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# ps11 - u x t ± u t 5 u x = 0 u x 0 = cos(3 x 4 Using the...

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MATH 405: Problem Set 11 due Friday 12/15/2006 1. Using Faraday’s Law of Induction ∇× E = B /∂t between the electric field E and the magnetic field B , find the line integral of the electric field C E · dr around a closed current loop C with area A due to a spatially homogeneous time- varying magnetic field B = B 0 sin ωt which is oriented orthogonally to the plane of the loop C . 2. Consider a harmonic function φ , which means that φ is a solution to Laplace’s equation, 2 φ = 0. Show that for a region B with boundary ∂B : ∂B g ∂g ∂n dA = B |∇ g | 2 dV 3. Using the method of characteristics (change of variables into a new frame, as done in class) solve the following PDE for
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Unformatted text preview: u ( x, t ): ± u t + 5 u x = 0 u ( x, 0) = cos(3 x ) 4. Using the method of characteristics solve the following PDE for u ( x, t ): ± u t-16 u x =-5 u u ( x, 0) = e-2 x 2 5. For the following one-dimensional wave equation ∂ 2 u ∂t 2 = 7 ∂ 2 u ∂x 2 show explicitly that u 1 = cos(2 x-2 √ 7 t ) is a solution, and so is u 2 = ( x + √ 7 t ) 3 . Is u 1 + u 2 also a solution? 6. For the same wave equation, given the Gaussian initial condition u ( x, 0) = 3 exp(-x 2 ), what do you expect will happen at later times? Sketch u ( x, t )....
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