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**Home Assignment 4 / Re: Problem 1**

« **on:**December 20, 2012, 02:12:38 AM »

In addition to Part c solution I posted above, here is Problem 1 Part e Bonus solution to prove eigenvalues are simple.

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In addition to Part c solution I posted above, here is Problem 1 Part e Bonus solution to prove eigenvalues are simple.

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In addition to Part c solution I posted above, here is Problem 2 Part d Bonus solution to prove eigenvalues are simple.

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I just find by myself online for a well-typed clear solution for Problem 4 and share solution attached

http://www.math.uiuc.edu/~rdeville/teaching/442/hw2S.pdf

http://www.math.uiuc.edu/~rdeville/teaching/442/hw2S.pdf

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Thanks for a and b. As no one has posted Problem 2 part c yet, I just did using IBP as attached.

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Thanks for parts a b d above. As no one has posted part c yet, I just post solution attached.

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Thanks Aida for posting Part b in December. As there is still NO solution posted for Part d yet, I spent 1 hour to write part d and post attached same 2 images, first by camera, second by scan.

Thanks professor. Here I used error function in the textbook, instead of that given in the homework, but yes they are same. In part d, I got limit u(x,t)=1 for Neumann condition.

Thanks professor. Here I used error function in the textbook, instead of that given in the homework, but yes they are same. In part d, I got limit u(x,t)=1 for Neumann condition.

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Problem 3 b Neumann Solution Attached

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http://www.math.toronto.edu/courses/apm346h1/20129/HA9.html#problem-9.3

Problem 3a Dirichlet Solution Attached

Problem 3a Dirichlet Solution Attached

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Problem 1 My Solution Attached

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Yes--fixed now in html (not in pdf)

Thanks for fixing Problem 2. Solution Attached.

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Thanks Calvin's solutions at 21:47, especially parts c and d bonus, which are not required. I only posted parts a and b at 21:32, which are all required work for Problem 1. By the way, note in my part b, r should be replaced by Ï. Part a posted t 21:31 is OK.

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Dear Ian: I strongly agree with you we should justify for math education purpose. For TA marking purpose only, I think what I posted at 21:30 before Calvin is correct to get full marks, as question part c only requires "to formulate similar statements".

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Problem 3 b my own solution attached for 2 counter examples, different from Calvin's solution, I just typed it in Paint thanks!

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Anyone knows full solution of Problem 4? One of my ideas is gravitational potential approaches zero at infinity to set B=0 as:

http://www.math.toronto.edu/courses/apm346h1/20129/L27.html

http://www.math.toronto.edu/courses/apm346h1/20129/L27.html