Mixed Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 160

u y ) ∂Ω = Example 4. Let the boundary conditions L ( u , u x , 0 in PDS (4.37) be

u x (

0 , y , t )= u x ( l , y , t )= 0 ,

(4.47)

u y ( x , 0 , t )= u y ( x , l , t )= 0 .

Find the Green function and write out its first four terms.

Solution . Based the given boundary conditions (4.47), we use the eigenfunctions in
Row 5 in Table 2.1 to expand G both in x and y ,

cos m

π x

cos n

π y

u 55 ( x , y )= .

l l

Therefore,

+ ∞
m , n = 0

τ 0 γ mn M mn cos m

1 πξ

l

cos m

π x

cos n

πη
l

cos n

π y

t − τ

2τ 0

e − G = sin γ mn ( t − τ ) ,

l l

4

τ 0 a 2 m

2

2 n

l 2
4 ,

1
2

l l

where M mn = M m M n = γ mn = + − 1.

τ 0

The first four terms are

t − τ

2τ 0

t − τ

2τ 0

l 2 1

τ 0

e − 1

τ 0

2 τ 0

i

4
l 2

i e

− 4

t − τ
2

t − τ

e − e −

G 00 =

τ 0

= − ,

2

γ 01 l 2 cos π l

4

cos π

y
l

t − τ
2

e − G 01 =

τ 0

sin γ 01 ( t − τ ) ,

τ

0

τ 0 γ 10 l 2 cos π l

4

cos π

x

t − τ
2τ 0

e −

= ( − τ ) ,

G 10 sin γ t 10

l

τ 0 γ 11 l 2 cos π l

4

cos π

x

cos π l cos π

y
l

t − τ
2τ 0

e −

G 11 = sin γ 11 ( t − τ ) .

l

It is always useful to perform a unit analysis. Here τ 0 a 2 =

L 2 and T − 1 so [ γ mn ]=

L − 2 .The u due to the nonhomogeneous term f

[ G ]= ( x , y , t ) reads

t

u = d τ Gf ( ξ , η , τ ) d σ ,

0

D

whose unit is [ u ]=[ d τ ][ G ][ f ][ d σ ]= Θ .