Outline
Generalization of PDEs and finite propagation
speed effects
Axel Hutt
Department of Physics, University of Ottawa, Canada
2007
Axel Hutt
Generalization of PDEs and finite propagation speed effects
Outline
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
Outline
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
Outline
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
Let us assume a 1D spatial domain with characteristic spatial interaction
range ξ
∂V (x, t)
∂t
=
Z
∞
−∞
K (x − y )V (y , t) dy
→
∂ Ṽ (k , t)
= K̃ (k )Ṽ (k , t)
∂t
with spatial Fourier transforms K̃ (k ), Ṽ (k ) and interaction kernel K (x) .
1 −|x|/σ
For K (x) = 2σ
→ K̃ (k ) = 1+σ12 k 2 = 1 − σ 2 k 2 + σ 4 k 4 + · · ·
∂ Ṽ (k , t)
∂t
∂V (x, t)
→
∂t
=
K̃ (k )Ṽ (k , t) ≈ (1 − σ 2 k 2 )Ṽ (k )
=
V (x, t) + D
∂2
V (x, t) , D = σ 2
∂x 2
relation between integral-differential equation (IDE) and partial differential
equation (PDE)
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
K(x)
x
spatial domain
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
More general:
Z
∞
−∞
K (x − y )V (y ) dy
=
Kn
=
∞
X
∂V (x, t)
∂ 2 V (x, t)
∂nV
···
+ K2
(−1)n Kn n = K0 − K1
∂x
∂x
∂x 2
n=0
Z ∞
1
x n K (x) dx
n! −∞
If K (x) = K (−x) : K2n+1 = 0.
−|x|/σ
1
: K0 = 1, K2 = σ 2 , K4 = σ 4
For K (x) = 2σ
R∞
2
4
V (x,t)
V (x,t)
→ −∞ K (x − y )V (y ) dy = V + σ 2 ∂ ∂x
+ σ 4 ∂ ∂x
+ ···
2
4
If Km ≈ 0 for m > n, then n gives the order of spatial interaction.
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
The reaction-diffusion equation
∂V (x, t)
∂t
∂V (x, t)
∂t
=
h[V (x, t)] +
Z
∞
−∞
K (x − y )V (y , t) dy
=
h[V (x, t)] + K0 V (x, t) + K2
∂ 4 V (x, t)
∂ 2 V (x, t)
+
K
···
4
∂x 2
∂x 4
≈
h[V (x, t)] + K0 V (x, t) + K2
∂ 2 V (x, t)
(RD equation)
∂x 2
∂ Ṽ (k , t)
= h̃[Ṽ (k , t)] + (K0 − K2 k 2 )Ṽ (k )
∂t
P
n
with ∞
n=4 (−1) Kn Ṽ (k ) small compared to first terms
→ necessary condition : |Kn+2 k n+2 | < Kn k n for n ≥ 2 (convergence of series)
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
The reaction-diffusion equation
|Kn+2 k n+2 | < Kn k n → σ < k for exponential kernels
If there is a maximum wave number km , then the spatial range is limited to
σ < 1/km .
Reasons for maximum wave number:
coarse-grained spatial field with unit length 2π/km
characteristic spatial range ξ
phase transition at a critical wave number
RD equation:
V̇ = h̄[V ] + D
∂2V
for D < 1/km2
∂x 2
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
The Swift-Hohenberg equation
∂V (x, t)
∂t
=
=
h[V ] −
Z
∞
−∞
K (x − y )V (y , t) dy
aV (x, t) − bV 3 (x, t) − K0 V (x, t) − K2
∂ 2 V (x, t)
∂ 4 V (x, t)
− K4
∂x 2
∂x 4
re-scaling in space and time:
∂V (x, t)
∂t
=
„
«2
∂2
εV (x, t) − V 3 (x, t) − 1 +
V (x, t)
∂x 2
ε
=
4K4 (a − K0 )/K22 + 1
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
The Swift-Hohenberg equation
the necessary condition:
∂ n+2 V
∂nV
| < |Kn n | for all n ≥ 4.
