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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 8 "PCMI-UFP" }}{PARA 18 "" 0 
"" {TEXT -1 67 "LAB 5:  The Diffusion Equation: Solution by Separation
 of Variables" }}{PARA 256 "" 0 "" {TEXT -1 14 "Janine Wittwer" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
3 "" 0 "" {TEXT -1 38 "Solving the Diffusion Equation given  " }}
{PARA 3 "" 0 "" {TEXT -1 29 "f(x) = 6 sin(15x) + 4/3sin(x)" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "restart;\nwith(plots):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 54 "f:=x -> 6*sin(15*x) + 4/3*sin(x);\nplot(f
(x), x=0..Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Set k=1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "u:=(x,t)-> 6*exp(-15^2*t)*si
n(15*x) + 4/3*exp(-t)*sin(x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 63 "ut:= diff(u(x,t),t);\nuxx:= diff(u(x,t),x$2);\nsimplify(ut-uxx
);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot3d(u(x,t), x=0.
.Pi,t=0..0.02,numpoints=5000, axes=boxed);" }}}{EXCHG {PARA 3 "" 0 "" 
{TEXT -1 53 "Solving the Diffusion Equation given a Fourier Series" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots):\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f:= x -> sum((1/n)*sin(n*x),
 n=1..20);\nplot(f(x), x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "u:=(x,t) -> sum((1/n)*sin(n*x)*exp(-n^2*t), n=1..20);
\n" }{TEXT -1 50 "Here we set kappa = 1 in the differential equation" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "ut:= diff(u(x,t),t):\nuxx
:= diff(u(x,t),x$2):\nsimplify(ut-uxx);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot3d(u(x,t),x=
0..Pi, t=0..1, axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "animate(u(x,t),x=0..Pi, t=0..3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 28 "Finding Fouri
er Coefficients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "with(plots):\nassume(n, integer
): assume(N,integer):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Set f and calculate its Fourier \+
coefficients for sin" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "f:= x->(Pi-
x)*x;\nplot(f(x),x=0..Pi);\n\na := [seq(int(f(x)*sin(n*x),x=0..Pi)*2/(
Pi),n=1..20)];" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 32 "Form the sin Fourier Series of f" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 56 "A:= x-> sum(a[k]*sin(k*x),k=1..20);\nplot(A
(x), x=0..Pi);" }}}{EXCHG }{EXCHG {PARA 3 "" 0 "" {TEXT -1 74 "Solving
 the differential equation with the calculated Fourier Coefficients" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Form the solution to the Diffusi
on Equation using the initial value f" }}{PARA 0 "" 0 "" {TEXT -1 8 "S
et k=1\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "u:= (x,t)-> su
m(exp(-k^2*t)*a[k]*sin(k*x),k=1..20);\nanimate(u(x,t),x=0..Pi,t=0..1);
\nplot3d(u(x,t),x=0..Pi,t=0..1, axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 16 "One more example" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r
estart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "assume(n, integer): assume(N,intege
r):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f:= x->(sin(x))^2
+ sin(3*x);\nplot(f(x),x=0..Pi);\n\nb := [seq(int(f(x)*sin(n*x),x=0..P
i)*2/(Pi),n=1..20)];" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Form the
 solution to the Diffusion Equation using the initial value g" }}
{PARA 0 "" 0 "" {TEXT -1 8 "Set k=1\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 131 "u:= (x,t)-> sum(exp(-k^2*t)*b[k]*sin(k*x),k=1..20);
\nanimate(u(x,t),x=0..Pi,t=0..0.5);\nplot3d(u(x,t),x=0..Pi,t=0..0.25, \+
axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "34" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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