
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
May 19th, 2018, 05:14 AM  #1 
Member Joined: Jan 2016 From: United Kingdom Posts: 35 Thanks: 0  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?

May 19th, 2018, 05:59 AM  #2 
Senior Member Joined: Jun 2015 From: England Posts: 891 Thanks: 269 
This is a non homogeneous PDE. The usual method is to apply Duhamel's Principle. http://people.math.gatech.edu/~xchen...atDuhamel.pdf 
June 3rd, 2018, 11:22 PM  #3 
Newbie Joined: Sep 2017 From: CA, USA Posts: 1 Thanks: 0 
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 oneform 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.

June 17th, 2018, 12:25 PM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 
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 12:28 PM. 

Tags 
equation, heat, separation, variables 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Solve differential equation with use of separation of variables  MathAboveMeth  Differential Equations  10  December 22nd, 2016 04:09 AM 
Separation of variables  Njprince94  Differential Equations  7  December 9th, 2015 07:20 AM 
Using Separation of variables  philm  Differential Equations  5  May 6th, 2015 11:41 AM 
Separation of Variables  engininja  Calculus  2  February 22nd, 2011 02:06 PM 
Separation of Variables  engininja  Calculus  4  September 23rd, 2010 12:58 AM 