Special Functions - Heat Conduction: Mathematical Models and Analytical Solutions - page 426

which has a homogeneous boundary condition. Assume U ( r , θ )= R ( r ) Θ ( θ ) . Sub-

stituting it into Eq. (A.3) yields

r 2 R + rR + λ r 2 R

= − Θ
Θ

.

R

Thus, with μ as the separation constant,

r 2 R + rR +( λ r 2

− μ ) R = 0 (A.4)

and

Θ ( θ )+ μΘ ( θ )= 0 .

Since Θ ( θ + 2 π )= Θ ( θ ) , μ n must be

n 2

μ n = , n = 0 , 1 , 2 , ··· .

Therefore Eq. (A.4) becomes
x 2 F (

)+ x 2

n 2 F xF (

)+ − ( )= ,

x x x 0 (A.5)

. Equation (A.5) is a linear homogeneous ordinary
differential equation of second order with variable coefficients. It is called the Bessel
equation of n-th order and also appears in solving two-dimensional wave equations
and Laplace equations in a cylindrical coordinate system.
The three-dimensional Laplace equation in a spherical coordinate system reads

R x

√

√

where x = λ r , F ( x )=

λ

r 2 ∂ sin

2 u
∂ϕ

r 2 ∂

1 u 1

r 2 sin

∂
∂θ

∂ u

∂θ

1
r 2 sin 2

∂

+ θ +

2 =

0 . (A.6)

∂ r ∂ r θ

θ

Let u = R ( r ) Θ ( θ ) Φ ( ϕ ) . Substituting it into Eq. (A.6) yields

r 2 d R
d r

sin

d 2

1
R

d
d r

1 d

d

d d 1 Φ

= − θ −

2 .

sin 2

Θ sin θ θ θ d ϕ

Φ θ

Thus with n ( n + as the separation constant,

r 2 d 2 R

1 )

2 r d R

d r 2 + d r −

n ( n + 1 ) R = 0 ,

sin

d 2

1
Θ

d
d

d d 1

Φ

Φ

sin 2 sin θ θ + n ( n + 1 ) θ = −

2 .

θ θ d ϕ

The former is the Euler equation. The latter leads to, with η as the separation con-

stant,

sin

1
sin

d
d

d d η

sin 2

θ −

θ Θ +

n ( n + 1 ) Θ = 0 ,

θ θ θ

d 2 Φ

2 + ηΦ =

0 , Φ ( ϕ + 2 π )= Φ ( ϕ ) .

d ϕ