 1 2 dy = π D 1 2 h f(h) D h 2gτ = π D 2 h D 1 2 h f(h) = πf(h)

30 Fractional Diffusion Theory and Applications Part II p. 9/2 Abel s Solution (1823) 2gτ = h f(y) (h y) 1 2 dy = π D 1 2 h D h 2gτ = π D 2 ( ) 1 2gτ = πf(h) π h f(h) h D 1 2 h f(h) = πf(h)

31 Fractional Diffusion Theory and Applications Part II p. 9/2 Abel s Solution (1823) 2gτ = h f(y) (h y) 1 2 dy = π D 1 2 h D h 2gτ = π D 2 ( ) 1 2gτ = πf(h) π h f(y) = 2g π f(h) h D 1 2 h τ 2 y f(h) = πf(h)

32 Fractional Diffusion Theory and Applications Part II p. 9/2 Abel s Solution (1823) 2gτ = h f(y) (h y) 1 2 dy = π D 1 2 h D h 2gτ = π D 2 ( ) 1 2gτ = πf(h) π h f(y) = 2g π f(h) h D 1 2 h τ 2 y f(h) = πf(h) q dx dy = f 2 (y) 1 = s 2gτ 2 π 2 y 1 8 < : x = a(θ + sin θ) y = a(1 cos θ) cycloid

33 Fractional Diffusion Theory and Applications Part II p. 1/2 Surfing Cycloids

34 Fractional Diffusion Theory and Applications Part II p. 11/2 Basic Properties of Fractional Calculus The Riemann-Liouville fractional derivative D q xf(x) satisfies D xf(x) = f(x) identity property D q xf(x) = f(x) is an ordinary derivative if q N Dx q [af(x) + bg(x)] = adxf(x) q + bdxg(x) q linearity property Dx[f(x)g(x)] α = α Dx m [f(x)]dx α m [g(x)] Leibniz rule m m=

35 Fractional Diffusion Theory and Applications Part II p. 11/2 Basic Properties of Fractional Calculus The Riemann-Liouville fractional derivative D q xf(x) satisfies D xf(x) = f(x) identity property D q xf(x) = f(x) is an ordinary derivative if q N Dx q [af(x) + bg(x)] = adxf(x) q + bdxg(x) q linearity property Dx[f(x)g(x)] α = α Dx m [f(x)]dx α m [g(x)] Leibniz rule m m= The Riemann-Liouville fractional integral Dx q f(x) q > satisfies the above together with D q x ( D p x f(x) ) = Dx q p f(x) semi-group property

36 Fractional Diffusion Theory and Applications Part II p. 12/2 Fourier and Laplace Transforms and Ordinary Calculus F denotes a Fourier transform with Fourier variable q L denotes a Laplace transform with Laplace variable u

37 Fractional Diffusion Theory and Applications Part II p. 12/2 Fourier and Laplace Transforms and Ordinary Calculus F denotes a Fourier transform with Fourier variable q L denotes a Laplace transform with Laplace variable u F (y(x)) = ŷ(q) = + e iqx y(x)dx, y(x) = 1 2π + e iqx ŷ(q)dx L(y(t)) = ŷ(u) = e ut y(t)dt, y(t) = c+i c i e ut ŷ(u)du

38 Fractional Diffusion Theory and Applications Part II p. 12/2 Fourier and Laplace Transforms and Ordinary Calculus F denotes a Fourier transform with Fourier variable q L denotes a Laplace transform with Laplace variable u F (y(x)) = ŷ(q) = + e iqx y(x)dx, y(x) = 1 2π + e iqx ŷ(q)dx L(y(t)) = ŷ(u) = e ut y(t)dt, y(t) = c+i c i e ut ŷ(u)du ( ) ( dy(x) d 2 ) y(x) F = iqŷ(q), F dx dx 2 = q 2 ŷ(q) ( ) ( dy(t) d 2 ) y(t) L = uŷ(u) y(), L dt dt 2 = u 2 ŷ(u) uy() y ()

39 Fractional Diffusion Theory and Applications Part II p. 13/2 Fourier and Laplace Transforms and Fractional Calculus Riemann-Liouville D α t y(t) = d dt ( 1 Γ(α) t ) y(s) ds (t s) 1 α < α < 1 L(D α t y(t)) = u α ŷ(u) ( D α 1 t y(t) ) t= F (D α xy(x)) = (iq) α ŷ(q)

