Refer to problem 7.19. Find a function that
satisfies the inhomogeneous boundary conditions. Define .
Find the PDE, BC and IC satisfied by .
Find suitable eigenfunctions in terms of which may be
written, and that satisfy the homogeneous boundary conditions. Work
out the Fourier coefficients of the relevant functions in the
problem for as far as possible.
Solve for using separation of variables in terms of
integrals of the known functions , , and .
Write the solution for completely.
Assume that , , and that at both
and . Work out the solution completely.
Plot the solution numerically at some relevant times. I suspect
that for large times the solution is approximately
Do your results agree?
Solve the problem of the unidirectional flow of a viscous fluid
if a plate at exerts a given shear stress on the fluid. The
PDE is:
and the initial and boundary conditions are
where and are constants and the force per unit area
is a given function.