Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 294

Proof. By the definition of W ψ ( r , θ , ϕ , t ) ,the W f τ ( r , θ , ϕ , t − τ ) satisfies

⎧
⎨

2 W f τ
∂

τ 0 ∂

1 W f τ

∂

+ ∂ B 2 ∂

∂

A 2

= Δ W f τ +

t Δ

W f τ (6.154a)

t t 2

r = a =

L W f τ , ∂

W f τ
∂

0 , (6.154b)

⎩

r

t = τ =

W f τ t = τ =

∂ W f τ

∂

0 , f ( r , θ , ϕ , τ ) . (6.154c)

t

By Eq. (6.154b)
L t
0

r = a = r = a

L W f τ ,

t t

∂
∂

∂
∂

W f τ d τ , W f τ d τ

r W f τ

d τ = 0 ,

r 0 0

so that the u ( r , θ , ϕ , t ) in Eq. (6.153) satisfies the boundary conditions of PDS (6.152).

Clearly, the u ( r , θ , ϕ , t ) in Eq. (6.153) satisfies the initial condition u ( r , θ , ϕ , 0 )=

0. Also

W f τ τ = t ,

t

∂ u ∂ W f τ

∂ t =

d τ +

∂ t 0

which shows that u t ( r , θ , ϕ , 0 )= 0 by Eq. (6.154c). Thus the u ( r , θ , ϕ , t ) in Eq. (6.153)

also satisfies the two initial conditions of PDS (6.152).
Since for the u ( r , θ , ϕ , t ) in Eq. (6.153),

t t

W f τ τ = t =

∂ u ∂ W f τ

∂

∂ W f τ

∂

t =

d τ + d τ ,

∂ t t 0 0

τ = t =

t

2 u 2 W f τ

∂

2 W f τ
∂

∂

t

∂

τ + ∂

W f τ
∂

∂

t 2 =

d d τ + f ( r , θ , ϕ , t ) ,

t 2 t 2 ∂ t 0 0

t
0 Δ

Δ u = W f τ d τ ,

t
0 Δ

t

W f τ τ = t

∂
∂

= ∂
∂

∂
∂ t Δ

u W f τ d τ =

t Δ

W f τ d τ + Δ

t 0

t

∂
∂

=

t Δ

W f τ d τ ,

0