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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 71 "We want to implement the sum on page 100 \+
of [Ball & Torous, 1984]. The " }{XPPEDIT 18 0 "n^th;" "6#)%\"nG%#thG
" }{TEXT -1 43 " term in the sum is a \"seperable\" function " }
{XPPEDIT 18 0 "C[n]*b[n](x)*T[n](t);" "6#*(&%\"CG6#%\"nG\"\"\"-&%\"bG6
#F'6#%\"xGF(-&%\"TG6#F'6#%\"tGF(" }{TEXT -1 21 ", where the constant \+
" }{XPPEDIT 18 0 "C[n];" "6#&%\"CG6#%\"nG" }{TEXT -1 13 " is given by \+
" }{XPPEDIT 18 0 "C[n] = b[n](0);" "6#/&%\"CG6#%\"nG-&%\"bG6#F'6#\"\"!
" }{TEXT -1 0 "" }}{PARA 200 "> " 0 "" {MPLTEXT 1 -1 48 "bfn:=(n,x)->s
qrt(2/(a+b))*sin(n*Pi*(x+b)/(a+b));" }{TEXT -1 0 "" }}{PARA 200 "> " 
0 "" {MPLTEXT 1 -1 47 "Tfn:=(n,t)->exp(-n^2*Pi^2*sigma^2*t/2/(a+b)^2);
" }}{PARA 200 "> " 0 "" {MPLTEXT 1 -1 53 "'p(x,t)'=Sum(b[n](0)*b[n](x)
*T[n](t),n=1..infinity); " }}{PARA 200 "> " 0 "" {MPLTEXT 1 -1 69 "pnf
n:=(n,x,t)->bfn(n,0)*bfn(n,x)*Tfn(n,t): 'pnfn(n,x,t)'=pnfn(n,x,t);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "For each " }{XPPEDIT 18 0 "n;" "6#
%\"nG" }{TEXT -1 15 ", the function " }{XPPEDIT 18 0 "p[n](x,t) = b[n]
(x)*T[n](t);" "6#/-&%\"pG6#%\"nG6$%\"xG%\"tG*&-&%\"bG6#F(6#F*\"\"\"-&%
\"TG6#F(6#F+F2" }{TEXT -1 48 " satisfies the \"heat equation\" on the \+
interval (" }{XPPEDIT 18 0 "-b,a;" "6$,$%\"bG!\"\"%\"aG" }{TEXT -1 31 
"), and the boundary conditions " }{XPPEDIT 18 0 "p[n](-b,t) = 0;" "6#
/-&%\"pG6#%\"nG6$,$%\"bG!\"\"%\"tG\"\"!" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "p[n](a,t) = 0;" "6#/-&%\"pG6#%\"nG6$%\"aG%\"tG\"\"!" }
{TEXT -1 1 "." }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 52 "Check to make \+
sure  the \"Heat equation\" is satisfied" }}}{EXCHG {PARA 200 "> " 0 "
" {MPLTEXT 1 -1 126 "Diff('pnfn(n,x,t)',t)=(sigma^2/2)*Diff('pnfn(n,x,
t)',x,x); diff( pnfn(n,x,t),t) =(sigma^2/2)*diff( pnfn(n,x,t),x,x); ev
alb(%);" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 -1 0 "" }}}{EXCHG 
{PARA 200 "> " 0 "" {MPLTEXT 1 -1 27 "p:=sum(pnfn(n,x,t),n=1..N);" }}}
{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 -1 19 "eval(subs(N=3,p)); " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Here, experiment with some evaluat
ions, plotting, etc. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "av
al:=2: bval:=1: sigmaval:=1/2: Nval:=20:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "evalf(subs(N=Nval,a=aval,b=bval,sigma=sigmaval,p)); p
lot3d(%,x=-1..2,t=0..1,grid = [50,50]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "OK the real p
urpose here is to form the joint PDF " }{XPPEDIT 18 0 "f(a,b,x);" "6#-
%\"fG6%%\"aG%\"bG%\"xG" }{TEXT -1 32 " for the high, low, and close  (
" }{XPPEDIT 18 0 "A,-B,X;" "6%%\"AG,$%\"BG!\"\"%\"XG" }{TEXT -1 92 ").
 For reasons that we can't go into right now, it turns out that his jo
int PDF is given by " }{XPPEDIT 18 0 "diff(p,a,b);" "6#-%%diffG6%%\"pG
%\"aG%\"bG" }{TEXT -1 22 ". So let's do that ..." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 49 "diff(pnfn(n,x,t),a,b): fn:=unapply(%,(n,a,b,x
)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "int(fn(n,a,b,x),x=-
b..a): gn:=unapply(%,(n,a,b)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=30:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f:=sum(fn(n,a,b,x),n=1..N): \+
g:=sum(gn(n,a,b,x),n=1..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "conditionalf :=f/g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "
aval:=2: bval:=1: sigmaval:=1: Nval:=20:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "int(subs(a=aval,b=bval,t=1,sigma=sigmaval,conditional
f),x=-bval..aval);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(
subs(a=aval,b=bval,t=1,sigma=sigmaval,conditionalf),x=-bval..aval);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "" 0 
"" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "13 0 0" 
49 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }