lecture 10: anomalous diffusion outline: generalized diffusion equation subdiffusion superdiffusion...
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Lecture 10: Anomalous diffusion
Outline:• generalized diffusion equation• subdiffusion• superdiffusion• fractional Wiener process
anomalous diffusionRecall derivation of Fokker-Planck equation:
anomalous diffusionRecall derivation of Fokker-Planck equation:
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
anomalous diffusionRecall derivation of Fokker-Planck equation:
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
But what if ?
€
s2r(x,s)ds∫ = ∞
anomalous diffusionRecall derivation of Fokker-Planck equation:
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
But what if ?
And what if the distribution of time steps has infinite mean?
€
s2r(x,s)ds∫ = ∞
anomalous diffusionRecall derivation of Fokker-Planck equation:
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
But what if ?
And what if the distribution of time steps has infinite mean?
Go back and reformulate the problem: €
s2r(x,s)ds∫ = ∞
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
€
η(x, t) = d ′ x d ′ t ψ (x − ′ x , t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
€
η(x, t) = d ′ x d ′ t ψ (x − ′ x , t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫= d ′ x r(x − ′ x ) d ′ t w(t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
€
η(x, t) = d ′ x d ′ t ψ (x − ′ x , t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫= d ′ x r(x − ′ x ) d ′ t w(t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫
€
P(x, t) = d ′ t 1− d ′ ′ t w( ′ ′ t )0
t− ′ t
∫[ ]0
t
∫ η ( ′ x , ′ t )
Then
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
€
η(x, t) = d ′ x d ′ t ψ (x − ′ x , t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫= d ′ x r(x − ′ x ) d ′ t w(t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫
€
P(x, t) = d ′ t 1− d ′ ′ t w( ′ ′ t )0
t− ′ t
∫[ ]0
t
∫ η ( ′ x , ′ t )
Then
______________ prob to survive from t’ to t without a jump
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
r(x) =1
4πξ 2exp −
x 2
ξ 2
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ r(k) = exp −k 2ξ 2
( ) ≈1− k 2ξ 2
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
r(x) =1
4πξ 2exp −
x 2
ξ 2
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ r(k) = exp −k 2ξ 2
( ) ≈1− k 2ξ 2
€
P(k,s) ≈sτ
s
1
1− 1− (kξ )2( ) 1− sτ( )
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
r(x) =1
4πξ 2exp −
x 2
ξ 2
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ r(k) = exp −k 2ξ 2
( ) ≈1− k 2ξ 2
€
P(k,s) ≈sτ
s
1
1− 1− (kξ )2( ) 1− sτ( )
≈1
s + ξ 2 τ( )k 2
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
r(x) =1
4πξ 2exp −
x 2
ξ 2
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ r(k) = exp −k 2ξ 2
( ) ≈1− k 2ξ 2
€
P(k,s) ≈sτ
s
1
1− 1− (kξ )2( ) 1− sτ( )
≈1
s + ξ 2 τ( )k 2=
1
s + Dk 2
Fourier-Laplace inversion
2 ways: (D = 1)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
(D = 1)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
€
P(k,s) =1
s + k 2⇒ P(k, t) = exp −k 2t( )
(D = 1)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
€
P(k,s) =1
s + k 2⇒ P(k, t) = exp −k 2t( )
P(x, t) =ds
2π∫ exp −ikx − k 2t( )
(D = 1)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
€
P(k,s) =1
s + k 2⇒ P(k, t) = exp −k 2t( )
P(x, t) =ds
2π∫ exp −ikx − k 2t( )
= exp −x 2
4 t
⎛
⎝ ⎜
⎞
⎠ ⎟
ds
2π∫ exp −ikx − k 2t +
x 2
4t
⎛
⎝ ⎜
⎞
⎠ ⎟
(D = 1)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
€
P(k,s) =1
s + k 2 ⇒ P(k, t) = exp −k 2t( )
P(x, t) =ds
2π∫ exp −ikx − k 2t( )
= exp −x 2
4 t
⎛
⎝ ⎜
⎞
⎠ ⎟
ds
2π∫ exp −ikx − k 2t +
x 2
4t
⎛
⎝ ⎜
⎞
⎠ ⎟
=1
4πtexp −
x 2
4 t
⎛
⎝ ⎜
⎞
⎠ ⎟
(D = 1)
other way:
2. Invert the Fourier transform first:
other way:
2. Invert the Fourier transform first:
€
P(k,s) =1
s + k 2⇒ P(x,s) =
1
2 sexp − x s( )
other way:
2. Invert the Fourier transform first:
€
P(k,s) =1
s + k 2⇒ P(x,s) =
1
2 sexp − x s( )
P(x, t) =ds
2πi∫ 1
2 sexp − x s + st( )
other way:
2. Invert the Fourier transform first:
€
P(k,s) =1
s + k 2⇒ P(x,s) =
1
2 sexp − x s( )
P(x, t) =ds
2πi∫ 1
2 sexp − x s + st( )
=du
2πexp −i x u − u2t( )∫ iu = s( )
other way:
2. Invert the Fourier transform first:
€
P(k,s) =1
s + k 2⇒ P(x,s) =
1
2 sexp − x s( )
P(x, t) =ds
2πi∫ 1
2 sexp − x s + st( )
=du
2πexp −i x u − u2t( )∫ iu = s( )
=1
4πtexp −
x 2
4 t
⎛
⎝ ⎜
⎞
⎠ ⎟
anomalous diffusion:
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)long waiting times:
anomalous diffusion:
