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′ +
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2
1
ρ
= f (ρ)u
2 sin 2 (φ) RT′′ P
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Bernd Schröder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations