14 04/25 F

1. Continuing the previous homework, solve for $v$ using separation of variables in terms of integrals of the known functions $f(x)$, $g_0(t)$, and $g_1(t)$. Write the solution for $u$ completely.

2. Assume that $f=0$, $k=\ell=1$, and that $u_x=t$ at both $x=0$ and $x=\ell$. Work out the solution completely.

3. Plot the solution numerically at some relevant times. I suspect that for large times the solution is approximately
   
   $$u = (x-{\textstyle\frac{1}{2}})t + {\textstyle\frac{1}{6}}(... ...{1}{2}})^3-{\textstyle\frac{1}{8}}(x-{\textstyle\frac{1}{2}})$$
   
   
   Do your results agree?