Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 114

⎧
⎨ a 2 u t = Δ u + f ( M , t ) , Ω × ( 0 , + ∞ ) ,

L u

∂Ω =

, ∂

u

(3.2)

0 ,

⎩

∂ n

( , )= ,

u M 0 0

⎧
⎨ a 2

= + ( , ) ,

Ω × (

, + ∞ ) ,

u t Δ u f M t 0

∂Ω =

L u , ∂

u

(3.3)

0 ,

⎩

∂ n

( , )= ϕ ( ) ,

u M 0 M

( , ) ( , , )

where M represents the point x , x y and x y z in one-, two- and three-dimensional

space. For the one-dimensional case, Δ u is defined as u xx .

Theorem. Suppose that u 1 = W ϕ ( M , t ) is the solution of (3.1), then

t

1. u 2 = W f τ ( M , t − τ ) d τ is the solution of (3.2), where f τ = f ( M , τ ) ;

0

t

2. u 3 = W ϕ ( M , t )+ W f τ ( M , t − τ ) d τ is the solution of (3.3).

0

Proof.
1. Since W f τ (

M , t − τ ) satisfies

⎧
⎨
⎩

∂ W f τ

∂

a 2

t =

Δ W f τ , Ω × ( 0 , + ∞ ) ,

∂Ω =

L W f τ , ∂

W f τ
∂

0 ,

n

W f τ t = τ =

f ( M , τ ) ,

then

t t

2 u 2
∂

∂

u 2 = ∂
∂

a 2 a 2

t 2 −

Δ W f τ ( M , t − τ ) d τ − Δ W f τ ( M , t − τ ) d τ

t 0 0

t

a 2 t

W f τ τ = t −

∂ W f τ

∂

= d τ +

0 Δ

W f τ d τ

t 0

∂ d

t

W f τ
∂

a 2

= −

τ +

( , )= ( , ) .

Δ W f τ f M t f M t

t

0

Therefore, u 2 satisfies the equation of (3.2).
Also,

L u 2

∂Ω =

L W f τ , ∂

∂Ω

t

, ∂

u 2
∂

W f τ
∂

τ =

d 0

n n

0