Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 314

For two-dimensional cases, let u t ( M , 0 )= ψ ( x , y ) . By Eq. (5.67), we obtain

u 0 ( M , t )= W ψ ( x , y , t )

c h A

2 2 2

t

( At ) − ( x − ξ ) − ( y − η )

1

e −

=

2 τ 0

ψ ( ξ , η ) d ξ d η .

2 π A

2 2 2

( At ) − ( x − ξ ) − ( y − η )

D At

Denote ∂
∂ t Δ

u 0 = f 1 ( x , y , t ) . By Eq. (6.207), we have

t

u 1 ( M , t )= W f 1τ ( M , t − τ ) d τ , f 1τ = f 1 ( x , y , τ ) .

0

Using this approach, we can also obtain u 2 ( M , t ) , u 3 ( M , t ) , ··· .

is
available in Eq. (5.120). Similar to the two-dimensional case, we can obtain the
perturbation solution of PDS (6.204)

For the three-dimensional case, let u t ( M , 0 )= ψ ( x , y , z ) , u 0 ( M , t )= W ψ ( M , t )

O

n + 1 2 u 2 n u n

u ( M , t , ε )= u 0 ( M , t )+ ε u 1 ( M , t )+ ε ( M , t )+ ··· + ε ( M , t )+ ε .

In applications, we normally take the first- or second-order perturbation solution
as the approximate solution of PDS (6.204). Once the approximate solution of
PDS (6.204) is available, an approximate solution of
⎧
⎨

∂
∂

A 2 R 2 or R 3 u t / τ 0 + u tt = Δ u + ε

t Δ

u + f ( M , t ) , × ( 0 , + ∞ ) × ( 0 , + ∞ ) ,

⎩

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

can be obtained by the solution structure theorem,

1 W ϕ (

t

τ 0 + ∂

u ( M , t )=

t − εΔ

M , t )+ W ψ ( M , t )+ W f τ ( M , t − τ ) d τ ,

∂

0

where f τ = f ( M , τ ) .

6.10 Thermal Waves and Resonance

In this section we examine thermal oscillation and resonance described by the dual-
phase-lagging heat-conduction equations. Conditions and features of underdamped,
critically-damped and overdamped oscillations are obtained and compared with
those described by the classical parabolic heat-conduction equation and the hyper-
bolic heat-conduction equation. Also derived is the condition for thermal resonance.
Both underdamped oscillation and critically-damped oscillation cannot appear if τ T

is larger than τ 0 . The modes of underdamped thermal oscillation are limited to a

region fixed by two relaxation distances for the case

τ T > 0, and by one relaxation

distance for the case τ T = 0.