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 Differential Equations Ordinary and Partial Differential Equations Math Forum

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?
Attached Images 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: 915 Thanks: 271 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.
Attached Images equation.jpg (26.7 KB, 0 views) June 17th, 2018, 11:25 AM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 You can "guess that the spatial (x) dependence follows a fourier cosine series" because the spatial dependence, "", 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. Tags equation, heat, separation, variables Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post MathAboveMeth Differential Equations 10 December 22nd, 2016 03:09 AM Njprince94 Differential Equations 7 December 9th, 2015 06:20 AM philm Differential Equations 5 May 6th, 2015 10:41 AM engininja Calculus 2 February 22nd, 2011 01:06 PM engininja Calculus 4 September 22nd, 2010 11:58 PM

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