Mixed Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 161

4.3.2 Circular Domain

Boundary conditions for mixed problems in a circular domain are separable in polar
coordinate systems. This is similar to the wave equations in Chapter 2. Therefore,
we consider the mixed problems here in a polar coordinate system:
⎧
⎨

a 2

+ τ = ( , θ , )+ ( , θ , ) , < < , < ,

u t 0 u tt Δ u r t F r t 0 r a 0 0 t

L ( u , u r ) | r = a 0 = 0 , | u ( 0 , θ , t ) | < ∞ , u ( r , θ + 2 π , t )= u ( r , θ , t ) , (4.48)

⎩

u ( r , θ , 0 )= Φ ( r , θ ) , u t ( r , θ , 0 )= Ψ ( r , θ ) .

where boundary condition includes all three kinds.

Preliminaries

1. By following the same approach of Remark 4 in 2.5.2, we can obtain the follow-
ing solution structure theorem.

Theorem. The solution of PDS (4.48) reads

1 W Φ (

t

τ 0 + ∂

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

∂ t

0

, θ , τ ) τ 0 , W Ψ (

Where F τ = F ( r r , θ , t ) is the solution of PDS (4.48) at Φ = F = 0.

2. Two S-L problems will follow from seeking W Ψ ( r , θ , t ) , the solution of

PDS (4.48) at Φ = F = 0. Consider u = T ( t ) U ( r , θ ) .Wehave

Δ U + λ U = 0 ,

(4.49)

τ 0 T + T + λ a 2 T = 0 ,

where − λ is the separation constant. Another separation of variables by letting

k 2 leads to

U = R ( r ) Θ ( θ ) and λ =

Θ + μΘ = (a) 0 , Θ ( θ + 2 π )= Θ ( θ ) .

The former has eigenvalues and eigenfunctions

n 2 Eigenvalues μ = , n = 0 , 1 , 2 , ··· ,

a n + b n =

Eigen f unctions Θ n ( θ )= a n cos n θ + b n sin n θ , 0 .

⎧
⎨

k 2

r 2 R n

n 2 1

r R n (

R n ( r )+ r )+ − ( r )= 0 ,

(b) (4.50)

⎩

R n (

( , ) | r = a 0 = , | ( ) | < ∞ , | ) | < ∞ .

L R R r 0 R n 0 0