Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 277

Proof. By following a similar approach in Section 6.5.1, we obtain u ( r , θ , t ) , satis-

fying the equation and the boundary conditions of PDS (6.101),

)= m , n e α mn t u ( r , θ , t [( A mn cos β mn t + B mn sin β mn t ) cos n θ

+( C mn cos β mn t + D mn sin β mn t ) sin n θ ] J n ( k mn r ) . (6.102)

Applying the initial condition u ( r , θ , 0 )= ϕ ( r , θ ) yields

π a

0 ϕ (

⎧
⎨

1
M n M mn

A mn = d θ r , θ ) rJ n ( k mn r ) cos n θ d r ,

− π

π a

0 ϕ (

(6.103)

1
M n M mn

⎩

C mn = d θ r , θ ) rJ n ( k mn r ) sin n θ d r .

− π

Applying the initial condition u t ( 0 leads to

α mn A mn + β mn B mn =

r , θ , 0 )=

0 ,

α mn C mn + β mn D mn = 0 ,

so that
⎧
⎨

B mn = − α mn
β mn

A mn

π a

0 ϕ (

= − α mn
β mn

1
M n M mn

d θ r , θ ) J n ( k mn r ) r cos n θ d r ,

− π

(6.104)

D mn = − α mn
β mn

⎩

C mn

π a

0 ϕ (

= − α mn
β mn

1
M n M mn

d θ r , θ ) J n ( k mn r ) r sin n θ d r .

− π

By W ψ ( r , θ , t ) in Eq. (6.99), we have

)= m , n α mn sin β mn t + β mn cos β mn t

∂
∂

t W ϕ (

r , θ , t

B mn cos n D mn sin n e α mn t J n

· (

θ + θ )

( ) ,

k mn r (6.105)

where

⎨

π a

0 ϕ (

1
M n M mn β mn

B mn = d θ r , θ ) J n ( k mn r ) r cos n θ d r ,

− π

(6.106) π a

0 ϕ (

⎩ 1

M n M mn β mn

D mn = d θ r , θ ) J n ( k mn r ) r sin n θ d r .

− π