n+2
∂x
∂x
For the Gaussian kernel :
2
2
1
n+2
K (x) = √
e−x /2σ : σ 2 <
km2
2πσ
|Kn+2
2
> 4th order (σ >4+2)
σ=2.3
σ=2.5
σ=1.8
2
(σk)
n
4th order (σ <4+2)
2
2nd order (σ <2+2)
1
0
2
4
6
order n
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
The Free Energy
the Swift-Hohenberg equation may be derived from an energy functional :
∂V (x, t)
∂t
=
−
F
=
Z
δF [V ]
δV
∞
−∞
dx
„
«2
1 4
∂2
V (x, t) + V (x, t) 1 +
V (x, t)
4
∂x 2
Considering the IDE, we obtain
Z
Z ∞
1 ∞
dy K (x − y )V (x, t)V (y , t)
F =
dx W [V (x, t)] +
2 −∞
−∞
Z ∞
∞
∂nV
1X
(−1)n Kn n
=
dx W [V (x, t)] +
2
∂x
−∞
n=0
„
«
Z ∞
∂ 2 V (x, t)
∂ 4 V (x, t)
1
=
K0 V (x, t) − K2
−
K
dx W [V (x, t)] +
4
2
∂x 2
∂x 4
−∞
with δW [V ]/δV = −h[V ]
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
further the Free Enery is a Lyapunov function:
«
„
Z ∞
dF [V (x, t)]
∂V 2
< 0
=−
dx
dt
∂t
−∞
Another model: The Kuramoto-Sivashinsky equation
Z ∞
“
”
∂V (x, t)
= h[V ] −
dy K (x − y )V 2 (y , t) + L(x − y )V (y , t)
∂t
−∞
→
∂V (x, t)
∂t
=
−ηV −
∂V 4 (x, t)
∂V (x, t)
∂ 2 V (x, t)
−
− V (x, t)
∂x 2
∂x 4
∂x
asymmetric spatial interaction K (x) = −K (−x) of order 1
symmetric spatial interaction L(x) = L(−x) of order 4
no Free Energy and no Lyapunov function known
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
The basic idea
Specific PDE-models
Importance of generalization
Importance of IDE formulation
bridge between two more or less distinct active reserach fields:
neural fields based on IDEs
pattern formation in physical/chemical/biological systems
based on PDEs
enrichment of fields by each other:
treatment of nonlocal interactions and propagation delays in
PDE-systems possible, well-studied in neural fields
nonlocal interactions e.g. modeled in plasma physics by
∂T ∂T
∂
∂T
χ(T ,
with nonlocal diffusivity χ
=
)
∂t
∂x
∂x ∂x
finite propagation speeds in diffusion systems, e.g.
Cattaneo-equation
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
The infinite propagation speed in diffusion systems
diffusion equation:
∂2ρ
∂x 2
∂ρ
∂t
=
D
initial condition: ρ(x, 0) = δ(x)
→
ρ(x, t) = √
Axel Hutt
2
1
e−x /4Dt
4πDt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
The Cattaneo-equation (Carlo Cattaneo, 1948)
The continuums equation in 1D:
∂ρ
∂t
=
Cattaneo: flux obeys Φ
=
∂ρ
∂t
=
→ parabolic PDE
∂Φ
∂x
∂ ∂ρ
∂ρ
+ κ2
−κ1
∂x
∂t ∂x
∂2ρ
∂ ∂2ρ
κ1 2 − κ2
∂x
∂t ∂x 2
−
Then
Φ
=
Φ
=
«
„
κ2 ∂
Φ
1+
κ1 ∂t
≈
1
1−
κ2 ∂
κ1 ∂t
→ hyperbolic PDE
Axel Hutt
«
„
κ2 ∂ ∂ρ
−κ1 1 −
κ1 ∂t ∂x
∂ρ
−κ1
∂x
−κ1
∂ρ
∂x
κ2 ∂ 2 ρ
∂ρ
∂2ρ
+
− κ1 2 = 0
κ1 ∂t 2
∂t
∂x
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
Extention of IDE by finite propagation delay
novel idea: considering finite propagation delay in IDE
K(x)
x
spatial domain
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
Extention of IDE by finite propagation delay
∂V (x, t)
∂t
=
h[V ] +
Z
∞
−∞
K (x − y )S[V (y , t −
|x − y |
)] dy
c
with nonlinear function S[V ] and propagation speed c.
Linear treatment with constant stationary state V0 and V (x, t) ∼ eλt+ikx
Z
δh
δS ∞
→λ =
+
dxK (x)e−λ|x|/c e−ikx
δV
δV −∞
Axel Hutt
Generalization of PDEs and finite propagation speed effects
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
PDEs are IDEs
Finite propagation delay
Summary
Extention of IDE by finite propagation delay
for large but finite propagation speeds, it is exp(−λ|x|/c) ≈ 1 − λ|x|/c and
λ+
1
Kn =
n!
