Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 292

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

⎧
⎨

1
M mnl

P n ( r 2 cos m A mnl = Φ ( r , θ , ϕ ) cos θ ) j n ( k nl r ) ϕ sin θ d θ d r d ϕ ,

r ≤ a

1
M mnl

⎩

P n ( r 2 sin m C mnl = Φ ( r , θ , ϕ ) cos θ ) j n ( k nl r ) ϕ sin θ d θ d r d ϕ .

r ≤ a

(6.149)

Applying the initial condition u t ( 0 yields

α nl A mnl + β nl B mnl =

r , θ , ϕ , 0 )=

0 ,

α nl C mnl + β nl D mnl = 0 ,

so that
⎧
⎨

B mnl = − α nl
β nl

α nl
M mnl β nl

A mnl = −

P n ( r 2 cos m · Φ ( r , θ , ϕ ) cos θ ) j n ( k nl r ) ϕ sin θ d θ d r d ϕ ,

r ≤ a

D mnl = − α nl
β nl

α nl
M mnl β nl

⎩

C mnl = −

P n ( r 2 sin m

·

Φ (

, θ , ϕ )

θ )

( )

ϕ . r cos j n k nl r ϕ sin θ d θ d r d

r ≤ a

By the structure of W ψ ( r , θ , ϕ , t ) in Eq. (6.146), we have

⎧
⎨

)= m , n , l α nl sin β nl t + β nl cosβ nl t
· (

∂
∂ t W Φ (

r , θ , ϕ , t

B mnl cos m D mnl sin m e α nl t P n ( ϕ + ϕ ) cos θ ) j n ( k nl r ) ,

A mnl
β nl =

1
M mnl β nl
·

B mnl =

P n ( r 2 cos m Φ ( r , θ , ϕ ) cos θ ) j n ( k nl r ) ϕ sin θ d θ d r d ϕ ,

⎩

r ≤ a

C mnl
β nl =

1
M mnl β nl
·

D mnl =

P n ( r 2 sin m Φ ( r , θ , ϕ ) cos θ ) j n ( k nl r ) ϕ sin θ d θ d r d ϕ .

r ≤ a

(6.150)