Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 119

Similar to the one-dimensional cases, we can also obtain the Green functions and
their physical implications by considering the case f ( x , y , t )= δ ( x − ξ , y − η , t − τ )

0.
By using the corresponding eigenfunctions in Table 2.1 to expand the solution,
we can also obtain the solutions for the other kinds of boundary conditions.

and ϕ ( x , y )=

Circular Domain

Find the solution of PDS
⎧
⎨

a 2 x 2 y 2 a 0 ,

u t = Δ u + f ( x , y , t ) + < 0 < t ,

L ( u , u n ) | r = a 0 = 0 , (3.10)

⎩

u ( x , y , 0 )= ϕ ( x , y ) .

Solution. Note that the boundary conditions in PDS (3.10) contain the first, the sec-
ond and the third kinds. It follows from Eq. (2.44) in Section 2.5.2 that the solution
for the case of f ( x , y , t )= 0is

⎧
⎨

+ ∞
∑

mn t

e − ω u = W Φ ( r , θ , t )=

0 (

b mn cos n θ + d mn sin n θ ) J n ( k mn r ) ,

m = 1 , n =

π a 0

1

b m 0 = Φ ( r , θ ) J 0 ( k m 0 r ) r d r d θ ,

2 π M m 0 − π 0

π a 0

(3.11)

1

⎩

b mn = Φ ( r , θ ) J n ( k mn r ) r cos n θ d r d θ ,

π M mn − π 0

π a 0

1

d mn = Φ ( r , θ ) J n ( k mn r ) r sin n θ d r d θ .

π M mn

− π 0

μ ( n m are the zero
points of Bessel functions which depend on the boundary conditions. The normal
squares M mn depend on the boundary conditions and are available in Chapter 2 for
the three kinds of boundary conditions. Therefore, the solution of PDS (3.10) is, by
the solution structure theorem,

, k mn = μ ( n ) Here Φ ( r , θ )= ϕ ( r cos θ , r sin θ ) / a 0 , ω mn = k mn a ,

m

t

u = W Φ ( r , θ , t )+ W F τ ( r , θ , t − τ ) d τ ,

0

where

f ( r cos θ , r sin θ , t )= F ( r , θ , t ) , F τ = F ( r , θ , τ ) .