Outline Outline Stationary Solution to Fokker Stationary - PowerPoint PPT Presentation

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 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Consider a single- -degree degree- -of of- -freedom system freedom system Stationary Density Function satisfies Consider a single Stationary Density Function satisfies with non- -linear spring linear spring with non ∂ ∂ ∂ 2 [ ] f ( ( ) ) f − + β + + = & & x x g x f D 0 ∂ ∂ ∂ & & 2 ( ) ( ) x x ⎧ ⎫ x & & & dX + β + = = & X X g X n t X ⎪ ⎪ or ⎪ ⎪ or dt ∂ ∂ ∂ ∂ ⎨ ⎬ ⎛ ⎞ ⎡ ⎤ f ( ) f f & ( ) ⎜ − + ⎟ + β + = & & x g x x f D 0 τ = δ τ d X ( ) ( ) ⎪ ⎢ ⎥ ⎪ ⎪ R nn 2 D ( ) = − β & − + ∂ ∂ ∂ ∂ ⎝ & ⎠ & ⎣ & ⎦ X g X n t x x x x ⎪ ⎩ ⎭ dt ∂ ∂ ∂ ⎛ ∂ ⎞ Fokker Fokker- -Planck Planck f ( ) f f + = & β ⎜ + ⎟ = & & x g x 0 x f D 0 ∂ ∂ ∂ ∂ & ⎝ & ⎠ x x x x ∂ ∂ ∂ ∂ 2 [ ] f ( ) ( ( ) ) f = − + β + + & & β ⎛ ⎞ x f x g x f D β 2 ( ) & x ( ) − & 2 ⎜ ⎟ x − + ∂ ∂ ∂ ∂ = G x ⎜ ⎟ & & 2 2 D f C x e = t x x x D ⎝ 2 ⎠ f C e 0 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1

( ) ( ) ( ) & & & & 2 ∂ ∂ ∂ ∂ + + = ( ) ⎡ ⎤ X f ( ) f ∫ x ( ) f X h H X g x n t = + η η Consider Consider Now set Now set − + = & + = H g d & x g x 0 ⎢ h H x f D ⎥ 0 ∂ ∂ 2 0 & ∂ ∂ x x & ⎣ & ⎦ x x ⎧ ⎫ dX & = ∂ ∂ ∂ ∂ X Assuming f = f(H f(H) ) ⎪ ⎪ Assuming f = f f f f ( ) ⎪ ⎪ = = & x g x dt ⎨ ⎬ ∂ ∂ ∂ ∂ & & x H x H ( ) d X ( ) ( ) ( ) ⎪ ⎪ ⎪ & = − + + g X h H X n t ⎪ ∂ ⎩ ⎭ ( ) f dt + = or or & & h H x f D x 0 ∂ H ∂ ∂ ∂ ∂ Fokker- -Planck Planck 2 Fokker f f [ ( ( ) ( ) ) ] f 1 ∫ H ( ) − ξ ξ = − + + + ∂ & & ( ) f df 1 ( ) dH h d x g x h H x f D = − = + = ∂ ∂ ∂ ∂ h H f D 0 h H D f C o e 0 & & 2 t x x x ∂ H f D ∂ ∂ ∂ ⎡ ∂ ⎤ ( ) ( ) ⎡ ⎤ f f f = ∫ ∫ ⎧− ⎫ Stationary Stationary + ∞ + ∞ 1 ( ) − + + + = H & & ∫ ξ ξ x g x ⎢ h H x f D ⎥ 0 ⎨ ⎬ & with C ⎢ exp h d dx d x ⎥ with ∂ ∂ ∂ ∂ ⎣ ⎦ & & & o ⎩ ⎭ x x x x ⎣ − ∞ − ∞ ⎦ D 0 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ) ∂ ( ) V X ( ) ∂ ∂ & & + β & + = ⎡ ⎤ Consider Consider H H Consider Consider X h H X n t ( ) H H ( ) = = & & + − & + = & & H x H x ⎢ x x ⎥ x i i i ∂ i X H h H D X n t ∂ & ∂ X & & x x x ⎣ ⎦ H H i & & x x ( ) ( ) = ∑ ( ) i = τ = δ δ τ 1 ( ) D > ⎧ ⎫ & R 2 D & i + F- -P P ⎡ ⎛ ⎞ ⎤ H x , x 0 2 F ∂ ∂ ∂ ⎪ ⎪ ∂ 2 H X V X const f f ( ) H H f n n i ij ⎜ ⎟ = − + − + + i j β & ⎨ ⎢ x & x & & x ⎥ ⎬ x H h H D x f D 2 ⎜ ⎟ ∂ ∂ ∂ & ∂ i ⎪ x ⎪ i & ⎝ ⎠ & 2 t x x ⎣ H H ⎦ x ⎩ ⎭ { } & & x x > H & 0 x ( ) ( ) Solution Solution ∫ H = − β ξ ξ f C exp / D f d ⎧ ⎛ ⎞ ⎫ ∂ ∂ 0 i i Stationary Stationary f H 0 ⎜ ⎟ − + = ⎪ & x ⎪ x f 0 ⎜ ⎟ ∂ ∂ ⎪ & ⎝ ⎠ ⎪ x x H & ⎨ x ⎬ Consider Consider + & ⎛ ⎞ 3 2 2 2 X X X ( ) ⎡ ⎛ ⎞ ⎤ ∂ ∂ & & + + & − & + = ⎜ 2 2 ⎟ ⎪ ( ) H f ⎪ X X 2 X 2 D X n t ⎜ ⎟ − & & + = ⎢ x x & ⎥ + & + & H h H x f 0 ⎝ 2 2 ⎠ 2 2 ⎜ ⎟ ⎪ ⎪ X 2 X X 2 X ∂ & ∂ x & ⎝ ⎠ & x ⎣ H x ⎦ ⎩ ⎭ & x { } ( ) ( ) τ = δ τ R nn 2 D ( ) { ( ) }( ) ∫ H = − ξ ξ Solution Solution Solution = − + + + Solution 4 & 4 2 & 2 2 & 2 f A exp x x x x x 2 x f C 0 exp h d H & x 0 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2

