Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 304

is, by the solution structure theorem,

1 W P m (

t

τ 0 + ∂

= , )+

W P l (

, )+

W P N (

, − τ )

τ . u x t x t x t d (6.178)

∂ t

0

Here

m
n = 0 b n x n

l
n = 0 a n x n

N
n = 0 c n x n

P m ( x )= , P l ( x )= , P N ( x )= .

Consider

N
n = 0 c n ( t ) x n

f ( x , t )= P N ( x , t )= , c n ( t ) are functions of t .

The solution of PDS (6.177) for the case of P m ( x )= P l ( x )= 0 is, by Eq. (5.33),

2 N

n d

I 0 b A 2

t x + A ( t − τ )

1
2 A

t − τ
2τ 0 d

e −

n = 0 c n ( τ ) ξ

= ( − τ )

2

− ( − ξ )

u τ t x ξ

0 x − A ( t − τ )

[ N / 2 ]
k = 1

d

1

τ 0 d

= τ 0 t
0

t

P ( 2 k )
N

t − τ

1
2 A

( x , τ )

t − τ

2τ 0

e − e −

P N ( x , τ ) − τ + G 2 k ( t − τ ) τ

( 2 k ) ! 0

(6.179)

or

t

u = W P N τ ( x , t − τ ) d τ , P N τ = P N ( x , τ ) . (6.180)

0

Here P ( 2 k )
N

with respect to x . It can be
shown that the unit in the above solutions is correct. For example, the unit of the
second term on the right side of Eq. (6.179) is

( x , τ ) is the 2 k -th derivative of P N ( x , τ )

A − 1 P ( 2 k )
N

T
L ·

Θ
T 2 L 2 k

L 2 k + 1 [ u ]= ( x , τ ) [ G 2 k ( t − τ )] [ d τ ]= · · T = Θ .

6.9.3 Perturbation Solutions of Dual-Phase-Lagging
Heat-Conduction Equations

Consider

u t

A 2 u xx R 1

/ τ + u tt = + ε u txx , × ( 0 , + ∞ ) ,

0

(6.181)

u ( x , 0 )= 0 , u t ( x , 0 )= P N ( x ) .