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Classification of 2nd order PDEs

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Classification of 2nd order PDEs. Classification of 2nd order PDEs. Classification of 2nd order PDEs. An important case:. the heat equation in 1D. Homogeneous state and its stability. the heat equation in 1D (Fourier approach). The heat equation in 1D. - PowerPoint PPT Presentation
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a 2 u t 2 + b 2 u t x + c 2 u x 2 + d u t + e u x + fu = g ( x , t) Δ= b 2 −4ac Classification of 2nd order PDEs
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Page 1: Classification of 2nd order PDEs

a∂ 2u

∂t 2+ b

∂ 2u

∂t∂ x+ c

∂ 2u

∂ x 2+ d

∂u

∂t+ e

∂u

∂ x+ f u = g(x, t)

Δ = b2 − 4ac

Classification of 2nd order PDEs

Page 2: Classification of 2nd order PDEs

Classification of 2nd order PDEs

Δ > 0 ⇒ hyperbolic equations

∂ 2u

∂t 2− c0

2 ∂ 2u

∂ x 2= h1 t, x,u,

∂u

∂t,∂u

∂ x

⎝ ⎜

⎠ ⎟ canonical form

e.g. ∂ 2u

∂t 2− c0

2 ∂ 2u

∂ x 2= 0 →

∂u

∂t+ c0

∂u

∂ x= 0

∂ 2u

∂t∂ x= h2 t, x,u,

∂u

∂t,∂u

∂ x

⎝ ⎜

⎠ ⎟ characteristic form

Page 3: Classification of 2nd order PDEs

Classification of 2nd order PDEs

Δ =0 ⇒ parabolic equations

∂ 2u

∂ x 2= h3 t,x,u,

∂u

∂t,∂u

∂ x

⎝ ⎜

⎠ ⎟ e.g.

∂u

∂t= μ

∂ 2u

∂ x 2

Δ < 0 ⇒ elliptic equations

∂ 2u

∂t 2+

∂ 2u

∂ x 2= h4 t, x,u,

∂u

∂t,∂u

∂ x

⎝ ⎜

⎠ ⎟

e.g. ∇ 2u = 0

e.g.∂u

∂t= μ∇ 2u

Page 4: Classification of 2nd order PDEs

Homogeneous state and its stability€

∂u

∂t= μ

∂ 2 u

∂ x 2

u = 0stable and attracting if μ > 0

unstable if μ < 0

⎧ ⎨ ⎩

for μ = 0 there is a bifurcation

An important case:

the heat equation in 1D

Page 5: Classification of 2nd order PDEs

∂u

∂t= μ

∂ 2u

∂ x 2

u x, t( ) =1

2πˆ u (k)

−∞

∫ exp ikx − μk 2t( ) dk

ˆ u (k) = u(x, 0)−∞

∫ exp −ikx( ) dx

ˆ u (k, t) = ˆ u (k) exp −μk 2t( ) = ˆ u (k) ˆ S (k)

the heat equation in 1D (Fourier approach)

Page 6: Classification of 2nd order PDEs

S(x, t) =1

2πˆ S (k, t)

−∞

∫ exp ikx( )dk

S(x, t) =1

4π μ texp −

x 2

4μ t

⎝ ⎜

⎠ ⎟

ˆ u (k) ˆ S (k, t) ⇒ u(x,0)∗S(x, t)

u(x, t) =1

4π μ texp −

x − y( )2

4μ t

⎝ ⎜ ⎜

⎠ ⎟ ⎟u(y,0) dy

−∞

The heat equation in 1D

Page 7: Classification of 2nd order PDEs

u(x,0) =δ(x)

uG (x, t) =1

4π μ texp −

x − y( )2

4 μ t

⎝ ⎜ ⎜

⎠ ⎟ ⎟δ(y) dy

−∞

uG (x, t) =1

4π μ texp −

x 2

4 μ t

⎝ ⎜

⎠ ⎟

Green’s function solution of the heat equation

Page 8: Classification of 2nd order PDEs

∂u

∂t= μ

∂ 2u

∂ x 2

u = u(z = x − ct)