40 Fractional Diffusion Theory and Applications Part II p. 13/2 Fourier and Laplace Transforms and Fractional Calculus Riemann-Liouville D α t y(t) = d dt ( 1 Γ(α) t ) y(s) ds (t s) 1 α < α < 1 Grunwald-Letnikov L(D α t y(t)) = u α ŷ(u) ( D α 1 t y(t) ) t= F (D α xy(x)) = (iq) α ŷ(q) Dt α 1 y(t) = lim h h α ( 1) j Γ(α + 1) y(t jh) Γ(j + 1)Γ(α j + 1) j= L(D α t y(t)) = u α ŷ(u)

41 Fractional Diffusion Theory and Applications Part II p. 14/2 Fourier and Laplace Transforms and Fractional Calculus Caputo D α t y(t) = ( 1 Γ(1 α) t d ds y(s) ) (t s) α ds L(D α t y(t)) = u α ŷ(u) ( u α 1 y() ) < α < 1

42 Fractional Diffusion Theory and Applications Part II p. 14/2 Fourier and Laplace Transforms and Fractional Calculus Caputo Riesz D α t y(t) = ( 1 Γ(1 α) t d ds y(s) ) (t s) α ds L(D α t y(t)) = u α ŷ(u) ( u α 1 y() ) < α < 1 α x y(x) = 1 2cos( πα 2 ) ( D α xy(x) + x D α y(x)), 1 < α < 2 Dx α and x D α are R-L-Weyl fractional derivatives. ) F ( α x y(x) = q α ŷ(q)

43 Fractional Diffusion Theory and Applications Part II p. 15/2 Special Functions for Fractional Calculus Mittag-Leffler Function E α (z) = k= z k Γ(αk + 1) α >

44 Fractional Diffusion Theory and Applications Part II p. 15/2 Special Functions for Fractional Calculus Mittag-Leffler Function L E α (z) = k= ( E α ( ( t ) τ )α ) z k Γ(αk + 1) = α > 1 α > u + u1 α τ α

45 Fractional Diffusion Theory and Applications Part II p. 15/2 Special Functions for Fractional Calculus Mittag-Leffler Function L E α (z) = k= ( E α ( ( t ) τ )α ) E 1 (z) = k= z k Γ(αk + 1) = α > 1 α > u + u1 α τ α z k Γ(k + 1) = k= z k k! = ez

46 Fractional Diffusion Theory and Applications Part II p. 15/2 Special Functions for Fractional Calculus Mittag-Leffler Function L E α (z) = k= ( E α ( ( t ) τ )α ) E 1 (z) = E α ( ( t τ )α ) exp k= ( z k Γ(αk + 1) = α > 1 α > u + u1 α τ α z k Γ(k + 1) = 1 Γ(1 + α) ( t τ k= ) α ) z k k! = ez t τ, < α < 1

47 Fractional Diffusion Theory and Applications Part II p. 15/2 Special Functions for Fractional Calculus Mittag-Leffler Function L E α (z) = k= ( E α ( ( t ) τ )α ) E 1 (z) = E α ( ( t τ )α ) exp E α ( ( t τ )α ) k= ( 1 Γ(1 α) z k Γ(αk + 1) α > 1 = α > u + u1 α τ α z k Γ(k + 1) = z k k! = ez k= ( ) 1 t α ) t τ, < α < 1 Γ(1 + α) τ ( ) t α t τ, < α < 1 τ

48 Fractional Diffusion Theory and Applications Part II p. 16/2 Generalized Mittag-Leffler Function E α,β (z) = k= z k Γ(αk + β) α >, β >

49 Fractional Diffusion Theory and Applications Part II p. 16/2 Generalized Mittag-Leffler Function E α,β (z) = k= z k Γ(αk + β) α >, β > E 1,1 (z) = e z

50 Fractional Diffusion Theory and Applications Part II p. 16/2 Generalized Mittag-Leffler Function E α,β (z) = k= z k Γ(αk + β) α >, β > E 1,1 (z) = e z E 1,2 (z) = ez 1 z

51 Fractional Diffusion Theory and Applications Part II p. 16/2 Generalized Mittag-Leffler Function E α,β (z) = k= z k Γ(αk + β) α >, β > E 1,1 (z) = e z E 1,2 (z) = ez 1 z E 2,2 (z) = sinh( z) z

52 Fractional Diffusion Theory and Applications Part II p. 16/2 Generalized Mittag-Leffler Function E α,β (z) = k= z k Γ(αk + β) α >, β > L E 1,1 (z) = e z E 1,2 (z) = ez 1 z E 2,2 (z) = sinh( z) z ( ) t αk+β 1 E (k) α,β (±atα ) = k!uα β (u α a) k+1 (k) denotes the kth derivative.

