Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 141

Now concentrate all particles Q to x = 0, so the initial density becomes

⎧
⎨

∞ ,

= ,

x 0

( , )=

N x 0

0 , x = 0 ,

⎩

0sothat ¯ and h → + ξ → 0. The density distribution reduces to

Q
2 √ π Dt e − ( x − ξ ) 2

Q
2 √ π

x 2
4 Dt

Dt e −

N ( x , t )= lim = . (3.63)

4 Dt

h → + 0

Remark. This method of arriving at Eq. (3.63) actually demonstrates again the
physical background of the Dirac function, which is beneficial for understanding
it. However, it would be more straightforward to obtain Eq. (3.63) if we apply the
Dirac function directly to the initial distribution. By using the Dirac function, the
PDS reads

N t =

DN xx , − ∞ < x < + ∞ , 0 < t ,

( , )=

δ (

) .

N x 0 Q x

Its solution is readily available from Section 3.4,

+ ∞

Q
2 √ π

x 2
4 Dt

Dt e −

N ( x , t )= Q δ ( ξ ) V ( x , ξ , t ) d ξ = ,

− ∞

which is the same as Eq. (3.63).

3.6.4 Diffusion Between Two Semi-Infinite Domains

The density of the mixed material is N 0 (constant) and zero in semi-infinite domains
x

< >

0, respectively. This density difference drives the diffusion between
the two domains. The density N

0and x

( x , t ) should thus satisfy

⎧
⎨

N t = DN xx , − ∞ < x < + ∞ , 0 < t ,

N 0 ,

x < 0 ,

⎩

N ( x , 0 )=

0 , x > 0 .