Time-Frequency Wigner Distribution Approach to Differential Equations - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 146

When the forcing term, f , is zero, one can show that the above reduces to

an equation that is of the same order, N, as the initial equation. In particular,

( x )

a α ( E )

B α Z u, u

A α ± a

α ( F )

( x, p ) = 0 . (4.118)

| α |≤ N

Appendix 4.5: Derivation of the Wigner Equation of Motion
for Diffusion

We show the derivation of the equation of motion for the Wigner distribution
for the diffusion equation. We work out the case of diffusion with drift:

2 u

∂ u

∂

c ∂

u

D ∂

t + x = x 2 ,

(4.119)

∂ ∂

where u is the field, c the drift coefficient, and D the diffusion

coefficient. To apply our method we first rewrite the equation as
∂
∂

= u ( x, t )

x 2 u

2

∂
∂

D ∂

t +

c

x −

( x, t ) = 0 . (4.120)

∂

We now apply the method described in Appendix 4.4 and obtain two equa-
tions for Z ( x, p, t, ω)

A t +

DA x Z

cA x − ( x, p, t, ω) = 0 , (4.121)

B t

DB x Z + cB x − ( x, p, t, ω) = 0 . (4.122)

Expanding the operators, we have
1

Z

2

2 ∂

c
2

∂
∂

D
4

∂

iDp ∂
∂

Dp 2

t −

i ω +

x −

icp −

x 2 +

+ ( x, p, t, ω) = 0 ,

∂ ∂ x

(4.123)

1 Z

2

2 ∂

c
2

∂
∂

D
4

∂

iDp ∂
∂

Dp 2

t +

i ω +

x +

icp −

x 2 +

− ( x, p, t, ω) = 0 .

∂ ∂ x

(4.124)

We add the two equations to have a real equation for the Wigner Z ( x, p, t, ω)

∂
∂

2 Dp 2 Z

2

∂
∂

D
2

∂

t +

c

x − x 2 +

( x, p, t, ω) = 0 . (4.125)

∂