Assignment 7 solutions

Assignment 7 solutions

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.085 - Mathematical Methods for Engineers I Prof. Gilbert Strang Solutions - Problem Set 7 3.4, Problem 4. Show that u = r cos ∂ + r −1 cos ∂ solves Laplace’s equation (13), and express u in terms of x and y . Find v = (ux , uy ) and verify that v · n = 0 on the circle x2 + y 2 = 1. This is the velocity of flow past a circle in Figure 3.18. Show u = r cos ∂ + r −1 cos ∂ solves Laplace’s equation: Laplace’s equation in r, ∂ is: �2u 1 �u 1 �2u + + 2 2 = 0. 2 �r r �r r �∂ (13) Now, if u = r cos ∂ + r −1 cos ∂, then �u = cos ∂ − r−2 cos ∂ �r �2u = 2r−3 cos ∂ � r2 Plugging this into the left-hand side of (13), we get 1 1 1 2r−3 cos ∂ + (cos...
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This note was uploaded on 04/12/2011 for the course MATH 18.085 taught by Professor Staff during the Fall '10 term at MIT.

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