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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 11 "Worksheet: " }{TEXT 258 12 "Lecture2.mws" }{TEXT 262 0 "" }}
{PARA 18 "" 0 "" {TEXT -1 10 "Lecture 2:" }{TEXT 260 1 " " }}{PARA 18 
"" 0 "" {TEXT 261 44 "Cooling of a Hot Bar: The Diffusion Equation" }}
{PARA 19 "" 0 "" {TEXT -1 17 "Andrew J. Bernoff" }}{PARA 18 "" 0 "" 
{TEXT 259 18 "PCMI,  Summer 2003" }}{PARA 0 "" 0 "" {TEXT -1 62 "This \+
worksheet  contains the examples from the second lecture." }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 40 "Some solutions to the diffusion equation
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Some  solutions  to the diffusion
 equation for a metal bar of length " }{XPPEDIT 18 0 "pi;" "6#%#piG" }
{TEXT -1 35 " with ends held at zero temperature" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 5 "K:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "Un:=(n,x,t)->sin(n*x)*exp(-(n^2)*K*t);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 68 "animate(Un(1,x,t),x=0..Pi,t=0..3,frames=100,colo
r=blue,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "ani
mate(Un(2,x,t),x=0..Pi,t=0..3,frames=100,color=blue,thickness=2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Un:=(n,x,t)->sin(n*x)*exp(-(
n^2)*K*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "U:=8/(Pi^2)*s
um((-1)^k*Un((2*k+1),x,t)/(2*k+1)^2,k=0..20):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 60 "animate(U,x=0..Pi,t=0..3,frames=100,color=blue,t
hickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot3d(U,x=
0..Pi,t=0..3,style=patchnogrid,shading=ZHUE,axes=boxed);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "Now, let's insulate the ends of the bar f
or the same initial condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Vn:=(n,x,t)->cos(n*x)*exp(-(
n^2)*K*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "V:=1/2-16/(Pi
^2)*sum(Vn((4*k+2),x,t)/(4*k+2)^2,k=0..20):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 60 "animate(V,x=0..Pi,t=0..3,frames=100,color=blue,thi
ckness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot3d(V,x=0.
.Pi,t=0..1,style=patchnogrid,shading=ZHUE,axes=boxed);" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 58 "Exercise 1: Solutions to the Homogeneous \+
Dirchlect Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 
"with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "Un:=" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "PDE:=(1+Zy^2)*Zxx+2*Zx*Zy*Zx
y+(1+Zx^2)*Zyy;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Zx:=diff
(Z,x);Zy:=diff(Z,y);Zxx:=diff(Zx,x);Zyy:=diff(Zy,y);Zxy:=diff(Zx,y);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "PDE;simplify(PDE);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot3d(Z,x=0..Pi,y=0..Pi,st
yle=patchcontour,shading=zhue,axes=boxed,grid=[100,100],title=\"Scherk
's Minimal Surface\",view=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "X:=r*cos(phi)-r^3*cos(3*phi)/3;Y:=r*sin(phi)+r^3*sin(
3*phi)/3;Z:=r^2*cos(2*phi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
123 "plot3d([X,Y,Z],r=0..3,phi=-Pi..Pi,style=patchnogrid,color=phi,axe
s=boxed,grid=[100,100],title=\"Enneper's Minimal Surface\");" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 42 "Exercise 2: Solution to the Cauch
y Problem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "F:=x->exp(-x^2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "xi:=x-C*t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "U:=F(xi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Ut:=diff(U,
t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Ux:=diff(U,x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "PDE:=Ut+C*Ux;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(PDE);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 114 "C:=1;animate(U,x=-10..10,t=0..10,numpoints=
200,color=blue,thickness=2,title=\"Solution to the Transport Equation
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