Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 298

Theorem 2 .Let u = W ψ ( M , t ) be the solution of PDS (6.156). The solution of

⎧
⎨

u t
τ 0 +

B 2 ∂
∂

A 2

u tt = Δ u +

t Δ

u + f ( M , t ) , Ω × ( 0 , + ∞ ) ,

(6.160)

⎩

u ( M , 0 )= 0 , u t ( M , 0 )= 0

is

t

u = W f τ ( M , t − τ ) d τ , (6.161)

0

where .

Proof. By the definition of W ψ (

f τ = f ( M , τ )

M , t ) ,the W f τ ( M , t − τ ) satisfies

⎧
⎨

2 W f τ
∂

τ 0 ∂

1 W f τ

∂ t + ∂

B 2 ∂
∂

A 2

− Δ W f τ −

t Δ

W f τ = 0 ,

t 2

t = τ =

(6.162)

− τ ) t = τ =

∂
∂

⎩

W f τ ( M , t 0 ,

t W f τ (

M , t − τ ) f ( M , τ ) .

Thus

u t
τ 0 +

B 2 ∂
∂

A 2 u tt − Δ u −

t Δ

u

t

t 2 t

t t

2

1
τ

∂
∂

τ + ∂ B 2 ∂

∂

A 2 = W f τ d W f τ d τ − Δ W f τ d τ −

t Δ

W f τ d τ

t ∂

0 0 0 0 0

t

W f τ τ = t

t

W f τ τ = t

1
τ 0

∂ W f τ

∂

+ ∂
∂

∂ W f τ

∂

=

τ + τ +

d d

t t t

0 0

A 2 t

t
0 Δ

B 2 ∂
∂

−

0 Δ

W f τ d τ − W f τ d τ

t

τ = t

t t

2 W f τ
∂

1
τ 0

∂ W f τ

∂

∂

τ + ∂

W f τ
∂

=

τ + d d

t t 2 t

0 0

B 2 t
0

W f τ τ = t A 2 t

∂
∂

−

0 Δ

W f τ d τ −

t Δ

W f τ d τ + Δ

1 d

t

2 W f τ
∂ τ 0 ∂

W f τ
∂ t + ∂

B 2 ∂
∂

A 2 = − Δ W f τ −

t Δ

W f τ τ + f ( M , t )

t 2

0

= f ( M , t ) .