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-1 19 "Homework Problem #3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 85 "In the above, we did the heat equation u_t = u_xx
.  Now let's consider u_t = c*u_xx. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 60 "a) Modify the above program so that you
 have incorporated c." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 104 "b) Pick one set of initial data.  Now pick three diff
erent values of c.  This determines three solutions" }}{PARA 0 "" 0 "
" {TEXT -1 101 "u_1, u_2, and u_3.  Plot the three solutions at a fixe
d time to graphically demonstrate the effect of" }}{PARA 0 "" 0 "" 
{TEXT -1 76 "your choice of diffusion constant.  Explain why the behav
ior is as expected." }}{PARA 257 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
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{MPLTEXT 1 0 44 "N := 10: a := array(1..N): b := array(0..N):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := x -> floor((x+Pi)/(2*Pi)):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "for j from 1 to N do a[j]:= evalf(
Int(f(x)*sin(j*x),x=0..2*Pi))/(Pi) od:\nfor j from 1 to N do b[j]:= ev
alf(Int(f(x)*cos(j*x),x=0..2*Pi))/(Pi) od:\nb[0] := evalf(Int(f(x),x=0
...2*Pi))/(2*Pi):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "c := 1/2: u1 \+
:= b[0]:\nfor j from 1 to N do u1 := u1 + exp(-c*j^2*t)*a[j]*sin(j*x) \+
od:\nfor j from 1 to N do u1 := u1 + exp(-c*j^2*t)*b[j]*cos(j*x) od:\n
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{MPLTEXT 1 0 44 "N := 10: a := array(1..N): b := array(0..N):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := x -> floor((x+Pi)/(2*Pi)):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "for j from 1 to N do a[j]:= evalf(
Int(f(x)*sin(j*x),x=0..2*Pi))/(Pi) od:\nfor j from 1 to N do b[j]:= ev
alf(Int(f(x)*cos(j*x),x=0..2*Pi))/(Pi) od:\nb[0] := evalf(Int(f(x),x=0
...2*Pi))/(2*Pi):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "c := 1: u2 :=
 b[0]:\nfor j from 1 to N do u2 := u2 + exp(-c*j^2*t)*a[j]*sin(j*x) od
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{MPLTEXT 1 0 44 "N := 10: a := array(1..N): b := array(0..N):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := x -> floor((x+Pi)/(2*Pi)):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "for j from 1 to N do a[j]:= evalf(
Int(f(x)*sin(j*x),x=0..2*Pi))/(Pi) od:\nfor j from 1 to N do b[j]:= ev
alf(Int(f(x)*cos(j*x),x=0..2*Pi))/(Pi) od:\nb[0] := evalf(Int(f(x),x=0
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 b[0]:\nfor j from 1 to N do u3 := u3 + exp(-c*j^2*t)*a[j]*sin(j*x) od
:\nfor j from 1 to N do u3 := u3 + exp(-c*j^2*t)*b[j]*cos(j*x) od:\n\n
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