13 04/17 F

  1. Refer to problem 7.19. Find a function $u_0(x,t)$ that satisfies the inhomogeneous boundary conditions. Define $v=u-u_0$. Find the PDE, BC and IC satisfied by $v$.

  2. Find suitable eigenfunctions in terms of which $v$ may be written, and that satisfy the homogeneous boundary conditions. Work out the Fourier coefficients of the relevant functions in the problem for $v$ as far as possible.

  3. 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.

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

  5. Plot the solution numerically at some relevant times. I suspect that for large times the solution is approximately

    \begin{displaymath}
u = (x-{\textstyle\frac{1}{2}})t + {\textstyle\frac{1}{6}}(...
...{1}{2}})^3-{\textstyle\frac{1}{8}}(x-{\textstyle\frac{1}{2}})
\end{displaymath}

    Do your results agree?

  6. Solve the problem of the unidirectional flow of a viscous fluid if a plate at $x=0$ exerts a given shear stress on the fluid. The PDE is:

    \begin{displaymath}
v_t = \nu v_{xx} \qquad x > 0 \quad t > 0
\end{displaymath}

    and the initial and boundary conditions are

    \begin{displaymath}
v(x,0) = 0 \qquad \rho \nu v_x(0,t) = f(t)
\quad v\sim 0\mbox{ for } x\to\infty
\end{displaymath}

    where $\nu$ and $\rho$ are constants and the force per unit area $f(t)$ is a given function.