1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

394 Chapter 6 Laplace Transform

Next, the equations are made dimensionless by introducing new vari-

ables:

C ̄=C−Ca

A , x ̄=

Vx

D( 0 ),

̄t= V

(^2) t
D( 0 ).
The new problem is
∂
∂x ̄

(

D(x ̄)

D( 0 )

∂C ̄

∂x ̄

)

−∂

C ̄

∂x ̄=

∂C ̄

∂ ̄t,^0 < ̄x,^0 <

̄t,

C ̄( 0 ,t ̄)=sin(ω ̄t), 0 < ̄t,

C ̄(x ̄, 0 )= 0 , 0 < ̄x,

whereω= 2 πD( 0 )/V^2 tC.

BecauseD(x ̄)depends in a complicated way onx ̄,anumericalsolution

was used. To check the numerical solution, the authors wished to find an

analytical solution of the problem corresponding to constant diffusivity,

D(x ̄)=D( 0 ).Letube the solution of

∂^2 u

∂x ̄^2

=∂u

∂x ̄

+∂u

∂ ̄t

, 0 <x ̄, 0 < ̄t,

u

(

0 , ̄t

)

=sin

(

ω ̄t

)

, 0 < ̄t,

u( ̄x, 0 )= 0 , 0 < ̄x,

ubounded asx→∞.

Find the Laplace transform of the solution of this problem.

28.The authors of the paper mentioned in Exercise 27 were particularly in-

terested in the persistent part of the solution. Use the methods of Sec-

tion 6.4 to show that the persistent part of the solution is

u 1 =

1

2 i

(

f(iω)−f(−iω)

)

,

where

f(iω)=exp

((

1

2

−

√

1

4

+iω

)

x ̄+iω ̄t

)

.

29.Find the square root required in the foregoing expression by setting

√

1

4

+iω=α+iβ