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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 53 "Numerik fuer Anfangswert
aufgaben - 14. Uebungsblatt \n" }{TEXT 256 0 "" }{TEXT -1 61 "Christin
e Schweinem (mit einigen Korrekturen von Lars Gruene)" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }{TEXT 257 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 15 "Implementierung" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "restart; with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 236 "aufgabe_i:=proc(T,hh,kk)\n\n  local h, k, m, n, r, x
, U, A, j, werte, i;\n\n  h:=evalf(hh); # Delta_x\n  m:=floor(1/h); \n
  k:=evalf(kk); # Delta_t\n  n:=floor(T/k);\n\n  r:=k/h^2;\n  if r > 0
.5 then WARNING(\"k/h^2 > 0.5 => nicht stabil\") fi;\n\n  " }{MPLTEXT 
1 0 67 "U.0:=vector(m-1,[seq(h*`i`*(1-h*`i`),`i`=1..m-1)]); # Anfangsw
erte\n" }{TEXT -1 1 "\n" }{MPLTEXT 1 0 54 "  A:=band([r,1-2*r,r],m-1);
\n\n  for j from 0 to n-1 do\n" }{MPLTEXT 1 0 315 "\n    U.(j+1):=map(
evalf, evalm( A&*U.j+k*vector(m-1,[seq(U.j[`i`]*(1-U.j[`i`]),`i`=1..m-
1)]) ) );   # (**)\n  \n  od;\n\n  werte:=[];\n\n  for j from 0 to n d
o\n\n    werte:=[op(werte),[0,seq(U.j[`i`],`i`=1..m-1),0]]; # Randwert
e U.j[0] und U.j[m] = 0\n\n  od;\n\n  RETURN(PLOT3D(GRID(0..T,0..1,wer
te),AXESSTYLE(BOX)));  \n\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 233 "aufgabe_ii:=proc(T,hh,kk)\n\n  local h, k, m, n, x, U, F, j, \+
werte, i, T_matrix;\n\n  h:=evalf(hh); # Delta_x\n  m:=floor(1/h); \n \+
 k:=evalf(kk); # Delta_t\n  n:=floor(T/k);\n\n  if k/h^2 > 0.5 then WA
RNING(\"k/h^2 > 0.5 => nicht stabil\") fi;\n" }{MPLTEXT 1 0 507 "\n  U
.0:=vector(m-1,[seq(h*`i`*(1-h*`i`),i=1..m-1)]); # Anfangswerte\n\n  T
_matrix:=band([1,-2,1],m-1);\n\n  F:=V->evalm( T_matrix&*V/h^2+vector(
m-1,[seq(V[`i`]*(1-V[`i`]),`i`=1..m-1)]) );\n\n  for j from 0 to n-1 d
o\n\n    U.(j+1):=map(evalf, evalm( U.j+ k/2*( F(U.j)+F( evalm(U.j+k*F
(U.j)) ) ) ) );   # (##)\n\n  od;\n\n  werte:=[];\n\n  for j from 0 to
 n do\n\n    werte:=[op(werte),[0,seq(U.j[`i`],`i`=1..m-1),0]]; # Rand
werte U.j[0] und U.j[m] = 0\n\n  od;\n\n  RETURN(PLOT3D(GRID(0..T,0..1
,werte),AXESSTYLE(BOX)));  \n\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 113 "h:=0.33; k:=h^2/2; T_max:=15;\nprint(`i mit T = 1:`)
; aufgabe_i(1,h,k); print(`ii mit T = 1:`); aufgabe_ii(1,h,k);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "for T from 5 by 5 to T_max \+
do\n  print(cat(`i mit T = `,T,`:`)); aufgabe_i(T,h,k); \n  print(cat(
`ii mit T = `,T,`:`)); aufgabe_ii(T,h,k);\nod;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 30 "Was passiert also fuer grosse " }{XPPEDIT 18 0 "T;" 
"6#%\"TG" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT 258 32 "Die Loesung \+
konvergiert gegen 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 1 3 0 1" 
250 }{VIEWOPTS 1 1 0 1 1 1803 }