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{SECT 0 {PARA 200 "" 0 "" {TEXT 202 188 "Vi l\277ser her varmeligninge
n p\214 intervallet 0 < x < a (der a spesifiseres) med randverdi 0 og \+
en gitt initialfunksjon, ved hjelp av den enkle, \"direkte\" metoden (
forlengs differanseskjema.)" }}{SECT 1 {PARA 201 "" 0 "" {TEXT -1 14 "
Initialization" }{TEXT -1 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 
1 203 12 "Digits := 6;" }{MPLTEXT 1 203 0 "" }}{PARA 203 "" 1 "" 
{XPPMATH 20 "6#>I'DigitsG6\"\"\"'" }{TEXT 204 0 "" }}}{EXCHG {PARA 
202 "> " 0 "" {MPLTEXT 1 203 33 "with(LinearAlgebra): with(plots):" }
{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 12 "a
 := 1.0;   " }{MPLTEXT 1 203 2 "# " }{TEXT 205 22 "endpoint of x-inter
val" }}{PARA 202 "> " 0 "" {TEXT 205 8 "interval" }{MPLTEXT 1 203 13 "
\nT := 0.2;   " }{MPLTEXT 1 203 2 "# " }{TEXT 205 13 "terminal time" }
{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 13 "\nN := 5;     " }{MPLTEXT 1 
203 2 "# " }{TEXT 205 21 "number of x-intervals" }{MPLTEXT 1 203 0 "" 
}{MPLTEXT 1 203 1 "\n" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I\"aG6\"$\"
#5!\"\"" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I\"TG6\"$
\"\"#!\"\"" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I\"NG6
\"\"\"&" }{TEXT 204 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 
22 "h := a/N;           # " }{TEXT 205 14 "step size in x" }{MPLTEXT 
1 203 0 "" }{MPLTEXT 1 203 23 "\nr := 1;             # " }{TEXT 205 
24 "Courant number r(=k/h^2)" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 23 
"\nk := r*h^2;         # " }{TEXT 205 14 "step size in t" }{MPLTEXT 1 
203 0 "" }{MPLTEXT 1 203 23 "\nM := round(T/k);    # " }{TEXT 205 20 "
number of time steps" }{MPLTEXT 1 203 0 "" }}{PARA 203 "" 1 "" 
{XPPMATH 20 "6#>I\"hG6\"$\"'++?!\"'" }{TEXT 204 0 "" }}{PARA 203 "" 1 
"" {XPPMATH 20 "6#>I\"rG6\"\"\"\"" }{TEXT 204 0 "" }}{PARA 203 "" 1 ""
 {XPPMATH 20 "6#>I\"kG6\"$\"'++S!\"(" }{TEXT 204 0 "" }}{PARA 203 "" 
1 "" {XPPMATH 20 "6#>I\"MG6\"\"\"&" }{TEXT 204 0 "" }}}{EXCHG {PARA 
202 "> " 0 "" {MPLTEXT 1 203 21 "f := x -> sin(Pi*x); " }{MPLTEXT 1 
203 2 "# " }{TEXT 205 32 "Initial temperature distribution" }{MPLTEXT 
1 203 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6#I\"xGF%F%6$
I)operatorGF%I&arrowGF%F%-I$sinGF%6#*&I#PiGI*protectedGF1\"\"\"9$F2F%F
%F%" }{TEXT 204 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 23 "
u := array(0..N,0..M); " }{MPLTEXT 1 203 2 "# " }{TEXT 205 25 "Array f
or storing results" }{MPLTEXT 1 203 0 "" }}{PARA 203 "" 1 "" {XPPMATH 
20 "6#>I\"uG6\"-I&arrayGI*protectedGF(6%;\"\"!\"\"&F*7\"" }{TEXT 204 
0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 50 "for i from 1 to \+
N do u[i,0] := evalf(f(i*h)) end; " }{MPLTEXT 1 203 2 "# " }{TEXT 205 
20 "Initialize u using f" }{MPLTEXT 1 203 0 "" }}{PARA 203 "" 1 "" 
{XPPMATH 20 "6#>&I\"uG6\"6$\"\"\"\"\"!$\"'&y(e!\"'" }{TEXT 204 0 "" }}
{PARA 203 "" 1 "" {XPPMATH 20 "6#>&I\"uG6\"6$\"\"#\"\"!$\"'d5&*!\"'" }
{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>&I\"uG6\"6$\"\"$\"
\"!$\"'e5&*!\"'" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>&
