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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Worksheet: " }{TEXT 257 
12 "Lecture8.mws" }}{PARA 256 "" 0 "" {TEXT 256 10 "Lecture 8:" }
{TEXT -1 24 " Sturm-Liouville, Part I" }}{PARA 257 "" 0 "" {TEXT -1 0 
"" }}{PARA 258 "" 0 "" {TEXT -1 32 "David Arnold and Katherine Socha" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 17 "PCMI,
 Summer 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 60 "This worksheet contains the example from the eighth lecture." }
{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 
48 "Heat Equation with Radiation Boundary Conditions" }{TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "K:=1: h:=1/100: T1:=75: T2:=-125: L:=10:N:=20:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "u_s:=x->(h*(T2-T1)/(1+h*L))*
x+T1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "surprise:=plot(u_s
(x),x=0..L):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "F:=x->T1-u_
s(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(F(x),x=0..L):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot([-tan(omega*L),ome
ga/h],omega=0..(2*N-1)*Pi/(8*L),0..150,numpoints=1000,discont=true);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ints:=seq((2*n-1)*Pi/(2*L
)..(2*n+1)*Pi/(2*L),n=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 63 "Omega:=seq(fsolve(-tan(omega*L)=omega/h,omega=ints[n]),n=1..N);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "X:=n->sin(Omega[n]*x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot([X(1),X(2)],x=0..L
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(X(1)*X(2),x=0..L)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "C:=n->int(F(x)*sin(Ome
ga[n]*x),x=0..L)/int(sin(Omega[n]*x)^2,x=0..L);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "c:=seq(C(n),n=1..N);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 65 "v:=(x,t)->sum(c[n]*exp(-Omega[n]^2*K*t)*sin(Omeg
a[n]*x),n=1..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p1:=pl
ot(v(x,0),x=0..L,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "p2:=plot(F(x),x=0..L,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "display(p1,p2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "u:=(x,t)->v(x,t)+u_s(x);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "plot(u(x,0),x=0..L,view=[0..L,0..80]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "animate(u(x,t),x=0..L,t=0..100,fram
es=100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot3d(u(x,t),x
=0..L,t=0..50,axes=boxed,grid=[50,50],style=patch,shading=zhue);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "uend:=plot(u(x,200),x=0..L,c
olor=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(surp
rise,uend);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 }
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