Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 137

is the solution of

u t

a 2 R 2 = Δ u + f ( x , y , t ) , × ( 0 , + ∞ ) ,

u ( x , y , 0 )= 0 .

The sum of Eqs. (3.53) and (3.54) is, by the principle of superposition, the solution
of

u t =

a 2 R 2 Δ u + f ( x , y , t ) , × ( 0 , + ∞ ) ,

u ( x , y , 0 )= ϕ ( x , y ) .

3.5.2 Three-Dimensional Case

We can apply a triple Fourier transformation followed by a triple inverse transfor-
mation to solve three-dimensional Cauchy problems. The method and the procedure
are similar to those in Section 3.5.1. Here, we simply list the final result. For PDS
u t

a 2 R 3

= + ( , , , ) , × ( , + ∞ ) ,

Δ u f x y z t 0

u ( x , y , z , 0 )= ϕ ( x , y , z ) .

The solution is

t

u = W ϕ ( x , y , z , t )+ W f τ ( x , y , z , t − τ ) d τ

0

= ϕ ( ξ , η , ζ ) V ( x , ξ , t ) V ( y , η , t ) V ( z , ζ , t ) d ξ d η d ζ

R 3

t

+ d τ f ( ξ , η , ζ , τ ) V ( x , ξ , t − τ ) V ( y , η , t − τ ) V ( z , ζ , t − τ ) d ξ d η d ζ .

0

R 3

3.6 Typical PDS of Diffusion

The heat-conduction equation is used not only to represent the diffusion of heat, but
also for other diffusions such as material diffusion. In this section, we discuss three
typical PDS and their solutions arising from material diffusion in manufacturing
silicon transistors to demonstrate the application of results in Section 3.4.