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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 0 "" }{TEXT 258 11 "Lecture 12:" }{TEXT 259 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" }
{TEXT 263 25 "Heat Transfer in the Ball" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 260 12 "John Polking" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }
{TEXT 261 17 "PCMI, Summer 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "
This worksheet contains a set of graphs and pictures that illustrate h
ow heat propagates inside a ball when the initial data" }}{PARA 0 "" 
0 "" {TEXT -1 33 "and boundary values are constant." }}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 33 "Fourier expansion of the solution" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 16 "In what follows " }{TEXT 264 1 "u" }
{TEXT -1 45 " is the solution, depending on the variables " }{TEXT 
265 2 "r " }{TEXT -1 14 "(radius) and  " }{TEXT 266 1 "t" }{TEXT -1 
24 " (time), and parameters " }{TEXT 267 1 "k" }{TEXT -1 9 " (thermal
" }}{PARA 0 "" 0 "" {TEXT -1 14 "diffusivity), " }{TEXT 268 2 "u0" }
{TEXT -1 40 " (initial temperature inside the ball), " }{TEXT 269 1 "f
" }{TEXT -1 29 " (boundary temperature), and " }{TEXT 277 1 "a" }
{TEXT -1 31 " (the ball's radius). The value" }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }{TEXT 270 1 "N" }{TEXT -1 53 " is the upper limit of the sum
 in the Fourier series." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 
"u:=(r,t,k,u0,f, a,N)->f+2*(u0-f)*(a/(Pi*r))*sum((-1)^(n+1)*exp(-k*n^2
*Pi^2*t/a^2)*sin(n*Pi*r/a)/n, n=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Temperature
 evolution inside the ball" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "We se
t " }{TEXT 271 6 "k=0.02" }{TEXT -1 2 ", " }{TEXT 272 5 "f=325" }
{TEXT -1 2 ", " }{TEXT 273 5 "u0=75" }{TEXT -1 2 ", " }{TEXT 274 3 "a=
1" }{TEXT -1 2 ", " }{TEXT 275 0 "" }{TEXT -1 4 "and " }{TEXT 276 5 "N
=200" }{TEXT -1 61 ". The first graph displays the temperature profile
 for times " }{TEXT 278 5 "t= 0," }{TEXT -1 0 "" }{TEXT 279 15 " 1, 2,
 3, 4, 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "k:=0.02: u0:=75: f:=325: \+
a:=1: N:=200:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "p[0]:=plot
(75, r=0..1, numpoints=400):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "for j from 1 to 5 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "  p[j
]:=plot(u(r,j,k,u0,f,a,N), r=0..1, numpoints=400);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "display([seq(p[j], j=0..5)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
113 "The second graph displays the temperature at the center of the ba
ll against time. We need to input a formula for " }{TEXT 280 8 "w=u(0,
t)" }{TEXT -1 7 " first." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 
"w:=t->f+2*(u0-f)*sum((-1)^(n+1)*exp(-k*n^2*Pi^2*t/a), n=1..N);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot(w(t),t=(.0001)..10, num
points=400);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "The last plot di
splays an animation of the temperature profile as times increases from
 zero to ten hours." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "q[0]
:=plot(75, r=0..1, numpoints=300):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "for j from 1 to 40 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "  q[j]:=plot(g(r,j/4), r=0..1, numpoints=300):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 47 "display([seq(q[j], j=0..40)], insequence=true);" }}{PARA 13 "
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