Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 117

When f ( x , t )= δ ( x − x 0 , t − t 0 ) ,

t l

u = G ( x , ξ , t − τ ) δ ( ξ − x 0 , τ − t 0 ) d ξ d τ = G ( x , x 0 , t − t 0 )

0 0

( , , − )

is thus the temperature distribution in a heat-conduction rod that is
due to a source term (the nonhomogeneous term f

G x x 0 t t 0

( x , t ) in the equation) of the unit

δ − or the temperature distribution caused by a unit point

source at time instant t 0 and spatial point x 0 . Therefore, u

function δ ( x − x 0 , t − t 0 )

= G ( x , ξ , t − τ ) satisfies

⎧
⎨

a 2 G xx + δ (

G t = x − ξ , t − τ ) , 0 < x < l , 0 < τ < t < + ∞ ,

G | x = 0 = G | x = l = 0 ,

⎩

G | t = τ = 0 .

The solution of PDS (3.4) is, by the principle of superposition, the sum of
Eqs. (3.5) and (3.6).

Boundary Condition of the Second Kind

Find the solution of PDS
⎧
⎨

a 2 u xx + u t = f ( x , t ) , ( 0 , l ) × ( 0 , + ∞ ) ,

u x ( 0 , t )= u x ( l , t )= 0 , (3.7)

⎩

u ( x , 0 )= ϕ ( x ) .

Solution. It follows from the corresponding solution of the wave equation in Sec-
tion 2.2.2 that the solution of PDS (3.7) is, for the case f ( x , t )= 0,

⎧
⎨

a 0
2 +

+ ∞
k = 1 a k e − (

2 t cos k

π x

k π a

)

=

u W ϕ ( x , t )= ,

l

l

(3.8)

l
0 ϕ (

2
l

cos k

π x

⎩

a k = x ) d x , k = 0 , 1 , 2 , ··· .

l

Therefore, the solution structure theorem yields the solution of PDS (3.7) for the
case ϕ (

)=

x 0,

t t l

u = W f τ ( x , t − τ ) d τ = G ( x , ξ , t − τ ) f ( ξ , τ ) d ξ d τ . (3.9)

0 0 0

Here the Green function is

+ ∞
k = 1

1
l +

2

− τ ) cos k

πξ
l

cos k

π x

2

l e − (

k π a
l

)

( t

G ( x , ξ , t − τ )= .

l

The solution of PDS (3.7) is thus the sum of Eqs. (3.8) and (3.9).