I\"uG6\"6$\"\"%\"\"!$\"'*y(e!\"'" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" 
{XPPMATH 20 "6#>&I\"uG6\"6$\"\"&\"\"!$F)F)" }{TEXT 204 0 "" }}}{EXCHG 
{PARA 202 "> " 0 "" {MPLTEXT 1 203 51 "for j from 0 to M do u[0,j] := \+
0; u[N,j] := 0 end: " }{MPLTEXT 1 203 2 "# " }{TEXT 205 33 "Initialize
 boundary values (zero)" }{MPLTEXT 1 203 0 "" }}}}{SECT 1 {PARA 201 ""
 0 "" {TEXT -1 13 "Find solution" }{TEXT -1 0 "" }}{EXCHG {PARA 202 ">
 " 0 "" {MPLTEXT 1 203 22 "for j from 0 to M-1 do" }{MPLTEXT 1 203 0 "
" }{MPLTEXT 1 203 25 "\n  for i from 1 to N-1 do" }{MPLTEXT 1 203 0 ""
 }{MPLTEXT 1 203 58 "\n    u[i,j+1] := r*u[i-1,j] + (1-2*r)*u[i,j] + r
*u[i+1,j];" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 7 "\n  end;" }
{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 5 "\nend;" }{MPLTEXT 1 203 0 "" }}
}}{SECT 1 {PARA 201 "" 0 "" {TEXT -1 42 "Plot results (animation: clic
k on picture)" }{TEXT -1 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 
203 17 "L := array(0..M):" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 22 "\n
R := Matrix(N+1,M+1):" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 38 "\nuexa
ct := (x,t) -> f(x)*exp(-Pi^2*t);" }{MPLTEXT 1 203 0 "" }}{PARA 203 ""
 1 "" {XPPMATH 20 "6#>I'uexactG6\"f*6$I\"xGF%I\"tGF%F%6$I)operatorGF%I
&arrowGF%F%*&-I\"fGF%6#9$\"\"\"-I$expGF%6#,$*&I#PiGI*protectedGF9\"\"#
9%F2!\"\"F2F%F%F%" }{TEXT 204 0 "" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 203 20 "for j from 0 to M do" }{MPLTEXT 1 203 0 "" }
{MPLTEXT 1 203 23 "\n  for i from 0 to N do" }{MPLTEXT 1 203 0 "" }
{MPLTEXT 1 203 25 "\n    R[i+1,j+1] := u[i,j]" }{MPLTEXT 1 203 0 "" }
{MPLTEXT 1 203 7 "\n  end:" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 36 "
\n  L[j] := matrixplot(R,axes=boxed):" }{MPLTEXT 1 203 0 "" }{MPLTEXT 
1 203 5 "\nend:" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 3 "\n  " }
{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 42 "d
isplay(seq(L[j],j=0..M),insequence=true);" }{MPLTEXT 1 203 0 "" }}
{PARA 204 "" 1 "" {GLPLOT3D 400 400 400 {PLOTDATA 3 "6%-%(ANIMATEG6(7#
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1 0 "Curve 1" }}{TEXT 206 0 "" }}}}{SECT 1 {PARA 201 "" 0 "" {TEXT -1 
45 "Comparison with exact solution (shown in red)" }{TEXT -1 0 "" }}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 22 "R2 := Matrix(N+1,M+1):" 
}{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 20 "
for j from 0 to M do" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 23 "\n  for
 i from 0 to N do" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 35 "\n    R2[i
+1,j+1] := uexact(i*h,j*k)" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 7 "\n
  end:" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 5 "\nend:" }{MPLTEXT 1 
203 0 "" }{MPLTEXT 1 203 3 "\n  " }{MPLTEXT 1 203 0 "" }}}{EXCHG 
{PARA 202 "> " 0 "" {MPLTEXT 1 203 55 "ExactPlot := matrixplot(R2,colo
r=red,transparency=0.5):" }{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 202 "> \+
" 0 "" {MPLTEXT 1 203 24 "display(L[M],ExactPlot);" }{MPLTEXT 1 203 0 
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