Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 381

u ∗ +

Proof. Substituting u = a into PDS (7.91) yields

⎧
⎨

u ∗ + u ∗ =

Δ (

)= Δ ( ) , ∈ Ω ,

a f x x

u ∗ + u ∗ | ∂Ω +

( a ) | ∂Ω = a = c + a ,

⎩

∂Ω ∂Ω

u ∗ + ∂ ( a )

∂ u

d S =

n d S

= A ,

∂ n ∂

u ∗ + so that u a are indeed solutions of PDS (7.91).

Let u 1 be a solution of PDS (7.91). The v

=

u ∗ must thus satisfy

= u 1 −

Δ

v = 0 , x ∈ Ω ,

c ,

∂ v

∂ n d S

v | ∂Ω = = 0 ,

∂Ω

where c is a undetermined constant. By the first Green formula in n -dimensional
space,

n
i = 1 ∂ u

d x

u ∂

v ∂ v

u Δ v d x =

n d S

− .

∂ ∂ x i ∂ x i

Ω ∂Ω Ω

Therefore

n d x

u ∂

v

i = 1 ∂ u

∂ v

n d S

= .

∂ ∂ x i ∂ x i

∂Ω Ω

Let u = v . Thus, by applying the boundary condition

n
i = 1

2 d x

∂

c

v i

v ∂

v

c ∂

v ∂ v

=

n d S

=

n d S

=

n d S

= 0 ,

∂ x i ∂ ∂ ∂ Ω ∂Ω ∂Ω ∂Ω

∂ v i

u ∗ must be equivalent to a con- so that

x i =

0, i = 1 , 2 , ··· , n . Therefore v = u 1 −

∂

stant a .