c∂u

∂z+ μ

∂ 2u

∂z2= 0 ⇒ u = A + Bexp −

cz

μ

⎝ ⎜

⎠ ⎟

B = 0

The heat equation does not havetravelling wave solutions

Page 9: Classification of 2nd order PDEs

∂u

∂t= f (u) + μ

∂ 2u

∂ x 2

Reaction-diffusion equations

∂ ru

∂t=

r f (

r u ) + μ∇ 2 r

u

in 1D, for a scalar field

Page 10: Classification of 2nd order PDEs

∂u

∂t= ν u + μ

∂ 2u

∂ x 2

u(x, t) =1

2πˆ u (k)exp ikx − iω t( )

−∞

∫ dk

u(x, t) =1

2πˆ u (k)exp ikx( )

−∞

∫ exp ν − μ k 2( ) t[ ] dk

instability (growth) for k <ν

μ

⎝ ⎜

⎠ ⎟

12

The simplest case:the KISS model for plankton blooms

Page 11: Classification of 2nd order PDEs

∂u

∂t= ν u 1−

u

K

⎝ ⎜

⎠ ⎟+ μ

∂ 2u

∂ x 2

u → ˜ u =u

K, t → ˜ t = ν t , x → ˜ x =

ν

μ

⎝ ⎜

⎠ ⎟

12

x

∂u

∂t=u 1− u( ) +

∂ 2u

∂ x 2

Need for a saturation mechanism:the Fisher equation

Page 12: Classification of 2nd order PDEs

The equation is now nonlinear…

∂u

∂t−u 1− u( ) −

∂ 2u

∂ x 2= 0

∂tˆ u (k)

−∞

∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩

⎫ ⎬ ⎭−

∂ 2

∂ x 2ˆ u (k)

−∞

∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩

⎫ ⎬ ⎭

− ˆ u (k)−∞

∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩

⎫ ⎬ ⎭

+ ˆ u (k)−∞

∫ ˆ u (k ')exp i k + k '( )x − i ω +ω'( ) t[ ] dk'dk ⎧ ⎨ ⎩

⎫ ⎬ ⎭= 0

Fourier decomposition does not help now

Page 13: Classification of 2nd order PDEs

Fisher equation: homogeneous solution(the logistic equation)

∂u

∂t=u 1− u( ) +

∂ 2u

∂ x 2

∂ x= 0

d u

dt=u 1− u( )

Page 14: Classification of 2nd order PDEs

the logistic equation

d u

dt=u 1− u( )

fixed points (stationary states)

u = 0 , u =1

linear stability of the fixed points

exact solution :

u(t) =u0 exp t

1+ u0 exp t −1( )

Page 15: Classification of 2nd order PDEs

back to the Fisher equation: travelling waves

∂u

∂t=u 1− u( ) +

∂ 2u

∂ x 2, U(z) = u(x − c t)

d2U

dz2+ c

dU

dz+ U 1−U( ) = 0

dU

dz= V

dV

dz= −cV −U 1−U( )

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Page 16: Classification of 2nd order PDEs

dynamical systems in a 2d phase space

dU

dz= f U,V( )

dV

dz= g U,V( )

⎨ ⎪ ⎪

⎩ ⎪ ⎪

types of fixed points and their linearized stability

bifurcations : pitchfork, saddle - node, Hopf

Page 17: Classification of 2nd order PDEs

dU

dz= f U,V( )

dV

dz= g U,V( )

