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May 19th, 2018, 04:14 AM   #1
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Heat Equation - separation of variables

Hello all , I have been asked to solve the problem in the image attached. I tried the standard separation of variables guess of u(x,t) = X(x)T(t) but it does not seem to separate at all. In the solution , it is mentioned that because the problem is accompanied by "Neumann" boundary conditions , we can immediately guess that the spatial (x) dependence follows a fourier cosine series ; that is our guess is of the form shown in the second image attached. How does one arrive at this conclusion?
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 tt.jpg (14.2 KB, 7 views) tt1.jpg (8.5 KB, 3 views)

 May 19th, 2018, 04:59 AM #2 Senior Member   Joined: Jun 2015 From: England Posts: 887 Thanks: 265 This is a non homogeneous PDE. The usual method is to apply Duhamel's Principle. http://people.math.gatech.edu/~xchen...at-Duhamel.pdf
June 3rd, 2018, 10:22 PM   #3
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Looks like this. It says heat flow so the final amount minus a initial amount will give you the rate but you don't want that, you want where the bouncing particles have flowed to. f, a map linear on Tangent, from derivative u is the one-form with components sin squared pi x. You want the determinant for p to f to give you the flow. It does not actually look like a square and a line coming out the left but the lines do represent the direction sin is going because x is telling it where to go. I don't know how to solve this because I don't know the X and Y parameters to u. The boundary conditions say on the right equation the x between vectors x and Y is a duel basis. It gives a direction and amount for anything tangent to the Y vector. The initial conditions say p=0. The function f is described as a derivative when it =1 and a basis when it =0. So I have no idea.
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 equation.jpg (26.7 KB, 0 views)

 June 17th, 2018, 11:25 AM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 You can "guess that the spatial (x) dependence follows a fourier cosine series" because the spatial dependence, "$sin^2(\pi x)$", is an even function. Don't worry about "separating" the variables, using X(x)T(t), just to ahead and write the solution as $\displaystyle u(x,t)= \sum_{n=0}^\infty T_n(t)cos(n\pi x)$ (your text gives an additional "$\displaystyle a_n$" but I would include that in the "$\displaystyle T_n(t)$"). You will need to write the right hand side of the equation in that form so you will need to find the Fourier cosine series for $\displaystyle sin^2(\pi x)$. Last edited by Country Boy; June 17th, 2018 at 11:28 AM.

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