53 Fractional Diffusion Theory and Applications Part II p. 17/2 Fractional Differential Equation Example D 1 2 t y(t) = y(t), D 1 2 t y(t) = C t=

54 Fractional Diffusion Theory and Applications Part II p. 17/2 Fractional Differential Equation Example D 1 2 t y(t) = y(t), D 1 2 t y(t) = C t= L ( ) D 1 2 t y(t) = L(y(t))

55 Fractional Diffusion Theory and Applications Part II p. 17/2 Fractional Differential Equation Example D 1 2 t y(t) = y(t), D 1 2 t L ( ) D 1 2 t y(t) u 1 2ŷ(u) D 1 2 t y(t) = C t= = L(y(t)) y(t) = ŷ(u) t=

56 Fractional Diffusion Theory and Applications Part II p. 17/2 Fractional Differential Equation Example D 1 2 t y(t) = y(t), D 1 2 t L ( ) D 1 2 t y(t) u 1 2ŷ(u) D 1 2 t ŷ(u) = y(t) = C t= = L(y(t)) y(t) = ŷ(u) t= C u 1 2 1

57 Fractional Diffusion Theory and Applications Part II p. 17/2 Fractional Differential Equation Example D 1 2 t y(t) = y(t), D 1 2 t L ( ) D 1 2 t y(t) u 1 2ŷ(u) D 1 2 t ŷ(u) = y(t) = C t= = L(y(t)) y(t) = ŷ(u) t= C u ( ) C y(t) = L 1 u = t 1 2E1 ( t) 2,1 2

58 Fractional Diffusion Theory and Applications Part II p. 18/2 Fox H Function Fox (1961) H m,n p,q (z) 1 2πi = H m,n p,q n k=1 Γ(1 a j + A j ζ) m j=1 Γ(b j B j ζ) q j=m+1 Γ(1 b j + B j ζ) p j=n+1 Γ(a j A j ζ) zζ dζ (a z 1,A 1 )... (a p,a p ) (b 1,B 1 )... (b q,b q ) C

59 Fractional Diffusion Theory and Applications Part II p. 18/2 Fox H Function Fox (1961) H m,n p,q (z) 1 2πi = H m,n p,q n k=1 Γ(1 a j + A j ζ) m j=1 Γ(b j B j ζ) q j=m+1 Γ(1 b j + B j ζ) p j=n+1 Γ(a j A j ζ) zζ dζ (a z 1,A 1 )... (a p,a p ) (b 1,B 1 )... (b q,b q ) C H 1,,1 z (b,1) = z b e z

60 Fractional Diffusion Theory and Applications Part II p. 18/2 Fox H Function Fox (1961) H m,n p,q (z) 1 2πi = H m,n p,q n k=1 Γ(1 a j + A j ζ) m j=1 Γ(b j B j ζ) q j=m+1 Γ(1 b j + B j ζ) p j=n+1 Γ(a j A j ζ) zζ dζ (a z 1,A 1 )... (a p,a p ) (b 1,B 1 )... (b q,b q ) C H 1,1 1,2 z H 1,,1 z (,1) (b,1) (,1),(1 β,α) = z b e z = E α,β ( z)

61 Fractional Diffusion Theory and Applications Part II p. 19/2 L tω Hp,q m,n = u ω 1 H m+1,n p,q+1 zt σ (a p,a p ) (b q,b q ) zu σ (a p,a p ) (1 + ω,σ)(b q,b q )

62 Fractional Diffusion Theory and Applications Part II p. 19/2 L tω Hp,q m,n = u ω 1 H m+1,n p,q+1 zt σ (a p,a p ) (b q,b q ) zu σ (a p,a p ) (1 + ω,σ)(b q,b q ) D ν z zα Hp,q m,n = z α ν H m,n+1 p+1,q+1 (az) β (a p,a p ) (b q,b q ) (az) β ( α,β), (a p,a p ) (b q,b q )(ν α,β)

