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

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Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation Separation of Variables 1. Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is a solution of the form u = R(ρ)T(θ)P(φ). 3. The special form of this solution function allows us to replace the original partial differential equation with several ordinary differential equations. 4. Key step: If f (ρ) = g(θ,φ), then f and g must be constant. 5. Solutions of the ordinary differential equations we obtain must typically be processed some more to give useful results for the partial differential equations. logo1 Bernd Schröder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation Separation of Variables 1. Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is a solution of the form u = R(ρ)T(θ)P(φ). 3. The special form of this solution function allows us to replace the original partial differential equation with several ordinary differential equations. 4. Key step: If f (ρ) = g(θ,φ), then f and g must be constant. 5. Solutions of the ordinary differential equations we obtain must typically be processed some more to give useful results for the partial differential equations. 6. Some very powerful and deep theorems can be used to formally justify the approach for many equations involving the Laplace operator. logo1 Bernd Schröder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations
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