! Special Functions ! Differential Equations ! Fourier Series and - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 437/537-Particle

• G. Ahmadi

ME 437/537-Particle

! Special Functions ! Differential Equations ! Fourier Series and Transforms ! Probability and Random Processes ! Linear System Analysis

• G. Ahmadi

ME 437/537-Particle

Unit Step Unit Step Function Function ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ < ≥ = − t t t t 1 t t u

u(t-to) to

• G. Ahmadi

ME 437/537-Particle

Dirac Dirac Delta Delta Function Function

( ) ( )

dt t t du t t − = − δ

δ(t-to) to

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ME 437/537-Particle

( ) ( )

1 dt t t dt t t

t t

= − δ = − δ

∫ ∫

ε + ε − +∞ ∞ −

( ) ( ) ( ) ( ) ( )

t t

t f dt t t t f dt t t t f = − δ = − δ

∫ ∫

ε + ε − +∞ ∞ −

( ) ( ) ( ) ( )

t 1 1 1

t t u t f dt t t t f − = − δ

∫ ∞

−

( ) [ ] ( )

t t a 1 t t a − δ = − δ

• G. Ahmadi

ME 437/537-Particle

Error Error Function Function

( )

∫

−

π =

x t dt

e 2 x erf

2

( ) ( )

∫

∞ −

π = − =

x t dt

e 2 x erf 1 x erfc

2

( ) ( )

∞ = = erfc erf

( ) ( )

x erf x erf − = −

( ) ∫

∞ −

=

1 xt n

dt t e x E

Exponential Exponential Integrals Integrals

( ) ∫ ∞

−

=

x t i

dt t e x E

• G. Ahmadi

ME 437/537-Particle

Linear First Linear First-

• Order

Order

( ) ( )

x Q y x P dx dy = +

( ) ( )

( )

∫

∫ + ∫ =

− − x 1 1 dx x P dx x P

dx x Q e ce y

x 1 x 2 1 x 1

( )

x Q y b dx dy = +

( )

∫

− −

=

x 1 1 ) x x ( b

dx x Q e y

1

• G. Ahmadi

ME 437/537-Particle

by dx dy a dx y d

2 2

= + +

2 1 2

m , m for Solve b am m → = + +

al Re m m

1 2

= ≠

x m 2 x m 1

2 1

e c e c y + =

m m m

1 2

= =

) x c c ( e y

2 1 mx

+ =

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ME 437/537-Particle

qi p m1 + = qi p m2 − =

2 a p − =

4

2

a b q − =

( )

qx sin c qx cos c e y

2 1 px

+ =

• G. Ahmadi

ME 437/537-Particle

Particular Solutions Particular Solutions

( )

x R by dx dy a dx y d

2 2

= + +

( ) ( )

∫ ∫

− −

− + − = dx x R e m m e dx x R e m m e y

x m 1 2 x m x m 2 1 x m P

2 2 1 1

( ) ( )

∫ ∫

− −

− = dx x R xe e dx x R e xe y

mx mx mx mx P

( ) ( )

∫ ∫

− −

− = dx qx sin x R e q qx cos e dx qx cos x R e q qx sin e y

px px px px P

• G. Ahmadi

ME 437/537-Particle

( )

x s by dx dy ax dx y d x

2 2 2

= + +

m

Ax y =

( )

1 = + + − b am m m

2 1

m 2 m 1

x A x A y + =

Let Let

• G. Ahmadi

ME 437/537-Particle

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = x y F dx dy

x y v =

( )

c v v F dv x ln + − = ∫ v dx dv x dx dy + =

Let Let

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ME 437/537-Particle

( ) ( )

dy y , x N dx y , x M = +

( )

y x y , x x N y M

2

∂ ∂ ϕ ∂ = ∂ ∂ = ∂ ∂ x M ∂ ϕ ∂ = y N ∂ ϕ ∂ =

( )

const y , x = ϕ

With With

• G. Ahmadi

ME 437/537-Particle

( )

y n x dx dy x dx y d x

2 2 2 2 2 2

= − β + +

( ) ( )

x Y C x J C y

n 2 n 1

β + β =

( )

x Jn β

( )

x Yn β

Solutions Solutions Bessel Bessel Functions Functions

• G. Ahmadi

ME 437/537-Particle

Fourier Cosine Series Fourier Cosine Series ( )

∑

∞ =

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + π + =

1 n n n

L x n sin b L x n cos a 2 a x f

( )

