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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "We will first solve the h
eat equation with the temperature at the ends fixed to be zero and the
 initial temperature given by " }}{PARA 0 "" 0 "" {TEXT -1 24 "       \+
    |   0   0<x<1" }}{PARA 0 "" 0 "" {TEXT -1 21 "f(x) =  |  1    1<x<
2" }}{PARA 0 "" 0 "" {TEXT -1 24 "           |  0    2<x<3" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=Heaviside(x-1)-Heaviside(
x-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&-%*HeavisideG6#,&%\"x
G\"\"\"F+!\"\"F+-F'6#,&F*F+\"\"#F,F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "plot(f,x=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
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F(7$$\"3')****\\i@OtBFapF(7$$\"3')**\\PfL'zV#FapF(7$$\"3>+++!*>=+DFapF
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{TEXT -1 38 "Next, define the Fourier coefficients:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "b:=n->2/3*int(sin(n*Pi/3*x),x=1..2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6#%\"nG6\"6$%)operatorG%&arrow
GF(,$-%$intG6$-%$sinG6#,$*(9$\"\"\"%#PiGF6%\"xGF6#F6\"\"$/F8;F6\"\"##F
=F:F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "And the solution:" }
}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "u:=(x,t)->sum(b(n)*sin(n*Pi/3*x)*exp(-n^2*Pi^2/9*t),n
=1..200);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"xG%\"tG6\"6$
%)operatorG%&arrowGF)-%$sumG6$*(-%\"bG6#%\"nG\"\"\"-%$sinG6#,$*(F4F5%#
PiGF59$F5#F5\"\"$F5-%$expG6#,$*()F4\"\"#F5)F;FEF59%F5#!\"\"\"\"*F5/F4;
F5\"$+#F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Plot the resulti
ng surface over the xt-plane." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 41 "plot3d(u(x,t),x=0..3,t=0..2,axes=framed);" }}{PARA 13 "" 1 "" 
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SG6%%\"xG%\"tGQ!F1" 1 2 0 1 10 0 2 1 1 3 2 1.000000 59.000000 
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n redo the problem with a different initial temperature, this time  f(
x)=-x(x-3)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=-x*(x-3);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,$*&%\"xG\"\"\",&F'F(\"\"$!
\"\"F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(f,x=0..3);
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7S7$$\"\"!F)F(7$$\"3s******\\i9Rl!#>$\"3q56KTM)*=>!#=7$$\"3/++vVA)GA\"
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$\"3z#*p8QTt+)*F07$$\"39+++bJ*[o#FO$\"3Ac9M7@Gg%)F07$$\"33++Dr\"[8v#FO
$\"36/G4VvFToF07$$\"3++++Ijy5GFO$\"3b58&3(>R=`F07$$\"31+]P/)fT(GFO$\"3
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DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "A formula for the fourie
r sine coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "c:=
n->2/3*int((-x*(x-3))*sin(n*Pi/3*x),x=0..3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"cGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,$-%$intG6$,$*
(%\"xG\"\"\",&F2F3\"\"$!\"\"F3-%$sinG6#,$*(9$F3%#PiGF3F2F3#F3F5F3F6/F2
;\"\"!F5#\"\"#F5F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "A formu
la for the solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "u2:
=(x,t)->sum(c(n)*sin(n*Pi/3*x)*exp(-n^2*Pi^2/9*t),n=1..200);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#u2Gf*6$%\"xG%\"tG6\"6$%)operatorG%&arrowG
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "A plot of the resulting surface ov
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u2(x,t),x=0..3,t=0..3,axes=framed);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 
300 300 {PLOTDATA 3 "6%-%%GRIDG6%;$\"\"!F($\"\"$F(F&X,%)anythingG6\"6
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