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
anomalous diffusion:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
=>
anomalous diffusion:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
≈(sτ )α
s
1
1− 1− (kξ )σ( ) 1− (sτ )α
( )
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
=>
anomalous diffusion:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
≈(sτ )α
s
1
1− 1− (kξ )σ( ) 1− (sτ )α
( )
≈1
s1−α
1
sα + ξ σ τ α( )kσ
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
=>
anomalous diffusion:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
≈(sτ )α
s
1
1− 1− (kξ )σ( ) 1− (sτ )α
( )
≈1
s1−α
1
sα + ξ σ τ α( )kσ
=1
s1−α
1
sα + ˜ D kσ
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
=>
Subdiffusion: long wait time distribution
€
P(k,s) =1
s1−α
1
sα + k 2
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
=1
2tα / 2
du
2πiexp −i x / tα / 2
( )uα / 2 + u( )∫ u = st( )
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
=1
2tα / 2
du
2πiexp −i x / tα / 2
( )uα / 2 + u( )∫ u = st( )
=1
tα / 2f x / tα / 2( )
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
=1
2tα / 2
du
2πiexp −i x / tα / 2
( )uα / 2 + u( )∫ u = st( )
=1
tα / 2f x / tα / 2( )
€
x 2(t) = x 2P(x)dx = tα∫ y 2 f (y)dy∫
Subdiffusion: long wait time distribution
Invert Fourier transform first:
α < 1: subdiffusion
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
=1
2tα / 2
du
2πiexp −i x / tα / 2
( )uα / 2 + u( )∫ u = st( )
=1
tα / 2f x / tα / 2( )
€
x 2(t) = x 2P(x)dx = tα∫ y 2 f (y)dy∫
long-tailed jump distribution:(α = 1, σ < 2)
€
P(k,s) =1
s + kσ
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ ) Sσ = stable distribution
of order σ
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ (superdiffusion: faster than √t ),
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ (superdiffusion: faster than √t ), but <x2(t)> = ∞.
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ (superdiffusion: faster than √t ), but <x2(t)> = ∞.fractional moments:
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ (superdiffusion: faster than √t ), but <x2(t)> = ∞.
€
x λ (t) = x λ P(x)dx = t λ /σ∫ y λ f (y)dy∫ < ∞, λ < σ
fractional moments:
Fractional Wiener process
For an ordinary Wiener process,
€
x 2(t) = σ 2t
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
Consider
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
€
x(t) = C d ′ t 1
(t − ′ t )a0
t
∫ ξ ( ′ t ); ξ (t)ξ ( ′ t ) = δ(t − ′ t ), a < 12
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
Consider
Then
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
€
x(t) = C d ′ t 1
(t − ′ t )a0
t
∫ ξ ( ′ t ); ξ (t)ξ ( ′ t ) = δ(t − ′ t ), a < 12
€
x 2(t) = σ 2C2 d ′ t 1
(t − ′ t )2a0
t
∫ =σ 2C2
1− 2at1−2a ⇒ a = 1
2 (1− H)
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
Consider
Then
Laplace-transformed:
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
€
x(t) = C d ′ t 1
(t − ′ t )a0
t
∫ ξ ( ′ t ); ξ (t)ξ ( ′ t ) = δ(t − ′ t ), a < 12
€
x 2(t) = σ 2C2 d ′ t 1
(t − ′ t )2a0
t
∫ =σ 2C2
1− 2at1−2a ⇒ a = 1
2 (1− H)
€
x(s) = C dt t−ae−st
0
∞
∫[ ]ξ (s) = C ⋅Γ(1− a)
s1−a⋅ξ (s)
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
Consider
Then
Laplace-transformed:
so choose
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
€
x(t) = C d ′ t 1
(t − ′ t )a0
t
∫ ξ ( ′ t ); ξ (t)ξ ( ′ t ) = δ(t − ′ t ), a < 12
€
x 2(t) = σ 2C2 d ′ t 1
(t − ′ t )2a0
t
∫ =σ 2C2
1− 2at1−2a ⇒ a = 1
2 (1− H)
€
x(s) = C dt t−ae−st
0
∞
∫[ ]ξ (s) = C ⋅Γ(1− a)
s1−a⋅ξ (s)
€
C =1
Γ(1− a)
fractional derivatives
€
x(s) =1
s1−aξ (s)
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t)
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
€
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟(1−a )
x(t) = ξ (t)or
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
€
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟(1−a )
x(t) = ξ (t)
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−a
˙ x (t) = ξ (t)
or
or
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
€
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟(1−a )
x(t) = ξ (t)
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−a
˙ x (t) = ξ (t)
1
Γ(a)d ′ t
1
(t − ′ t )1−a0
t
∫ ˙ x ( ′ t ) = ξ (t)
or
i.e.,
or
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
€
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟(1−a )
x(t) = ξ (t)
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−a
˙ x (t) = ξ (t)
1
Γ(a)d ′ t
1
(t − ′ t )1−a0
t
∫ ˙ x ( ′ t ) = ξ (t)
or
i.e.,
or
nonlocal!