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This question concerns the separation of variables method. Let $u(x,t)$ be the solution of the diffusion equation $u_t=Du_{xx}+f(x,t)$ with initial data $u(x,0)=\phi(x)$ and boundary conditions $u_x(0,t)=u_x(1,t)=0$ for $t>0$. Assume $f(x,t)=1$ and $\phi(x)=1$. Can you give a solution without using separation of variables?

I have just did a problem that was identical to this, except with $f(x,t)=0$, in which I used the even-extension of the Fourier series to find the Fourier coefficients $a_0$ and $a_n$ for $u(x,t)$. I found out that $a_0$ happened to be equal to $1$, and $a_n=0$ for $n\in \Bbb{N}^+$. This gave the solution $u(x,t)=a_0=1$ for that particular problem. I am unsure how to solve the question with $f(x,t)=1$ though, appreciate any pointers I can get - thanks!

Leonard Kuan
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  • You could try looking for a solution depending only on $t$. – Hans Lundmark Mar 03 '21 at 09:19
  • @HansLundmark thanks for the tip. I guess the trivial solution $u(x,t)=t+1$ would work, but is the problem implying that we simply make an educated guess that fits all the conditions? – Leonard Kuan Mar 03 '21 at 10:13
  • That's what I would do anyway. I can't know for sure what the person constructing the problem was thinking. – Hans Lundmark Mar 03 '21 at 10:17
  • @HansLundmark Ok, I think that's what I'm gonna go for. Thank you so much! – Leonard Kuan Mar 03 '21 at 14:24
  • Find the homogeneous solution then use the method of eigenfunction expansion. The homogeneous Neumann problem has a solution like $u(x,t) = \sum_{n=0}^{\infty} B_n \exp(-D(n\pi)^2t) \cos(n \pi x)$ then set $u(x,0) = \phi(x) = \sum_{n=0}^{\infty} B_n \exp(-D(n\pi)^2t) \cos(n \pi x)$ and get the coefficients $B_n$ – Ryan Howe Mar 04 '21 at 17:26

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