Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 309

Remark 4. Note that

P ( 2 m )
N

S m

m m

( u 0 ) ε = ( x ) q ( t ) ε , (6.194)

where N = 2 m or 2 m + 1, m = 0 , 1 , 2 , ··· , q ( t ) is a function to be determined. By

introducing an operator

t = τ

I 0 b A 2 u 2 d

d t

t A ( t − τ )

1
2 A

t − τ

2τ 0 d

e −

B = τ ( t − τ )

2

− d u

0 − A ( t − τ )

t = τ

0 t
0

1

τ 0

d
d t

t − τ

e −

= τ − d τ ,

we have

B m τ 0 1

τ 0

t

e −

q ( t )= − . (6.195)

Therefore,

B τ 0 1

τ 0

= τ 0 t
0

1

τ 0 e − τ 0 d

t t − τ

e − e −

− − τ

= τ 0

τ 0

t

e −

τ 0 − ( τ 0 + t ) ,

= τ 0 t
0

B 2 τ 0 1

τ 0

1

τ 0

t t − τ

e − τ 0 d

e − e −

− − τ τ

e −

τ 0

t 2
2

t

2
0

= τ τ 0 − τ 0 + t + ,

τ 0

= τ 0 t
0

B 3 τ 0 1

τ 0

1

τ 0 τ

2
2! e − τ 0 d

t t − τ

e − e −

− − τ

e −

τ 0

t 2
2!

t 3
3!

t

3
0

= τ τ 0 − τ 0 + t +

τ 0 +

.

2
0

τ

For the real computation, m is normally not larger than 4. By the recurrence method,
we have

= τ 0 t
0

B m τ 0 1

τ 0

1

τ 0

m − 1

τ

t t − τ

! e − τ 0 d

e − e − q ( t )= − − τ

( m − 1 )

e −

t 2
2!

t m

t
τ 0

0

= τ τ 0 − τ 0 + t +

τ 0 + ··· +

.

m −

1

m ! τ

0

L 2
T

m

]= TL 2 m · S m

m

T m + 1

The unit of the last term is [ ( u 0 ) ε · = Θ , which is cor-

rect. Therefore, the last term of the solution is independent of x if P N ( x ) is even. It