Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 297

Substituting Eq. (6.158) into the equation of PDS (6.157) and using Eq. (6.159a)
yields

u t
τ 0 +

B 2 ∂
∂

A 2 u tt − Δ u −

t Δ

u

1 W ϕ + ∂

t 2 1

W ϕ

2

1
τ 0

∂
∂ τ 0 + ∂ τ 0 + ∂

B 2 B 2 =

t −

Δ

t −

Δ

t ∂ ∂ ∂

1 W ϕ − 1 W ϕ

τ 0 + ∂

B 2 ∂
∂ τ 0 + ∂

A 2 B 2 B 2 − Δ

t −

Δ

t Δ t −

Δ

∂ ∂

1

2 W ϕ
∂

1
τ 0 τ 0 ∂

W ϕ
∂ t + ∂

B 2 ∂
∂

A 2 = − Δ W ϕ −

t Δ

W ϕ

t 2

1

2 W ϕ
∂

+ ∂
∂ τ 0 ∂

W ϕ
∂ t + ∂

B 2 ∂
∂

A 2 − Δ W ϕ −

t Δ

W ϕ

t 2

t

1

2 W ϕ
∂ τ 0 ∂

W ϕ
∂ t + ∂

B 2 ∂
∂

B 2 A 2 − Δ − Δ W ϕ −

t Δ

W ϕ = 0 .

t 2

Therefore, the u in Eq. (6.158) satisfies the equation of PDS (6.157).

By Eq. (6.159b), we have

( M , t )

1 W ϕ t = 0

τ 0 + ∂

B 2 u ( M , 0 )=

t −

Δ

∂

t = 0 −

W ϕ t = 0 = ϕ (

1
τ

)+ ∂

W ϕ
∂

B 2 = W ϕ ( M , 0 Δ M )

t

0

and

1 W ϕ (

∂ u

t = ∂ τ 0 + ∂

B 2

t −

Δ M , t )

∂ ∂ t ∂

2 W ϕ
∂

τ 0 ∂

1 W ϕ

∂ t + ∂

B 2 ∂
∂

A 2 = −

t Δ

W ϕ = Δ W ϕ ,

t 2

such that

t = 0 =

W ϕ t = 0 =

∂ u

A 2

Δ 0 .

∂ t

in Eq. (6.158) also satisfies the two initial conditions of
PDS (6.157), thus verifying that it is the solution of Eq. (6.157).

Therefore, the u ( M , t )