Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 295

we have

t

t 2 t

t t

2

1
τ 0

∂
∂

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

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 ( r , θ , ϕ , t )

t 2 0

= f ( r , θ , ϕ , t ) .

Therefore, the u ( r , θ , ϕ , t ) in Eq. (6.153) is indeed the solution of PDS (6.152).

Remark 1. Let u be the solution of PDS (6.141) (Eq. (6.146)). The

solution of PDS (6.140) is, by the principle of superposition,

= W ψ ( r , θ , ϕ , t )

1 W Φ (

τ 0 + ∂

B 2 W k nl Φ (

u = r , θ , ϕ , t )+ r , θ , ϕ , t )

∂ t

t

+ W ψ ( r , θ , ϕ , t )+ W f τ ( r , θ , ϕ , t − τ ) d τ , (6.155)

0

1
2 )

1
2 )

( n + ( n +

where f τ = f ( r , θ , ϕ , τ ) , k nl = μ / a .The μ are available in Section 2.6.2.

l l

Remark 2. All results for the hyperbolic heat-conduction equations are recovered
as the special case of dual-phase-lagging heat-conduction equations at B = 0.

Remark 3. By using the structure of W ψ in Eq. (6.146), we obtain the u ( r , θ , ϕ , t ) in

Eq. (6.153).

t

=

W f τ (

, θ , ϕ , − τ )

u r t d τ

0

t

r ; θ , θ ; ϕ , ϕ ; t r , θ , ϕ , τ ) = G ( r , − τ ) f ( d Ω d τ .

0

r ≤ a

This is called the integral expression of the solution of PDS (6.152). The triple series
G r ; θ , θ ; ϕ , ϕ ; t

( , − τ )

is called the Green function of dual-phase-lagging heat-
conduction equations in a spherical domain .When f

r

( r , θ , ϕ , t )= δ ( r − r 0 , t − t 0 ) ,

in particular, the solution of PDS (6.152) reduces

u = G ( r , r 0 ; θ , θ 0 ; ϕ , ϕ 0 ; t − t 0 ) .

where r =( r , θ , ϕ ) , r 0 =( r 0 , θ 0 , ϕ 0 ) . Thus the Green function G ( r , r 0 ; θ , θ 0 ; ϕ , ϕ 0 ; t ,

t 0 ) is the solution due to a source term δ ( r − r 0 , t − t 0 ) .