1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

2.5 Example: Different Boundary Conditions 169

construct the functionG(x)with these properties:

G(x)=g(x), 0 <x<a,

G(x)=g( 2 a−x), a<x< 2 a.

Show thatG(x)corresponds to the series

G(x)∼

∑∞

N= 1

BNsin

(Nπx

2 a

)

, 0 <x< 2 a.

12.Show that theBNof the series in the preceding equation satisfy

BN= 0 (Neven), BN=

2

a

∫a

0

g(x)sin

(Nπx

2 a

)

dx (Nodd).

13. a.Solve this problem over the interval 0<x< 2 a.

∂^2 u

∂x^2 =

1

k

∂u

∂t,^0 <x<^2 a,^0 <t,

u( 0 ,t)=T 0 , u( 2 a,t)=T 0 , 0 <t,

u(x, 0 )=g(x), 0 <x< 2 a.

Afunctionfis given over the interval 0<x<a,andgis an extension

offdefined by

g(x)=

{

f(x), 0 <x<a,

f( 2 a−x), a<x< 2 a.

b.Explain why the solution of the problem comprising Eqs. (1)–(4) is

exactly the same as the solution of the problem in parta.

14.In the ceramics industry, the following problem has to be analyzed for
parameter measurement. A cylindrical rod of uniform porous material is
suspended vertically so that its lower end is immersed in water. The cylin-
drical surface and the upper end are sealed — with wax, for example. The
concentration of water in the rod (weight per unit volume) is a function
C(x,t)that satisfies the boundary value problem

D∂

(^2) C
∂x^2 =

∂C

∂t,^0 <x<L,^0 <t,

C( 0 ,t)=C 0 , ∂∂Cx(L,t)= 0 , 0 <t,

C(x, 0 )= 0 , 0 <x<L.