⎨ ⎪ ⎪

⎩ ⎪ ⎪

fixed points : f U ,V ( ) = g U ,V ( ) = 0

small perturbation : U = U + u , V = V + v

du

dz= f U + u,V + v( ) ≈

∂ f

∂U

⎝ ⎜

⎠ ⎟

U ,V

u +∂ f

∂V

⎝ ⎜

⎠ ⎟

U ,V

v

dv

dz= g U + u,V + v( ) ≈

∂g

∂U

⎝ ⎜

⎠ ⎟

U ,V

u +∂g

∂V

⎝ ⎜

⎠ ⎟

U ,V

v

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Page 18: Classification of 2nd order PDEs

du

dz= f U + u,V + v( ) ≈

∂ f

∂U

⎝ ⎜

⎠ ⎟

U ,V

u +∂ f

∂V

⎝ ⎜

⎠ ⎟

U ,V

v

dv

dz= g U + u,V + v( ) ≈

∂g

∂U

⎝ ⎜

⎠ ⎟

U ,V

u +∂g

∂V

⎝ ⎜

⎠ ⎟

U ,V

v

⎨ ⎪ ⎪

⎩ ⎪ ⎪

d

dz

u

v

⎝ ⎜

⎠ ⎟=

∂ f

∂U

∂ f

∂V∂g

∂U

∂g

∂V

⎜ ⎜ ⎜

⎟ ⎟ ⎟

U ,V

u

v

⎝ ⎜

⎠ ⎟

det

∂ f

∂U

∂ f

∂V∂g

∂U

∂g

∂V

⎜ ⎜ ⎜

⎟ ⎟ ⎟

U ,V

− λ1 0

0 1

⎝ ⎜

⎠ ⎟

⎨ ⎪

⎩ ⎪

⎬ ⎪

⎭ ⎪= 0

Page 19: Classification of 2nd order PDEs

d

dz

u

v

⎝ ⎜

⎠ ⎟=

∂ f

∂U

∂ f

∂V∂g

∂U

∂g

∂V

⎜ ⎜ ⎜

⎟ ⎟ ⎟

U ,V

u

v

⎝ ⎜

⎠ ⎟

Tr

∂ f

∂U

∂ f

∂V∂g

∂U

∂g

∂V

⎜ ⎜ ⎜

⎟ ⎟ ⎟

U ,V

=∂ f

∂U+

∂g

∂V= λ 1 + λ 2

dissipative and conservative dynamical systems

Page 20: Classification of 2nd order PDEs

real eigenvalues

λ +,λ− < 0 stable node

λ + > 0 , λ− < 0 saddle point

λ +,λ− > 0 unstable node

complex conjugate eigenvalues : λ + = λ R + iλ I , λ− = λ R − iλ I

λ R < 0 stable focus

λ R > 0 unstable focus

pure imaginary eigenvalues : λ + = iλ I , λ− = −iλ I center

Page 21: Classification of 2nd order PDEs

pitchfork bifurcation

dU

dz= α U −U 3 supercritical

dU

dz= α U + U 3 subcritical

types of bifurcation:normal form

Page 22: Classification of 2nd order PDEs

saddle - node bifurcation

dU

dz= α −U 2 supercritical

dU

dz= α −U 2 subcritical

types of bifurcation:normal form

Page 23: Classification of 2nd order PDEs

Hopf bifurcation (supercritical)

dU

dz= α U −V −U U 2 + V 2

( )

dV

dz= U + α V −U U 2 + V 2

( )

radial form

dz= ρ α − ρ 2

( )

dz=1

types of bifurcation: normal form

Page 24: Classification of 2nd order PDEs

Fisher equation: travelling wave

dU

dz= V

dV

dz= −cV −U 1−U( )

⎨ ⎪ ⎪

⎩ ⎪ ⎪

fixed points

U,V( ) = (0,0) ; U,V( ) = (1,0)

(0,0) → λ ± =1

2−c ± c 2 − 4( )

12 ⎡

⎣ ⎢ ⎤ ⎦ ⎥ ⇒

stable node if c 2 ≥ 4

stable focus if c 2 < 4

⎧ ⎨ ⎩

(1,0) → λ ± =1

2−c ± c 2 + 4( )