∫

−

π =

L L n

dx L x n cos x f L 1 a

( )

∫−

π =

L L n

dx L x n sin x f L 1 b

( ) ( )

x f x f − =

( )

∑

∞ =

π + =

1 n n

L x n cos a 2 a x f

( )

∫

π =

L n

dx L x n cos x f L 2 a

• G. Ahmadi

ME 437/537-Particle

Fourier Exponential Series Fourier Exponential Series

( ) ( )

x f x f − − = ( ) ∑

∞ =

π =

1 n n

L x n sin b x f

( )

∫

π =

L n

dx L x n sin x f L 2 b

( ) ∑

∞ −∞ = ω

=

n x i n

n

e c x f

( )

∫−

ω −

=

L L x i n

dx e x f L 2 1 c

n

L n

n

π = ω

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ME 437/537-Particle

( ) ∑

∞ −∞ = π

=

n L x in ne

c x f

L x L < < −

( )

∫−

π −

′ =

L L L x in n

dx x f e L 2 1 c

( )

( ) ( )

∑∫

+∞ ∞ − − ′ − ω

′ =

L L x x i

dx x f e L 2 1 x f

n

L n

n

π = ω L π = ω ∆ ∞ → L

∫ ∑

ω = ω ∆ gd gn

FES FES FES

Replacing for cn

• G. Ahmadi

ME 437/537-Particle

( )

( ) ( )

∫ ∫

∞ + ∞ − ∞ + ∞ − ′ − ω

ω ′ ′ π = d x d x f e 2 1 x f

x x i

( ) ( )

∫

+∞ ∞ − ′ ω −

′ ′ = ω x d x f e f

x i

( ) ( )

∫

∞ + ∞ − ω

ω ω π = d f e 2 1 x f

x i

Fourier Integral Representation Fourier Integral Representation Fourier Transform (Exponential) Fourier Transform (Exponential)

• G. Ahmadi

ME 437/537-Particle

( ) ( )

∫

∞

′ ′ ′ ω = ω

c

x d x f x cos f

( ) ( )

∫

∞

ω ω ω π =

c

d f x cos 2 x f

( ) ( )

∫

∞

′ ′ ′ ω = ω

s

x d x f x sin f

( ) ( )

∫

∞

ω ω ω π =

s

d f x sin 2 x f Fourier Cosine Fourier Cosine Transform Transform Fourier Sine Fourier Sine Transform Transform

• G. Ahmadi

ME 437/537-Particle

( ) ( )

ω ω = = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ℑ

∫

∞ + ∞ − ω −

f i dx dx x df e dx df

x i

( )

ω ω − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ℑ f dx f d

2 2 2

( ) ( )

ω ω = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ℑ f i dx f d

n n n

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• G. Ahmadi

ME 437/537-Particle

( )

2 2

x x bf dx df a dx f d − δ = + + +∞ < < ∞ − x

( ) ( ) ( )

x i 2

e f b f ai f

ω −

= ω + ω ω + ω ω −

Taking Fourier Transform Taking Fourier Transform

( )

ω + ω − = ω

ω −

ia b e f

2 x i

( )

( )

∫

∞ + ∞ − − ω

ω ω + ω − π = d ia b e 2 1 x f

2 x x i

• G. Ahmadi

ME 437/537-Particle

( ) ( ) ( ) ( )

∫

+∞ ∞ −

ξ ξ − ξ = d x f f x f * x f

2 1 2 1

( ) ( )

ω ω

2 1

f f

( )

x x − δ

x i

e

ω −

x

e

α −

2 2

2 α + ω α

x cos ω ( ) ( ) [ ]

ω + ω δ + ω − ω δ π x cos e

x

β

α −

( ) ( )

2 2 2 2 2 2 2 2 2

4 2 ω α + α − β − ω β + α + ω α

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ β β α + β

α −

x sin x cos e

x

( ) ( )

2 2 2 2 2 2 2 2

4 4 ω α + α − β − ω β + α α

( )

x f ( )

ω f ( )

1

x x f +

( )

ω

ω f

e

x i

• G. Ahmadi

ME 437/537-Particle

x cos e

2 2x

β

α −

( ) ( )

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ α β − ω − + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ α β + ω − α π

2 2 2 2

4 exp 4 exp 2

( )

x f

( )

ω f

2 2x

e α

−

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ α ω − α π

2 2

4 exp

( )

x dx d

n n

δ

( )

n

iω

( )

x J0

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ < ω ω − elsewhere 1 1 2

2

• G. Ahmadi

ME 437/537-Particle

FY(y) y 1

( ) { }

y Y P y F

Y

≤ =

( )

