Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 289

fore, in applications, we can directly write out the solutions of the 189 PDS based
on Eqs. (6.123) and (6.139) without going through the individual details.

Remark 2 .If β mnk is purely imaginary for some m , n and k such that β mnk =

e iβ mnk t e − iβ mnk t

2i

−

i γ mnk ( γ mnk is real ) , we can change sin β mnk t into by using the for-

e i z e − i z

−

mula sin z = for any imaginary variable z . We thus have the term

2i

1
β mnk

1 1

γ mnk s h

e γ mnk t e − γ mnk t

sin β mnk t =

γ mnk (

− )= γ mnk t .

2

6.7 Mixed Problems in a Spherical Domain

Boundary conditions of all the three kinds for mixed problems in a spherical domain
are separable with respect to the spatial variables in a spherical coordinate system. In
this section we apply the separation of variables to find solutions of mixed problems
in a spherical coordinate system
⎧
⎨

u t
τ 0 +

B 2 ∂
∂

A 2

u tt = Δ u ( r , θ , ϕ , t )+

t Δ

u ( r , θ , ϕ , t )

+ f ( r , θ , ϕ , t ) , 0 < r < a , 0 < θ < π , 0 < ϕ < 2 π , 0 < t ,

⎩

L ( u , u r ) | r = a = 0 ,

u ( r , θ , ϕ , 0 )= Φ ( r , θ , ϕ ) , u t ( r , θ , ϕ , 0 )= ψ ( r , θ , ϕ ) ,

(6.140)

where Ω stands for a sphere of radius a , with ∂Ω as its boundary. The boundary

condition L ( u , u r ) | r = a = 0 contains all three kinds. We will also examine the relation

among solutions from Φ ( r , θ , ϕ ) , ψ ( r , θ , ϕ ) and f ( r , θ , ϕ , t ) , respectively.

6.7.1 Solution from ψ ( r , θ , ϕ )

The solution due to ψ ( r , θ , ϕ ) satisfies

⎧
⎨

u t
τ 0 +

B 2 ∂
∂

A 2 u tt = Δ u ( r , θ , ϕ , t )+

t Δ

u ( r , θ , ϕ , t ) ,

0 < r < a , 0 < θ < π , 0 ≤ ϕ ≤ 2 π , 0 < t ,

(6.141)

⎩

L ( u , u r ) | r = a = 0 ,

u ( r , θ , ϕ , 0 )= 0 , u t ( r , θ , ϕ , 0 )= ψ ( r , θ , ϕ ) .