
Announcements
 Next week, we will look at partial differential equations, linearization and the chain rule.
There is a Handout on PDE's.
 An exam solution draft and basic stats are posted.
If you should see typos, mail them in. We don't assign letter grades to scores
but over 90 is definitely an A range. Below 50 you should contact your TF to
look over the exam.
 The final exam schedule has been released
here.
It takes place Thursday, 12/17/2015 at 9 AM.
Code for problem 4a). Find the value U[0.6.7]
f[x_]:=Sin[Pi 7x];
g[x_]:=5 Sin[5 Pi x];
U = NDSolveValue[
{D[u[t,x],{t,2}]D[u[t,x],{x,2}]==0,
u[0,x]==f[x],
Derivative[1,0][u][0,x] == g[x],
DirichletCondition[u[t,x]==f[0],x==0],
DirichletCondition[u[t,x]==f[1],x==1]},
u,{t,0,1},{x,0,1}];
Animate[Plot[U[t,x],{x,0,1},
PlotRange>{2,2}],{t,0,1,0.01}]
Plot[U[t,0.5], {t, 0, 1}]
Code for problem 4b) We want to see U[t,0.6,0.7]
A = Rectangle[{0, 0}, {1, 1}]; Clear[t, x, y];
f[x_, y_] := Sin[2 Pi x] Abs[Sin[3 Pi y]];
g[x_, y_] := 3 Sin[Pi x] Sin[Pi y];
U = NDSolveValue[{D[u[t, x, y], {t, 2}] 
Inactive[Laplacian][u[t, x, y], {x, y}] == 0,
u[0, x, y] == f[x, y], Derivative[1, 0, 0][u][0, x, y] == g[x, y],
DirichletCondition[u[t, x, y] == 0, True]},
u, {t, 0, 2 Pi}, {x, y} [Element] A];
Plot3D[U[4, x, y], {x, 0, 1}, {y, 0, 1}]
Animate[ContourPlot[U[t, x, y], {x, 0, 1}, {y, 0, 1}], {t, 0, 2 Pi}]
Example code for problem 5:
f[t_,x_]:=(1/Sqrt[t])*Exp[x^2/(4t)];
Simplify[ D[f[t,x],t] == D[f[t,x],{x,2}]]
