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Partial Differential Equations and Applied Mathematics

by: Ila Haag

Partial Differential Equations and Applied Mathematics MATH 5220

Marketplace > University of Virginia > Mathematics (M) > MATH 5220 > Partial Differential Equations and Applied Mathematics
Ila Haag
GPA 3.5

Lawrence Thomas

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Lawrence Thomas
Class Notes
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This 2 page Class Notes was uploaded by Ila Haag on Monday September 21, 2015. The Class Notes belongs to MATH 5220 at University of Virginia taught by Lawrence Thomas in Fall. Since its upload, it has received 52 views. For similar materials see /class/209572/math-5220-university-of-virginia in Mathematics (M) at University of Virginia.

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Date Created: 09/21/15
Math 522 Take home nal exam 2009 You may use your books and notes but no discussing among each other Please sign honor pledge Explain your work The exam is due within 48 hours after you receive it and no later than in class May 2nd I would appreciate early submissions There is a total of 60 points 1 Consider the problem of nding a real function uzy bounded and har monic in the upper half plane D E R2 y gt 0 and having boundary values on the real line given by u 0 1 Here7s an approach to constructing u a 5 points Show that kxy eikmequotkly k any real number is harmonic in D as is 1 ikm ilk A ultzy 7 e e W dk lt2 7r for any integrable function b 10 points In the above k is the Fourier transform of the data 90 Wm dz lt3 Stick this integral into the previous equation Eq2 to get a double integral but then do the k integral rst which you can do explicitly You should be left with an integral of the function g convolved with another function ie you should be looking at something like my hyltz e zltgltz dz lt4 Find c 10 points Solve the above boundary value problem for uy with boundary data g given as follows 0 for z lt 0 9z 1for0 x 1 5 0forgt1 Math 522 Take home exam on Legendre functions You may use your books and notes but no discussing among each other Please sign honor pledge Explain your work The exam is due in class April 16 There are a total of 40 points We know that 1 1 3 5 2722 723 1 71 12 2 8 16 for lt 1 by power series expansion a 10 points Use the generating function for the Legendre functions and this power series expansion to determine the rst four Legendre polynomials 13000713101 P201 P301 b 5 points Determine 1360 Set u 0 in the generating function expansion What do you learn Recall that the Pmpm071727m comprise a complete set of functions on 171 Any square integrable function u can be expanded mo gamma lt2 We also know that 1 7r L1 PmpPmIu dp 0 Pmcos 6Pmcos 6 sin6d6 2 76m W 3 2m 1 Suppose that given f we set M 1Pmufudu lt4


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