Z
∞
λX
(−ik )n Pn
c
=
n=0
∞
dx x n K (x)
−∞
,
∞
δh
δS X
+
(−ik )n Kn
δV
δV
n=0
Z ∞
1
Pn =
dx |x|x n K (x)
n! −∞
In the case of a Gaussian kernel K (x) =
2
2
√ 1 e −x /2σ :
2πσ
Kn ∼ σ n , Pn ∼ σ n+1
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
Outline
1
PDEs are IDEs
The basic idea
Specific PDE-models
Importance of generalization
2
Finite propagation delay
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
3
Summary
Axel Hutt
Generalization of PDEs and finite propagation speed effects
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
PDEs are IDEs
Finite propagation delay
Summary
The RD equation
consideration of terms up to the third order of σ:
K0 = 1
r
2
σ
P0 =
π
c → ∞ : λ = −D 2 k 2
,
,
σ2
, K3 = 0
2
r
2 3
P1 = 0, P2 =
σ
π
K1 = 0, K2 =
field u(x,t)
0.4
-1
0
1
λ(k)
σ=2.0
wave number k
0
0.3
0.2
-0.5
0.1
0
k m=1
-5
0
5
0.4
wave number k
0
-20
0.3
0
20
λ(k)
σ=0.1
0.2
-100
0.1
0
-5
0
space x
Axel Hutt
5
Generalization of PDEs and finite propagation speed effects
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
PDEs are IDEs
Finite propagation delay
Summary
c<∞:
propagation delay τ
λ=
−Dk 2
p
1 + τ (1 − σ 2 k 2 ) 2/π
,
τ = σ/c, τth =
√
√
π/3 2
unstable
stable
τth
1
wave number k
Axel Hutt
2/σ
Generalization of PDEs and finite propagation speed effects
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
PDEs are IDEs
Finite propagation delay
Summary
The extended RD equation
∂V
+τ
∂t
r
2
π
«
„
∂V
∂2V
∂2
=D 2
1 + σ2 2
∂x
∂t
∂x
activity V(x,t) at x=8
initial activity V(x,0)
0.04
0.4
τ=2.08x10
-5
τ=10
0.02
-3
0.2
measurement
point
0
-8
-4
0
4
space x
8
Axel Hutt
0
0
30
time t
60
Generalization of PDEs and finite propagation speed effects
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
PDEs are IDEs
Finite propagation delay
Summary
relation to Cattaneo-equation
„
«
∂2
∂V
∂V
+ τ α 1 + 2D 2
∂t
∂x
∂t
=
D
∂V
∂t
=
D
1 + ατ
∂V
∂t
=
with Φ
=
∂2V
∂x 2
„
«
∂ ∂2V
1 − 2ατ
∂t ∂x 2
∂Φ
∂x
2Dατ ∂ ∂V
D ∂V
+
−
1 + ατ ∂x
1 + τ ∂t ∂x
−
→ equivalence to Cattaneo-ansatz
hyperbolic equation : 2ατ
∂2V
∂V
D ∂2V
+
=0
−
∂t 2
∂t
1 + ατ ∂x 2
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
The extended Swift-Hohenberg equation
consideration of terms up to the 5th order of σ:
K0 = −1
r
2
P0 = −
σ
π
,
,
σ4
σ2
, K4 = −
2
8
r
√
2 3
2
P2 = −
σ , P4 = − √ σ 5
π
3 π
K2 = −
c < ∞, τth = 1/7α
ε<0
unstable
delay τ
4/α
λSH =
stable
1/α
τth
1
„
«
4 ∂4
∂2
∂V
∂V
− τα 1 + 2 2 +
∂t
∂x
3 ∂x 4 ∂t
„
«2
∂2
= εV − V 3 − 1 +
V
∂x 2
ε>0
delay τ
4/α
1/α
τth
(ǫ − 1 + 2(σk )2 − (σk )4 )/2
1 − τ α(1 − 2(σk )2 + 4(σk )4 /3)
unstable
stable
ε− 1 ε+
scaled wavenumber l
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Basic idea
The Cattaneo equation
The novel idea
Application to specific models
The extended Kuramoto-Sivashinsky equation
λ
-1
1
wave number l
„
«
∂V
∂2
∂V
− τα 1 + 4 2
∂t
∂x
∂t
delay τ
= −ηV −
τth
-1
-0.5
stable
stable
stable
∂V
∂V
∂2V
−
−V
∂x 2
∂x
∂x
unstable
0
0.5
scaled wave number l
1
Axel Hutt
Generalization of PDEs and finite propagation speed effects
PDEs are IDEs
Finite propagation delay
Summary
Summary
integral-differential equations generalize some partial differential
equations
advantages:
natural consideration of nonlocal interactions
natural consideration of finite propagation speeds
extention of reaction-diffusion equations, the Swift-Hohenberg equation
and the Kuramoto-Sivashinsky equation by finite propagation delay
effects
large but finite propagation speeds in diffusion systems yield
Cattaneo-equation
Axel Hutt
Generalization of PDEs and finite propagation speed effects