Mixed Problems of Hyperbolic Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 169

k 2 U

Tabl e 4 . 2 Eigenfunctions of Δ U + = 0inasphere

S-L
problem

Eigenvalues Eigenfunctions Normal square Weight

function

1
2

m 2
m = 0 , 1 , ···

(4.62) a m cos m ϕ + b m sin m ϕ

a m + b m = 0

π , m = 0
1
π ,

1

m = 1 , 2 , ···

( n + m ) ! 2

P n ( (4.63) n ( n + 1 ) cos θ ) , m ≤ n , sin θ

( n − m ) ! 2 n + 1

d n
d x n (

1
2 n n !

n = 0 , 1 , 2 , ···

x 2

n

P n ( x )= − 1 )

d m P n (

x )

m
2

P n ( x 2 x )=( 1 − )

d x m

l a 0 2
n = 0 , 1 , 2 , ···

j n r a 0 μ (

1
2

μ (

1
2

)

n + n +

r 2 (4.64) , M nl

l

2 x J n +

j n ( x )=

2 (

x ) 1

Remark 3. It can be readily shown that the solution structure theorem is also valid
in spherical coordinate systems.

Example 3. Solve
⎧
⎨

a 2

u t + τ 0 u tt = Δ u ( r , θ , ϕ , t )+ F ( r , θ , ϕ , t ) ,

0 < r < a 0 , 0 < t ,

L ( u , u r ) | r = a 0 = 0 , | u ( 0 , θ , ϕ , t ) | < ∞ , (4.65)

⎩

u ( r , θ , ϕ + 2 π , t )= u ( r , θ , ϕ , t ) ,

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

Solution. We first seek u = W Ψ ( r , θ , ϕ , t ) , the solution for the case F = Φ = 0. After

separating variables, the T ( t ) -equation reads

τ 0 T + T +( λ nl a

2 T

) = 0 ,

a 0 . Thus the solution of PDS (4.65) for the case F

λ nl = μ (

2

)

n +

where = Φ = 0

l

can be written as

a ( 1 )

γ nl t cos m

∞
∑

t
2τ 0

e − b ( 1 )

u =

mnl cos

γ nl t +

mnl sin

ϕ

m , n = 0 , l = 1

⎛
⎝ μ (

⎞

2

)

a ( 2 )

γ nl t sin m

P n (

n +

b ( 2 )

l

⎠ ,

+

mnl cos

γ nl t +

mnl sin

ϕ cos θ ) j n r

a 0

4

1

2

where γ nl = τ 0 ( λ nl a ) − 1.

2 τ

0