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M257-316Notes_Lecture26

# M257-316Notes_Lecture26 - Chapter 22 Lecture 26 Laplaces...

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Chapter 22 Lecture 26 - Laplace’s Equation Steady State Solutions of the Heat or Wave Equations that do not vary with time so that ∂u ∂t =0= 2 u 2 . 2D: Δ u = 2 u ∂x 2 + 2 u ∂y 2 =0 . (22.1) 3D: Δ u = 2 u 2 + 2 u 2 + 2 u ∂z 2 . (22.2) No initial conditions required. Only boundary conditions. In Polar Coordinates: Δ u = 2 u ∂r 2 + 1 r + 1 r 2 2 u ∂θ 2 =0. 129

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Lecture 26 - Laplace’s Equation 22.1 Summary In this course we have studied the solution of the second order linear PDE. ∂u ∂t = α 2 Δ u 2 u 2 = c 2 Δ u Δ u =0 Heat: Parabolic T = αX 2 Wave: Hyperbolic T 2 c 2 X 2 = A Laplace’s Eq.: Elliptic X 2 + Y 2 = A. (22.3) Important: 1. These equations are second order because they have at most 2nd par- tial derivatives. 2. These equations are all linear so that a linear combination of solutions is again a solution. 22.2 Laplace’s Equation (1) 2D Steady-State Heat Conduction and (2) Static Deﬂection of a Mem- brane.
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M257-316Notes_Lecture26 - Chapter 22 Lecture 26 Laplaces...

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