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PDE-Examples

PDE-Examples - u x = v y u y =-v x Maxwell’s equations in...

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Math 425, Spring 2011 Jerry L. Kazdan Classical Examples of PDEs Laplace equation: Δ u : = 3 j = 1 2 u x 2 j = 0 (some write Δ u = · u = 2 u ) Poisson equation: - Δ u = f ( x ) Helmholtz (or eigenvalue) equation: - Δ u = λ u Transport equation: u t = 3 j = 1 b j u x j Heat (or diffusion) equation: u t - k Δ u = 0 Schr¨odinger equation iu t = - Δ u + V ( x ) u Wave equation: u tt - c 2 Δ u = 0 Cauchy-Riemann equations:
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Unformatted text preview: u x = v y , u y =-v x Maxwell’s equations in a vacuum: E t = c ∇ × H , H t =-c ∇ × E , div H = , div E = Euler’s (nonlinear) equations for incompressible inviscid flow: u t + u · ∇ u =-∇ p , div u =...
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