Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 263

Its characteristic roots are r 1 , 2 = α mn ± β mn i, where

4

1
τ

B 2

1
τ 0 +( λ m + λ n )

B 2 2

1
2

1
2

A 2 α mn = −

0 +( λ m + λ n )

, β mn = ( λ m + λ n ) − .

Thus the equation (6.70) becomes

)= m , n e α mn t u ( x , y , t ( A mn cos β mn t + B mn sin β mn t ) X m ( x ) Y n ( y ) (6.71)

sin

β mn t , when β mn = 0 ,

where sin β mn t = A m and B m are constants to be deter-

t , when β = 0 . mn

mined from the initial conditions. Applying the initial condition u ( x , y , 0 )= 0 yields

A mn = 0. Also

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

where

β mn , when β mn = 0 ,

β mn =

1 , when β = 0 .

mn

To satisfy the initial condition u t ( x , y , 0 )= ψ ( x , y ) , B mn must be determined such

that

m , n B mn β mn X m ( x ) Y n ( y )= ψ ( x , y ) .

Thus we obtain W ψ (

, , )

x y t , the solution of PDS (6.64)

⎧
⎨

)= m , n B mn e α mn t sin β mn t · X m ( x ) Y n ( y ) , u = W ψ ( x , y , t

1
M mn

(6.72)

⎩

B mn = ψ ( x , y ) X m ( x ) Y n ( y ) d x d y .

β mn

D

Similarly, the solution of PDS (6.68) can be written as

= mn e α mn t u ( C mn cos β mn t + D mn sin β mn t ) X m ( x ) Y n ( y ) . (6.73)

( , , )= ϕ ( , )

Applying the initial condition u x y 0 x y yields

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