I completed the separation of variables step but I am very confused on how to apply the initial any boundary conditions to solve the problem. Please give me some advice or help to go about solving this problem, thank you!


Problem 1: Solution to homogeneous PDEs by separation of variables
a. Apply separation of variables to derive the general solution $u(x,t)$ to the following homogeneous PDE over the interval $0\leq x\leq 1$, with boundary conditions $u(0,t) = u(1,t) = 0$: $$u = c\frac{\partial^3u}{\partial t\,\partial x^2}$$ b. Find the particular solution with initial conditions $u(x,0)=1$ for $0<x<1$.

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    $\begingroup$ The boundary conditions arise in selecting the eigenfunctions. The initial conditions determine the "Fourier coefficients" of the eigenfunctions in the expansion. $\endgroup$ – Ian Sep 30 '16 at 21:47
  • $\begingroup$ Formatting tips here. $\endgroup$ – Em. Sep 30 '16 at 23:32

Since you did the separation of variables, you have something like $u(x,t) = a(x)b(t)$. The condition that $u(0,t) = 0$ means that $a(0)b(t) =0$. So you must have $a(0)=0$, because if you set $b(t) = 0$ it forces $u(x,t) =0$ and we have no use for the trivial solution. Likewise the second condition gives us $a(1) =0$. Similarly the initial condition says $b(0) =1$.

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