Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 139

For the slow diffusion which we consider in this section, the domain of variation
of x can be viewed as . When manufacturing the silicon

transistors, there is no source of mixed material inside the silicon plates. Therefore,
we only consider Cauchy problems of homogeneous diffusion equations.

− ∞ < x < + ∞ or 0 ≤ x < + ∞

3.6.2 Diffusion from a Constant Source

Diffusion from a constant source refers to the particle diffusion in a semi-infinite
domain without initial distribution of particles from a constant source of density N 0
at x = 0. Diffusion in closed tubes and boxes are all of this kind. The particle density

N ( x , t ) should thus satisfy the PDS

⎧
⎨

N t = DN xx , 0 < x < + ∞ , 0 < t ,

N ( 0 , t )= N 0 , (3.57)

⎩

N ( x , 0 )= 0 .

Let N ( x , t )= v ( x , t )+ N 0 ; thus v ( x , t ) is the solution of a PDS with homogeneous

boundary conditions

⎧
⎨

= , < < + ∞ , < ,

v t Dv xx 0 x 0 t

( , )= ,

v 0 t 0 (3.58)

⎩

v ( x , 0 )= − N 0 .

By an odd continuation, PDS (3.58) becomes
⎧
⎨

v t = Dv xx , − ∞ < x < + ∞ , 0 < t ,

⎧
⎨

− N 0 , x > 0 ,

(3.59)

⎩

v ( 0 , t )= 0 , v ( x , 0 )= 0 , x = 0 ,

⎩

N 0 , x < 0 .

Note that the boundary homogenization and the continuation of the initial contribu-
tion indeed preserve the CDS in (3.57).
The solution of PDS (3.59) is, by the result in Section 3.4,

0 + ∞

v ( x , t )= N 0 V ( x , ξ , t ) d ξ + ( − N 0 ) V ( x , ξ , t ) d ξ .

− ∞ 0

− ξ
2 √ Dt

x

η = ξ −

x
2 √ Dt

After performing the variable transformation of η = and for the