1 y F

Y

≤ ≤

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• G. Ahmadi

ME 437/537-Particle

fY(y) y

( ) ( )

dy y dF y f

Y Y

=

( ) ( )

∫

+∞ ∞ −

= = ∞ dy y f 1 F

Y Y

{ } ( ) ( ) ( )

1 Y 2 Y y y Y 2 1

y F y F dy y f y Y y P

2 1

− = = ≤ <

∫

• G. Ahmadi

ME 437/537-Particle

{ } ( )

∫

+∞ ∞ −

= = dy y yf Y Y E

Y

( ) { } ( ) ( ) ( )

∫

+∞ ∞ −

= = dy y f y g Y g Y g E

Y

( )

{ }

{ }

2 2 2 2 Y

Y Y E Y Y E − = − = σ

Expected Expected Value Value Variance Variance

• G. Ahmadi

ME 437/537-Particle

X(t) t

( ) { } ( )

∫

+∞ ∞ −

= dx t , x xf t X E

X

• G. Ahmadi

ME 437/537-Particle

( ) ( ) ( ) { }

t X E dt t X T 1 t X

T

≈ = ∫

( ) ( ) ( ) { } ( ) ( )

∫

τ + = τ + = τ

T xx

dt t X t X T 1 t X t X E R

( ) ( )

{ }

( )

t X t X E R

2 2 xx

= = Time Time Averaging Averaging Autocorrelation Autocorrelation

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• G. Ahmadi

ME 437/537-Particle

( ) ( )

∫

+∞ ∞ − ωτ −

τ τ = ω d R e S

xx i xx

( ) ( )

∫

∞ + ∞ − ωτ

ω ω π = τ d S e 2 1 R

xx i xx

( ) ( )

2 xx

X ~ T 1 S ω = ω

( ) ( )

∫

+∞ ∞ − ω −

= ω dt t X e X ~

t i

• G. Ahmadi

ME 437/537-Particle

f(t) X(t)

( ) ( )

ω H t h ( ) ( ) ( )

∫

τ τ τ − =

t

d f t h t X

( ) ( ) ( ) ( ) ( )

t f * t h d f t h t X = τ τ τ − = ∫

+∞ ∞ −

h(t)=Impulse Response h(t)=Impulse Response H( H(ω ω)=System Function )=System Function

( ) ( )

∫

+∞ ∞ − ω −

= ω dt t h e H

t i

• G. Ahmadi

ME 437/537-Particle

( ) ( ) ( )

ω ω = ω f ~ H x ~

( ) ( ) ( ) ( ) ( ) ( )

ω ω = ω ω = ω = ω

ff 2 2 2 2 xx

S H f H T 1 x T 1 S

( ) ( ) ( )

ω ω = ω

ff 2 xx

S H S

Fourier Fourier Transform Transform Spectral Spectral Relationship Relationship

• G. Ahmadi

ME 437/537-Particle

( )

t f x x = α + &

( )

t

e t h

α −

=

( )

t f x x 2 x

2

= ω + ζω + & & &

( )

t sin e 1 t h

d t d

ω ω =

ζω −

2 d

1 ζ − ω = ω

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• G. Ahmadi

ME 437/537-Particle

( )

t n x x = α + &

i 1 i i 1 i

n x t x x = α + ∆ −

+ +

) t ( x x

i i = i i 1 i

n t 1 t x t 1 1 x ∆ α + ∆ + ∆ α + =

+

Finite difference

1 t for n t x x

i i 1 i

<< ∆ α ∆ + =

+

1 t for n 1 x t 1 x

i i 1 i

>> ∆ α α + ∆ α =

+

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ME 437/537-Particle

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 2.5 3 White Noise 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)

• G. Ahmadi

ME 437/537-Particle

• 150000
• 100000
• 50000

50000 100000 Brownian Force 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)

• G. Ahmadi

ME 437/537-Particle

• 0.003
• 0.002
• 0.001

0.001 0.002 0.003 0.004 Particle Velocity 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)

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• 2E-08

2E-08 4E-08 6E-08 8E-08 1E-07 1.2E-07 1.4E-07 1.6E-07 Particle Position 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)

• G. Ahmadi

ME 437/537-Particle

• 4E-08
• 2E-08

2E-08 4E-08 6E-08 8E-08 Particle Position 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)