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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.)" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}
{PARA 200 "" 0 "" {TEXT 202 139 "I dette eksemplet velger vi r=1, og m
ed valget N=8 (antall x-intervaller) og slutt-tid T=0.2 bryter metoden
 sammen, som vi ser fra grafene." }}{PARA 200 "" 0 "" {TEXT 202 55 "Me
toden konvergerer kun hvis r er mindre eller lik 1/2." }}{SECT 0 
{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(L
inearAlgebra): 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-interval" }}{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 := 8;     " }{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\"$\"'+]
7!\"'" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I\"rG6\"\"
\"\"" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I\"kG6\"$\"'
]i:!\"(" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I\"MG6\"
\"#8" }{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$I)operatorGF%I&arrowGF%F%-I$sin
G6$I*protectedGF/I(_syslibGF%6#*&I#PiGF/\"\"\"9$F4F%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 for storing resul
ts" }{MPLTEXT 1 203 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>I\"uG6\"-
I&arrayGI*protectedGF(6%;\"\"!\"\");F+\"#87\"" }{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 "Init
ialize u using f" }{MPLTEXT 1 203 0 "" }}{PARA 203 "" 1 "" {XPPMATH 
20 "6#>&I\"uG6\"6$\"\"\"\"\"!$\"'$o#Q!\"'" }{TEXT 204 0 "" }}{PARA 
203 "" 1 "" {XPPMATH 20 "6#>&I\"uG6\"6$\"\"#\"\"!$\"'2rq!\"'" }{TEXT 
204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>&I\"uG6\"6$\"\"$\"\"!$\"'
\")Q#*!\"'" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>&I\"uG
6\"6$\"\"%\"\"!$\"\"\"F)" }{TEXT 204 0 "" }}{PARA 203 "" 1 "" 
{XPPMATH 20 "6#>&I\"uG6\"6$\"\"&\"\"!$\"'#)Q#*!\"'" }{TEXT 204 0 "" }}
{PARA 203 "" 1 "" {XPPMATH 20 "6#>&I\"uG6\"6$\"\"'\"\"!$\"'5rq!\"'" }
{TEXT 204 0 "" }}{PARA 203 "" 1 "" {XPPMATH 20 "6#>&I\"uG6\"6$\"\"(\"
\"!$\"'(o#Q!\"'" }{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 en
d: " }{MPLTEXT 1 203 2 "# " }{TEXT 205 33 "Initialize boundary values \+
(zero)" }{MPLTEXT 1 203 0 "" }}}}{SECT 0 {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 
0 {PARA 201 "" 0 "" {TEXT -1 42 "Plot results (animation: click on pic
ture)" }{TEXT -1 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 17 "
L := array(0..M):" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 22 "\nR := Mat
rix(N+1,M+1):" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 38 "\nuexact := (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&arrow
GF%F%*&-I\"fGF%6#9$\"\"\"-I$expG6$I*protectedGF6I(_syslibGF%6#,$*&I#Pi
GF6\"\"#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 "" }}
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0 {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 ""
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{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 "E
xactPlot := matrixplot(R2,color=red,transparency=0.5):" }{MPLTEXT 1 
203 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 203 24 "display(L[M]
,ExactPlot);" }{MPLTEXT 1 203 0 "" }}{PARA 204 "" 1 "" {GLPLOT3D 400 
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