Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 81

Generalized Fourier Expansion of u = W Ψ ( r , θ , ϕ , t )

a 0 ,the T

With k nl = μ (

1
2

)

n +

( t ) -equation becomes

l

T
nl

nl T nl =

+ ω 0 ,

a

a 0 2

1
2 )

( n +

2
nl

where ω = μ . Its general solution reads

l

c ( 1 ) c ( 2 )

T nl ( t )=

nl cos

ω nl t +

nl sin

ω nl t .

Therefore

∞
∑

u =

1 { (

a mnl cos ω nl t + b mnl sin ω nl t ) cos m ϕ

m , n = 0 , l =

⎛
⎝ μ (

⎞

2

)

n +

P n (

l

⎠

+( c mnl cos ω nl t + d mnl sin ω nl t ) sin m ϕ } cos θ ) j n r

a 0

satisfies the wave equation and the boundary conditions of PDS (2.54).
Rewriting Eqs. (2.56) and (2.58) in the form of Eq. (2.48) shows that the eigen-
function sets are orthogonal with respect to weight functions r 2 and sin θ , respec-

tively.
Applying the initial condition u ( r , θ , ϕ , 0 )= 0 leads to a mnl = c mnl = 0 . To satisfy

the initial condition u t ( r , θ , ϕ , 0 )= Ψ ( r , θ , ϕ ) , b mnl and d mnl must be determined

such that

⎛
⎝ μ (

⎞

1
2

)

n +

∞
∑

⎠ .

P n (

l

Ψ ( r , θ , ϕ )=

1 (

b mnl ω nl cos m ϕ + d mnl ω nl sin m ϕ ) cos θ ) j n r

a 0

m , n = 0 , l =