Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 253

which implies

>

0 , h 1 + h 2 = 0 ,

λ =

0 , h 1 + h 2 = 0 .

However, the latter will lead to a trivial solution for PDS (6.43), which cannot in
fact occur because physically h 1 >

0and h 2 > 0 for boundary conditions of the third

kind (if h 1 = h 2 = 0, the case reduces to the boundary conditions of the second kind

at both ends).

6.3.3 Solve Mixed Problems with Table 2.1

We can use the eigenfunctions in Table 2.1 and the solution structure theorem to
effectively solve mixed problems of dual-phase-lagging heat-conduction equations
for nine combinations of boundary conditions.

Example 1. Solve
⎧
⎨

A 2 u xx + B 2 u txx + u t / τ 0 + u tt = f ( x , t ) , ( 0 , l ) × ( 0 , + ∞ ) ,

u x ( 0 , t )= 0 , u ( l , t )= 0 , (6.46)

⎩

( , )= ϕ ( ) , ( , )= ψ ( ) .

u x 0 x u t x 0 x

Solution. To obtain the solution of PDS (6.46), by the solution structure theorem
we first seek its solution at f 0. Based on the given boundary

conditions, we should use the eigenfunctions in Row 4 in Table 2.1 to expand the
solution,

( x , t )= 0and ϕ ( x )=

+ ∞

m = 0 T m ( t ) cos ( 2 m + 1 ) π x

u ( x , t )= .

2 l

Substituting it into the equation yields

T m ( 1

τ 0 +

2 T m (

(

+ ∞
m = 0

2 m + 1 ) π B

t )+ t )

2 l

cos (

( 2

2 m + 1 ) π A 2 m + 1 ) π x

+ T m ( t ) = 0 ,

2 l 2 l

which leads to, by the completeness and the orthogonality of cos (

,

2 m + 1 ) π x

2 l

1
τ 0 +

2 T m (

( ( 2

2 m + 1 ) π B 2 m + 1 ) π A

T m ( t )+ t )+ T m ( t )= 0 .

2 l 2 l