Outline Outline Stationary Solution to Fokker Stationary - - PowerPoint PPT Presentation

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1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Stationary Solution to Fokker

Stationary Solution to Fokker-

• Planck Equation

Planck Equation

• Generalized Stationary Solutions

Generalized Stationary Solutions

• Additional Exact Solutions

Additional Exact Solutions

• Non

Non-

• linear Systems

linear Systems

• Equations with Random

Equations with Random coefficients coefficients

• G. Ahmadi

ME 529 - Stochastics

Consider a single Consider a single-

• degree

degree-

• of
• f-
• freedom system

freedom system with non with non-

• linear spring

linear spring Fokker Fokker-

• Planck

Planck

( ) ( )

t n X g X X = + + & & & β ( )

) ( D Rnn τ δ τ 2 =

( ) ( )⎪

⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ + − − = = t n X g X dt X d X dt dX & & & β

( ) ( ) ( ) [ ]

2 2

x f D f x g x x f x x t f & & & & ∂ ∂ + + ∂ ∂ + ∂ ∂ − = ∂ ∂ β

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ME 529 - Stochastics

Stationary Density Function satisfies Stationary Density Function satisfies

• r
• r

( ) ( ) [ ]

2 2

= ∂ ∂ + + ∂ ∂ + ∂ ∂ − x f D f x g x x x f x & & & & β

( )

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − x f D f x x x f x g x f x & & & & & β ( )

= ∂ ∂ + ∂ ∂ x f x g x f x & & = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ x f D f x x & & & β

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + −

=

2

2

x x G D

e C f

& β

( )

2

2 x D

e x C f

& β −

=

SLIDE 2

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• G. Ahmadi

ME 529 - Stochastics

Stationary Stationary Consider Consider Fokker Fokker-

• Planck

Planck ( ) ( ) ( )

t n x g X H h X = + + & & &

( )

∫

+ =

x

d g X H

2

2 η η &

( ) ( )

( )

( )⎪

⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ + + − = = t n X H h X g dt X d X dt dX & & &

( ) ( ) ( ) [ ]

2 2

x f D f x H h x g x x f x t f & & & & ∂ ∂ + + ∂ ∂ + ∂ ∂ − = ∂ ∂

( ) ( )

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ − x f D f x H h x x f x g x f x & & & & &

• G. Ahmadi

ME 529 - Stochastics

Now set Now set Assuming f = Assuming f = f(H f(H) )

• r
• r

( )

= ∂ ∂ + ∂ ∂ − x f x g x f x & &

( )

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ x f D f x H h x & & & x H f x f & & ∂ ∂ = ∂ ∂

( )

x g H f x f ∂ ∂ = ∂ ∂ ( )

= ∂ ∂ + x H f D f x H h & &

( )

= ∂ ∂ + H f D f H h

( )dH

H h D f df 1 − =

( )

∫ =

−

H

d h D

• e

C f

1 ξ ξ

( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧− = ∫ ∫

∫

∞ + ∞ − ∞ + ∞ −

x d dx d h D exp C

H

• &

1 ξ ξ

with with

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ME 529 - Stochastics

Solution Solution Consider Consider

( ) ( ) ( )

t n X V X H h X

i i i i i

= ∂ ∂ + + X & & & β

( ) ( )

τ δ δ τ

ij i n n

D R

j i

2 =

( )

X V X H

i i +

= ∑

2

2 1 & const D

i i =

β

( ) ( )

{ } ∫

− =

H i i

d f D C f / exp ξ ξ β

Solution Solution Consider Consider

( )

t n X X X X X X D X X X X X = + + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − + +

2 2 2 3 2 2 2 2

2 2 2 2 2 2 & & & & & & &

( ) ( )

τ δ τ D Rnn 2 =

( ) { }( )

2 2 2 2 4 4

2 exp x x x x x x A f & & & + + + − =

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ME 529 - Stochastics

Stationary Stationary Consider Consider F F-

• P

P

( ) ( ) t n H H X D H H H h H X

x x x x x x

= + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − +

& & & & &

& & & x H H x ∂ ∂ = x H H x &

&

∂ ∂ = ( )

2 2

x f D f H H x D H H H h H x x f x t f

x x x x x x

& & & &

& & & & &

∂ ∂ + ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∂ ∂ + ∂ ∂ − = ∂ ∂

( )

, > x x H & >

x

H &

( )

⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − x f f x H H H h H x f H H x x f x

x x x x x x

& & & & &

& & & & &

( )

{ }

x H

H d h C f

&

ξ ξ

∫

− =

0 exp

Solution Solution

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

Consider the Nonlinear system given as Consider the Nonlinear system given as

( ) ( )

t n X g X D X X = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + & & & & β β 1 sgn ( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − =

∫

x x

d g D X D C f

2 0 exp

ξ ξ β β &

• G. Ahmadi

ME 529 - Stochastics

Fokker Fokker-

• Planck Equation

Planck Equation Consider the nonlinear stochastic equation Consider the nonlinear stochastic equation with random coefficient with random coefficient (Yong & Lin, 1987) (Yong & Lin, 1987)

( ) ( ) [ ] ( ) [ ] ( )

t n X t n X t n h X

• 3

2 2 1

1 = + + + + ω Γ & & &

2 2 2

2 1 2 1 X X

• ω

Γ + = &

( )

[ ] { } ( ) [ ]

33 2 22 2 11 4 2 2 2 22

= + + ∂ ∂ + + − ∂ ∂ + ∂ ∂ − f D X D X D X f X X D X h X x f X

• &

& & & & & ω ω Γ

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ME 529 - Stochastics

Corresponding Ito’s Equation Corresponding Ito’s Equation

dt X dX & =

( ) ( )

{ } ( ) W

d D X D X D dt X X D h X d ˆ 2

33 2 22 2 11 4 2 22

+ + + + − Γ − = & & & ω ω

( )

{ } dt

W ˆ d E =

2

( ) ( ) ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + − + =

∫

Γ

π Γ

33 22 33 22 3

2 2 D u D du u h exp D D C x , x f &

( )

α + Γ β = Γ h

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧− ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − 22 1 2 2 1 33 22

2 2

22 2 22 33

D exp D D C x , x f

D D D

Γ β π Γ

α β

&

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧− =

22

2D exp C x , x f

• Γ

β &

For For Or Or

• G. Ahmadi

ME 529 - Stochastics

Solution Solution For For For For ( )

( ) [ ] ( )

t n X t n X X X

• 2

1 2 2

1 = + + + + ω β α & & &

β α =

11 22

D D

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + − =

2 2 2 11 4

2 X X D exp C f

• ω

β &

( ) ( ) ( ) ( )

t X t X t n X X h X

j n j j ij n j j ij i i i i

ξ η ω = + + + Γ +

∑ ∑

= = 1 1 2

& & & &

( )

∑

=

ω + = Γ

n 1 j 2 j 2 j 2 j

X X 2 1 &

( ) ( ) ( ) τ δ = τ +

11 ij ij

D 2 t n t n ( ) ( ) ( ) τ δ τ η η

22

2D t t

ij ij

= + ( ) ( ) ( ) τ δ = τ + ξ ξ

33 j j

D 2 t t ( ) ( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − + Γ =

∫

Γ − 33 22 2 1 33 22 5

D U D 2 dU U h exp D D 2 C f

Solution Solution