Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 75

Proof

1. Since W Φ ( r , θ , t ) satisfies

r = R =

L W Φ , ∂

⎧
⎨

2 W Φ
∂

∂ W Φ

∂

a 2

− Δ W Φ = 0 , 0 ,

t 2 r

⎩

∂
∂

W Φ ( r , θ , 0 )= 0 ,

t W Φ (

r , θ , 0 )= Φ ( r , θ ) .

We have

∂

2 u 1
∂

2 W Φ
∂

∂

u 1 = ∂
∂

a 2 a 2

t 2 −

Δ − Δ W Φ = 0 ,

t 2 t

r = R = ∂ L W Φ , ∂ r = R

, ∂

u 1
∂

W Φ
∂

L ( u 1 ) = 0 ,

r ∂ t r

)= ∂
∂

( , θ ,

t W Φ (

, θ , )= Φ ( , θ ) ,

u 1 r 0 r 0 r

2

∂
∂

)= ∂ a 2

t u 1 (

r , θ , 0

t 2 W Φ (

r , θ , 0 )= Δ W Φ ( r , θ , 0 )= 0 .

∂

Therefore u 1 = ∂

W Φ
∂

is the solution for the case of Ψ = F = 0.

t

2. Since W F τ ( r , θ , t − τ ) satisfies

⎧
⎨

r = R =

2 W F τ
∂

∂

W F τ , ∂

W F τ
∂

a 2

− Δ W F τ = 0 , L (

r )

0 ,

t 2

t = τ =

, ∂

W F τ
∂

⎩

W F τ | t = τ = 0 F ( r , θ , τ ) ,

t

we have

2 u 3
∂

∂

a 2

t 2 −

Δ u 3

t

− τ ) | τ = t a 2 t = ∂

∂

∂ W F τ

∂

d τ + W F τ ( r , θ , t −

0 Δ

W F τ d τ

t t 0

τ = t −

t t

2 W F τ
∂

∂

τ + ∂

W F τ
∂

a 2 = d Δ W F τ d τ

t 2 t 0 0

∂ d τ = t =

t

2 W F τ
∂

τ + ∂

W F τ
∂

a 2 = − Δ W F τ F ( r , θ , t ) ,

t 2 t 0