Consider the two classes of partial differential equations that we have studied in ME

$391$:

a$?$u

$)$

$02=224$

ax$?"$

Our physical interval of interest is $0<$ x $<1.$ The BCs for both equations are

u$(0,$ t$)=1,$ u$(1,$ t$)=1.$

a$)$

$(35$ pointe$)$ Without yet applying any initial condition or conditions, write, for each equation $*)$ and

$(4)$ the general solution that satisfies the BCs$.$

b$)$

$(10$ points$)$ Now apply the IC u$($x$,0)=1+0\mathrm{.}5$sin$2$x and solve the appropriate equation, giving

your final solution. Do not solve the problem for which this TC$.$ is inappropriate.

c$)(5$ points$)$ Draw the evolution in time of the final solution. Write the scales on the axes. Does it or does it not approach a steady state? Describe the equation and solution physics.