Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 271

6.4.4 Summary and Remarks

Summary

The solution of

⎧
⎨

u t
τ 0 +

B 2 ∂
∂

A 2

u tt = Δ u +

t Δ

u + f ( M , t ) ,

Ω × ( 0 , + ∞ ) ,

(6.90)

⎩

L ( u , u n ) | ∂Ω = 0 ,

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

is

1
τ

W ϕ (

0 + ∂

B 2 W λ mnk ϕ (

u = M , t )+ M , t )

∂ t

t

+ W ψ ( M , t )+ W f τ ( M , t − τ ) dτ (6.91)

0

where

⎧
⎨

0 < x < l 1 , one-dimensional case ,

Ω : 0 < x < l 1 , 0 < y < l 2 , two-dimensional case ,

⎩

0 < x < l 1 , 0 < y < l 2 , 0 < z < l 3 , three-dimensional case .

Ω . ∂Ω : the boundary of

⎧
⎨

λ , one-dimensional case ,

m

λ mnk = λ m + λ n , two-dimensional case ,

⎩

λ m + λ n + λ k , three-dimensional case .

⎧
⎨

M m , one-dimensional case ,

M mnk = M m M n , two-dimensional case ,

⎩

M m M n M k , three-dimensional case .

⎧
⎨

X m ( x ) , one-dimensional case ,

F mnk ( M )= X m ( x ) Y n ( y ) , two-dimensional case ,

⎩

X m ( x ) Y n ( y ) Z k ( z ) , three-dimensional case .