12 ⎡

⎣ ⎢ ⎤ ⎦ ⎥ ⇒ saddle point

Page 25: Classification of 2nd order PDEs

Fisher equation: phase-plane picture

dV

dU=

−cV −U 1−U( )V

Page 26: Classification of 2nd order PDEs

Fisher equation: travelling wave

http://commons.wikimedia.org/wiki/File:Travelling_wave_for_Fisher_equation.svg

z

c ≥ 2

Page 27: Classification of 2nd order PDEs

Fisher equation: travelling wavegeneral initial condition

u(x,0) =1 x < x1

u(x,0) = 0 x ≥ x2

u → U(x − 2t)

u(x,0) ≈ Aexp −ax( ) x → ∞

u(x, t) ≈ Aexp −a x − ct( )[ ] x → ∞

c = a +1

aif 0 < a ≤1 ; c = 2 if a >1

Page 28: Classification of 2nd order PDEs

Fisher equation: travelling wave

asymptotic form

stability of the travelling wave€

dV

dU=

−cV −U 1−U( )V

no general solution ∀c

expand in ε =1

c 2

Page 29: Classification of 2nd order PDEs

A different form of nonlinearity…

∂u

∂t+ α u

∂u

∂ x= 0

∂tˆ u (k)

−∞

∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩

⎫ ⎬ ⎭

+ α ˆ u (k)−∞

∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩

⎫ ⎬ ⎭

∂ xˆ u (k ')

−∞

∫ exp ik ' x − iω t( )dk' ⎧ ⎨ ⎩

⎫ ⎬ ⎭= 0

Again, Fourier decomposition does not help

Page 30: Classification of 2nd order PDEs

Heuristic, just to understand…

∂u

∂t+ α u

∂u

∂ x= 0

ω(k) = c k =α uk

c = α u

This is an hyperbolic equation

Singularity in finite time

Page 31: Classification of 2nd order PDEs

The Burgers equation

∂u

∂t+ α u

∂u

∂ x= μ

∂ 2 u

∂ x 2

Shock solutions

Page 32: Classification of 2nd order PDEs

the Cole-Hopf transformation

∂u

∂t+ α u

∂u

∂ x= μ

∂ 2 u

∂ x 2

u = −2μϕ x

ϕ

u =ψ x → ψ t +1

2ψ x( )

2= μ ψ xx

ψ = −2μ logϕ

ϕ t = μϕ xx

ϕ x,0( ) = exp −1

2μu η ,0( ) dη

0

x

∫ ⎡

⎣ ⎢

⎦ ⎥

u =

x −η

texp −

G

⎝ ⎜

⎠ ⎟dη

−∞

exp −G

⎝ ⎜

⎠ ⎟dη

−∞

∫; G η ;x, t( ) = u η ',0( ) dη '

0

η

∫ +x −η( )

2

2 t

Page 33: Classification of 2nd order PDEs

The Burgers equation:

shock structure

general solution and confluence of shocks

test for numerical methods

stochastically-forced Burgers eq.

Page 34: Classification of 2nd order PDEs

The Korteweg-de Vries equation

∂u

∂t+ α u

∂u

∂ x+

∂ 3 u

∂ x 3= 0

Travelling wave solutions:“solitary waves”

on the periodic: “cnoidal waves”

u x, t( ) = 3κ 2 sech2 κ x −κ 3t −θ0

2

⎝ ⎜

⎠ ⎟

Page 35: Classification of 2nd order PDEs

The inverse scattering transformfor the Korteweg-de Vries equation

u x, t( ) =ψ xx

ψ+ λ

ψ xx + λ − u x, t( )[ ] ψ = 0

∂u

∂t− 6u

∂u

∂ x+

∂ 3 u

∂ x 3= 0

Page 36: Classification of 2nd order PDEs

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