63 Fractional Diffusion Theory and Applications Part II p. 2/2 A Levy stable law can be expressed in closed form in terms of MANU Fox H functions (Schneider 1986) F 1 (exp( D α t q α )) = 1 α x H1,1 2,2 x (D α t) 1 α (1, 1 α ), (1, 1 2 ) (1,1)(1, 1 2 ) E T MENTE

64 Fractional Diffusion Theory and Applications Part II p. 2/2 A Levy stable law can be expressed in closed form in terms of MANU Fox H functions (Schneider 1986) F 1 (exp( D α t q α )) = 1 α x H1,1 2,2 x (D α t) 1 α (1, 1 α ), (1, 1 2 ) (1,1)(1, 1 2 ) E T MENTE To evaluate Fox H functions see Metzler and Klafter, Physics Reports (2)

65 Fractional Diffusion Theory and Applications Part II p. 21/2 Fractional Calculus for Fractals? West, Bologna, Grigolini, (23) Functions with fractal graphs are not (integer) differentiable

66 Fractional Diffusion Theory and Applications Part II p. 21/2 Fractional Calculus for Fractals? West, Bologna, Grigolini, (23) Functions with fractal graphs are not (integer) differentiable Functions with fractal graphs may be fractionally differentiable

67 Fractional Diffusion Theory and Applications Part II p. 21/2 Fractional Calculus for Fractals? West, Bologna, Grigolini, (23) Functions with fractal graphs are not (integer) differentiable Functions with fractal graphs may be fractionally differentiable The Mandelbrot-Weierstrass function W(t) is everywhere continuous, nowhere differentiable, but fractionally differentiable for orders < 2 D, fractionally integrable for orders < D 1.

68 Fractional Diffusion Theory and Applications Part II p. 21/2 Fractional Calculus for Fractals? West, Bologna, Grigolini, (23) Functions with fractal graphs are not (integer) differentiable Functions with fractal graphs may be fractionally differentiable The Mandelbrot-Weierstrass function W(t) is everywhere continuous, nowhere differentiable, but fractionally differentiable for orders < 2 D, fractionally integrable for orders < D 1. Fractional calculus operators change the fractal dimension dim[dt α W(t)] = dim[w(t)] α dim[d α t W(t)] = dim[w(t)] + α

69 Fractional Diffusion Theory and Applications Part II p. 22/2 Fractional Diffusion Standard Diffusion X 2 (t) scales linearly with time. The probability density function is Gaussian The evolution equation for the probability density function is the diffusion equation Random walks occur at constant discrete time intervals.

70 Fractional Diffusion Theory and Applications Part II p. 22/2 Fractional Diffusion Standard Diffusion X 2 (t) scales linearly with time. The probability density function is Gaussian The evolution equation for the probability density function is the diffusion equation Random walks occur at constant discrete time intervals. Fractional Diffusion X 2 (t) does not scale linearly with time. Where does the anomalous scaling come from? What is the probability density function? What is the evolution equation for the probability density function? What are the properties of the random walks?

71 Fractional Diffusion Theory and Applications Part II p. 23/2 Diffusion on Fractals Random walks on self-similar fractal lattices with dimension d f r 2 t 2 dw, d w > 2 d w is the dimension of the walk.

72 Fractional Diffusion Theory and Applications Part II p. 23/2 Diffusion on Fractals Random walks on self-similar fractal lattices with dimension d f r 2 t 2 dw, d w > 2 d w is the dimension of the walk. Re-write r 2 D(r)t where D(r) = r 2 d w

73 Fractional Diffusion Theory and Applications Part II p. 23/2 Diffusion on Fractals Random walks on self-similar fractal lattices with dimension d f r 2 t 2 dw, d w > 2 d w is the dimension of the walk. Re-write r 2 D(r)t where D(r) = r 2 d w O Shaugnessy & Procaccia (1985) Consider radially symmetric diffusion equation with space dimension d d f and diffusion constant D D(r) c t = 1 r d f 1 r ( r d f 1 r 2 d c ) w r fractal diffusion equation

74 Fractional Diffusion Theory and Applications Part II p. 24/2 Fractional Brownian Motion Diffusion Process Consider a time dependent diffusion constant D D(t) = αt α 1 D c t = αtα 1 D 2 c x 2

75 Fractional Diffusion Theory and Applications Part II p. 24/2 Fractional Brownian Motion Diffusion Process Consider a time dependent diffusion constant D D(t) = αt α 1 D c t = αtα 1 D 2 c x 2 Solution non-markovian probability density function ( ) 1 c(x,t) = exp x2 4πDt α 4Dt α Rescaled Gaussian

