Separation of Variables -- Legendre Equations - Louisiana Tech ...

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
Bring all terms that depend on ρ to the right side:
T ′′
T = ρ2 f (ρ)
P ′′ cos(φ) P
+
P sin(φ)
′ 1
+
P sin 2 T
(φ)
′′
T = ρ2 2 R′′ R′
f (ρ) − ρ − 2ρ
R R ,
Both sides must be constant.
ρ 2 2 R′′ R′
f (ρ) − ρ − 2ρ = −λ, or
R R
ρ 2 R ′′ + 2ρR ′ − λR + ρ 2 f (ρ) R = 0.
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Bernd Schröder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations