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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 23 "Section 10.6, Problem 9" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "The probl
em is to solve the heat equation in a 20 cm aluminum bar (" }{XPPEDIT 
18 0 "alpha^2 = .86;" "6#/*$%&alphaG\"\"#$\"#')!\"#" }{TEXT -1 1 ")" }
{TEXT -1 56 ", with initial value 25,\n u(0,t)=0, u(20,t)=60 for t>0.
\n" }}{PARA 0 "" 0 "" {TEXT -1 281 "The first step is to find a steady
 state solution v with v(0,t)=0, v(20,t)=60. The solution is v=3x.  Wr
ite u=v+w, so w=u-v solves the heat equation with homogeneous boundary
 conditions and initial value 25-3x.  Let b(n) be the nth coefficient \+
in the Fourier sine series of 25-3x.\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "assume(n,integer);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "b := proc(n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "  \+
integrate(2*(25-3*x)*sin(n*Pi*x/20)/20,x=0..20);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b(n)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "u := proc(n,x,t)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "    3*x+ sum(b(k)*exp(-0.86*k^2*Pi^
2*t/400)*sin(k*Pi*x/20),k=1..n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "
end;" }}}{PARA 0 "" 0 "" {TEXT -1 107 "The first plot is shows the ini
tial value, the steady state solution, and the solution at two other t
imes.\n" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "pl
ot(\{3*x,25,u(3,x,5),u(3,x,50)\},x=0..20,numpoints=500);" }}}{PARA 0 "
" 0 "" {TEXT -1 33 "What happens if we take 10 terms?" }{TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " plot(\{3*x,25,u(10,x,5),u
(10,x,50)\},x=0..20,numpoints=500);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "plot(u(10,x,50),x=0..20);" }}}{PARA 0 "" 0 "" {TEXT 
-1 58 "Next we plot u as a function of t for certain values of x." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "plot(u(3,5,t),t=0..50,numpoints=500);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 " plot(u(3,10,t),t=0..50,numpoints=500);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(u(3,15,t),t=0..50,numpo
ints=500);\n" }}}{PARA 0 "" 0 "" {TEXT -1 117 " Next find where the fi
rst term in the series is within 1% of the steady state temperature at
 x=5 (which is 3*5=15).\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "
u(1,5,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(15-%=0.
15,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "22" 
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