76 Fractional Diffusion Theory and Applications Part II p. 24/2 Fractional Brownian Motion Diffusion Process Consider a time dependent diffusion constant D D(t) = αt α 1 D c t = αtα 1 D 2 c x 2 Solution non-markovian probability density function ( ) 1 c(x,t) = exp x2 4πDt α 4Dt α Anomalous subdiffision x 2 = Rescaled Gaussian x 2 c(x,t)dx = 2D t α = 2D t 2H, H Hurst exponent

77 Fractional Diffusion Theory and Applications Part II p. 24/2 Fractional Brownian Motion Diffusion Process Consider a time dependent diffusion constant D D(t) = αt α 1 D c t = αtα 1 D 2 c x 2 Solution non-markovian probability density function ( ) 1 c(x,t) = exp x2 4πDt α 4Dt α Anomalous subdiffision x 2 = Rescaled Gaussian x 2 c(x,t)dx = 2D t α = 2D t 2H, H Hurst exponent Note D(t) = D 1 α t (Γ(α)D ),

78 Fractional Diffusion Theory and Applications Part II p. 25/2 Fractional Brownian Motion Stochastic Process Mandelbrot and Van Ness (1968) Let B H (t) denote a fractional Brownian motion stochastic process with Hurst exponent H [,1] (i) E(B H (t)b H (s)) = 1 2 ( t 2H + s 2H t s 2H) (ii) B H (at) a H B H (t). (iii) Realizations x B (t) are continuous but nowhere differentiable x B (t) versus t is a fractal graph with dimension d = 2 H (iv) Standard Brownian motion corresponds to H = 1/2

79 Fractional Diffusion Theory and Applications Part II p. 26/2 Fractional Brownian Motion and Fractional Calculus Fractional Brownian motion is also generated by an evolution equation x(t) = x + Dt α F(t) where D α t is the RL fractional integral.

80 Fractional Diffusion Theory and Applications Part II p. 26/2 Fractional Brownian Motion and Fractional Calculus Fractional Brownian motion is also generated by an evolution equation x(t) = x + Dt α F(t) where D α t is the RL fractional integral. x(t) denotes the position of a random walker at time t given that it started at x

81 Fractional Diffusion Theory and Applications Part II p. 26/2 Fractional Brownian Motion and Fractional Calculus Fractional Brownian motion is also generated by an evolution equation x(t) = x + Dt α F(t) where D α t is the RL fractional integral. x(t) denotes the position of a random walker at time t given that it started at x F(t) is Gaussian white noise with F(t)F(s) = δ(t s).

82 Fractional Diffusion Theory and Applications Part II p. 26/2 Fractional Brownian Motion and Fractional Calculus Fractional Brownian motion is also generated by an evolution equation x(t) = x + Dt α F(t) where D α t is the RL fractional integral. x(t) denotes the position of a random walker at time t given that it started at x F(t) is Gaussian white noise with F(t)F(s) = δ(t s). Fractional Brownian motion x(t) x is a fractional integral, of order α, of white noise and standard Brownian motion is an ordinary integral of white noise.

83 Fractional Diffusion Theory and Applications Part II p. 27/2 Fractional Langevin Equation Wang (1992), Lutz (21) Dissipative memory kernel m dv t dt = F(t) m γ(t t )v(t )dt F(t) = F()F(t) = D α t α coloured noise

84 Fractional Diffusion Theory and Applications Part II p. 27/2 Fractional Langevin Equation Wang (1992), Lutz (21) Dissipative memory kernel m dv t dt = F(t) m γ(t t )v(t )dt F(t) = F()F(t) = D α t α coloured noise Equilibrium fluctuation-dissipation theorem F(t)F() = mk b Tγ(t) γ(t) = D α mk b T t α m dv dt = F(t) D α k B T D α 1 t v(t) < α < 1

85 Fractional Diffusion Theory and Applications Part II p. 27/2 Fractional Langevin Equation Wang (1992), Lutz (21) Dissipative memory kernel m dv t dt = F(t) m γ(t t )v(t )dt F(t) = F()F(t) = D α t α coloured noise Equilibrium fluctuation-dissipation theorem F(t)F() = mk b Tγ(t) γ(t) = D α mk b T t α m dv dt = F(t) D α k B T D α 1 t v(t) < α < 1 The probability density function for trajectories satisfies a fractional Brownian motion diffusion equation.