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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 258 "" 0 "" {TEXT -1 101 "input the data functions; try f
or example:  f=sinx g=0; f=hat g=0, f=sinx  g neq 0; f=0 g= small step
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "f:=x->(Pi/4-abs(x-Pi/2))*(Heaviside(x-Pi/4)-Heaviside
(x-3*Pi/4));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "g:=x->0;" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6
#%\"xG6\"6$%)operatorG%&arrowGF(*&,&%#PiG#\"\"\"\"\"%-%$absG6#,&9$F0F.
#!\"\"\"\"#F8F0,&-%*HeavisideG6#,&F6F0F.#F8F1F0-F<6#,&F6F0F.#!\"$F1F8F
0F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"\"!" }}}{EXCHG {PARA 
257 "" 0 "" {TEXT -1 58 "input the number of terms and generate the Fo
urier coeffs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 6 "N:=10;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b:=a
rray(1..N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "B:=array(1..N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "for k to N do b[k]:=evalf(2/Pi*int(sin(k*x)*f(x),x=0..Pi)) od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for k to N do B[k]:=evalf(2/Pi*int(
sin(k*x)*g(x),x=0..Pi)) od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG
\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%&arrayG6$;\"\"\"\"#57
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%&arrayG6$;\"\"\"\"#57\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 55 "generate the partial sums of
  the solution and the data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "u:=(x,t)->sum('(evalf(b[n])*
cos(n*t)+B[n]/n*sin(n*t))*sin(n*x)','n'=1..N);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "sf:=x->sum('evalf(b[n])*sin(n*x)','n'=1..N);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sg:=x->sum('evalf(B[n])*sin(n*x)','
n'=1..N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "G:=x->int(g(v),v=0..x)
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sG:=x->sum('evalf(B[n])*int(si
n(n*w), w=0..x)','n'=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG:
6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)-%$sumG6$.*&,&*&-%&evalfG6#&%\"b
G6#%\"nG\"\"\"-%$cosG6#*&F:F;9%F;F;F;*(&%\"BGF9F;F:!\"\"-%$sinGF>F;F;F
;-FF6#*&F:F;9$F;F;/.F:;F;%\"NGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#sfG:6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sumG6$.*&-%&evalfG6#&%\"bG6
#%\"nG\"\"\"-%$sinG6#*&F7F89$F8F8/.F7;F8%\"NGF(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#sgG:6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sumG6$.*&-%&
evalfG6#&%\"BG6#%\"nG\"\"\"-%$sinG6#*&F7F89$F8F8/.F7;F8%\"NGF(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG:6#%\"xG6\"6$%)operatorG%&arrowG
F(-%$intG6$-%\"gG6#%\"vG/F2;\"\"!9$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#sGG:6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sumG6$.*&-%&evalfG6#&
%\"BG6#%\"nG\"\"\"-%$intG6$-%$sinG6#*&F7F8%\"wGF8/F@;\"\"!9$F8/.F7;F8%
\"NGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 68 "generate the plots o
f the data, their partial sums, and the solution" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot3d(u(x,
t),x=0..Pi,t=0..10,title=`u(x,t)`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "plot(sG(x),x=0..Pi,title=`Partial sums of G`,color=red);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 54 "plot(G(x),x=0..Pi,title=`antiderivative G
`,color=red);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(sg(x),x=0..Pi
,title=`Partial sum of f(x)`,color=orange);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "plot(g(x),x=0..Pi,title=`g(x)`,color=orange);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot(sf(x),x=0..Pi,title=`Partial s
um of f(x)`,color=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(f(
x),x=0..Pi,title=`f(x)`,color=blue);" }}}{EXCHG {PARA 256 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 45 "animate the traveling w
aves given by the data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
84 "animate(sf(x-t),x=0..Pi,t=0..10,frames=50,color=blue,title=`f movi
ng to the right`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "animate(sf(x+
t),x=0..Pi,t=0..10,frames=50,color=yellow,title=`f moving to the left`
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "animate(sG(x-t),x=0..Pi,t=0..
10,frames=50,color=blue,title=`G moving to the right`);" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 85 "animate(sG(x+t),x=0..Pi,t=0..10,frames=50,colo
r=yellow,title=`G moving to the left`);" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 258 55 "animation of the solution and the
 solution and the data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "a1:=animate(u(x,t),x=0..Pi,t=0..10,
frames=50,color=green,title=`solution and data`):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 62 "a2:=animate(1/2*sf(x-t),x=0..Pi,t=0..10,frames=50,c
olor=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a3:=animate(1/2*sf(x
+t),x=0..Pi,t=0..10,frames=50,color=yellow):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "a4:=animate(1/2*sG(x-t),x=0..Pi,t=0..10,frames=50,col
or=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a5:=animate(1/2*sG(x+t
),x=0..Pi,t=0..10,frames=50,color=yellow): " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "display(\{a1,a2,a3,a4,a5\});" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "animate(u(x,t),x=0..Pi,t=0..10,frames=50,color=green,
title=`solution`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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