Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 305

Let the solution of PDS (6.181) be

2 n

u = u 0 ( x , t )+ u 1 ( x , t ) ε + u 2 ( x , t ) ε + ··· + u n ( x , t ) ε + ··· , (6.182)

where u n ( are functions to be determined. Substituting Eq. (6.182)

into (6.181) and comparing the coefficients of

x , t )( n = 0 , 1 , 2 , ··· )

n

ε ( n = 0 , 1 , 2 , ··· ) yields the PDS of

u n ( x , t )( n = 0 , 1 , 2 , ··· )

0 : u 0 t / τ 0 +

A 2 u 0 xx , R 1 u 0 tt = × ( 0 , + ∞ )

ε (6.183)

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

1 : u 1 t / τ 0 +

A 2 u 1 xx + R 1 u 1 tt = u 0 txx , × ( 0 , + ∞ )

ε (6.184)

u 1 ( x , 0 )= 0 , u 1 t ( x , 0 )= 0 .

2 : u 2 t / τ 0 +

A 2 u 2 xx + R 1 u 2 tt = u 1 txx , × ( 0 , + ∞ )

ε (6.185)

u 2 ( x , 0 )= 0 , u 2 t ( x , 0 )= 0 .

··· ··· ··· ···

( , ) −

The solution u 0 0)-

th polynomial of x . The nonhomogeneous term u 0 txx in PDS (6.184) can be obtained
and is a ( N

x t of (6.183) is the same as that in Eq. (6.176), which is a ( N

− 2)-th polynomial of x .The u 1 ( x , t ) can be obtained by Eq. (5.33) by

replacing f ( x , t ) by u 0 txx and is a ( N − 2)-th polynomial of x .The u 1 txx is thus a ( N −

4)-th polynomial of x . Therefore, u n ( x , t )( n = 0 , 1 , 2 , ··· ) is a ( N − 2 n )-th polynomial

of x and u n ( x , t )= 0( n =[ N / 2 ]+ 1 , [ N / 2 ]+ 2 , ··· ). Therefore, the analytical solution

of PDS (6.181) can be expressed by

2

) ε [ N / 2 ] . u = u 0 ( x , t )+ u 1 ( x , t ) ε + u 2 ( x , t ) ε + ··· + u [ N / 2 ] ( x , t (6.186)

Let

I 0 b A 2

2

x = ξ
t = τ

t x + A ( t − τ )

3

1
2 A

∂

t − τ
2τ 0 d

e −

S ()= τ ( t − τ )

2

− ( x − ξ ) d ξ ,

∂ t ∂ x 2

0 x − A ( t − τ )

(6.187)

S 2 , S [ N / 2 ] ( so that S ( u 0 )= u 1 , S ( u 1 )= ( u 0 )= u 2 , ··· u 0 )= u [ N / 2 ] . Thus the solution

of PDS (6.181) is

S 2

2

S [ N / 2 ] ( u 0 ) ε [ N / 2 ] .

u = u 0 + S ( u 0 ) ε + ( u 0 ) ε + ··· + (6.188)