Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 264

Thus

1
M mn

C mn = ϕ ( x , y ) X m ( x ) Y n ( y ) d x d y .

D

Also,

)= m , n e α mn t α mn ( C mn cos β mn t + D mn sin β mn t )

u t ( x , y , t

β mn t X m (

+ − C mn β mn sin β mn t + D mn β mn cos x ) Y n ( y ) .

Applying the initial condition u t ( x , y , 0 )= 0 yields

α mn C mn + D mn β mn = 0 ,

which leads to

= − α mn
β mn

D mn C mn

1
τ 0 +( λ m + λ n )

B 2

1
M mn

= ϕ ( x , y ) X m ( x ) Y n ( y ) d x d y . (6.74)

2 β mn

D

By Eq. (6.72), we have
⎧
⎨

)= m , n D mn e α mn t α mn sin β mn t + β mn cos β mn t X m ( x ) Y n ( y ) ,

∂
∂ t W ϕ (

x , y , t

1
M mn β mn

⎩

D mn =

ϕ ( x , y ) X m ( x ) Y n ( y ) d x d y .

D

Thus

m , n C mn e α mn t cos β mn t · X m ( x ) Y n ( y )= ∂

t W ϕ (

x , y , t )

∂

⎛ ⎞

1
τ 0 +( λ

B 2 + λ )

1
M mn β mn

m n

⎝ ⎠

+ m , n

ϕ ( x , y ) X m ( x ) Y n ( y ) d x d y

2

D

e α mn t sin

· β mn t · X m ( x ) Y n ( y ) . (6.75)

Substituting Eqs. (6.74) and (6.75) into Eq. (6.73) and using the structure of
W ψ ( x , y , t ) in Eq. (6.72) leads to the solution of PDS (6.68)

1
τ

W ϕ (

0 + ∂

B 2 W ( λ m + λ n ) ϕ ( u = x , y , t )+ x , y , t ) .

∂ t