Consider the Nonlinear system given as Consider the Nonlinear system given as Consider the nonlinear stochastic equation Consider the nonlinear stochastic equation β ⎛ + ⎞ with random coefficient (Yong & Lin, 1987) with random coefficient (Yong & Lin, 1987) ( ) ( ) & & & & + β + = ⎜ ⎟ X sgn X 1 X g X n t [ ( ) ( ) ] [ ( ) ] ( ) ⎝ ⎠ D & & & + Γ + + ω + = 2 X h n t X 1 n t X n t 1 o 2 3 1 1 & Γ = + ω 2 2 2 X X o 2 2 ⎧ ⎫ ⎪ β ⎛ β ⎞ 2 x ⎪ ( ) ∫ x & = − − ξ ξ ⎜ ⎟ Fokker- -Planck Equation Planck Equation ⎨ ⎬ Fokker f C 0 exp X g d ⎪ ⎝ ⎠ ⎪ D D ⎩ 0 ⎭ ∂ ∂ { [ ] } ∂ [ ( ) ] 2 ( ) f − & + Γ & − & + ω + ω + & + = 2 4 2 2 0 X h X D X X f D X D X D f ∂ ∂ & ∂ & 22 o 2 o 11 22 33 x X X ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ) [ ] ( ) ( ) α & & & + α + β + ω + = D 2 2 Corresponding Ito’s Equation For = Corresponding Ito’s Equation = & For X X X 1 n t X n t 22 dX X dt o 1 2 β D 11 { } ( ) W ( ( ) ) & & & = − Γ − + ω + ω + + ˆ ⎧ ⎫ 2 4 2 2 β ( ) d X h D X X dt 2 D X D X D d Solution Solution = − & + ω ⎨ 2 2 2 ⎬ 22 0 0 11 22 33 f C exp X X 4 o ⎩ ⎭ 2 D 11 ( ) ⎧ ⎫ { } dt ( ) C Γ h u du ∫ ( ) = − & 3 ⎨ ⎬ f x , x exp ( ) ( ) 2 ˆ = + & & + Γ & + ω Γ + For ⎩ ⎭ For 2 E d W 2 D D 0 2 D u D X h X X 22 33 22 33 i i i i ( ) ( ) n 1 ∑ π = ξ Γ = & + ω n n ∑ ( ) ∑ ( ) t 2 2 2 X X + & + η n t X t X j j j j 2 For For ij j ij j = j 1 = = j 1 j 1 ⎛ ⎞ α ( ) 1 D ⎜ β − − ⎟ 33 ⎧− ⎫ Γ = β Γ + α ⎛ Γ + ⎞ 1 β Γ ⎜ ⎟ ( ) 2 D D h 2 ⎝ 2 D ⎠ 2 D = ⎜ ⎟ 22 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) & 22 33 22 ⎨ ⎬ f x , x C exp + τ = δ τ η η + τ = δ τ ξ ξ + τ = δ τ π ⎝ ⎠ n t n t 2 D t t 2 D t t 2 D ⎩ ⎭ 2 D ij ij 11 ij ij 22 j j 33 22 ( ) ⎧− ⎫ ⎡ ⎤ β Γ Solution Or ( ) Solution ( ) 1 Γ h U dU Or ∫ = = Γ + − − & ⎨ ⎬ f x , x C exp f C 2 D D exp ⎢ ⎥ 2 + o 5 22 33 ⎩ ⎭ ⎣ ⎦ 2 D 0 2 D U D 22 22 33 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3