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Sitzung" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Erinnerung an die 11.te Sitzung" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "a) Gehe zurueck zur 11. S itzung, falls diese noch nicht abgeschlossen war. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "b) Es sind in den ersten 11 Sitzungen die folgend en Operatoren und Funktionen aufgetaucht:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 257 31 "$ , ` ` , ' ' , . , &*" }{TEXT 261 1 " \+ " }{TEXT 260 107 ", add , alias , allvalues, array, assign, BesselJ, b inomial, cat , convert , copy , cost , debug , diff , " }{TEXT 274 4 "Diff" }{TEXT 275 3 " , " }{TEXT 266 7 "display" }{TEXT 267 150 " , ev al , evalf , evaln , evalm , expand , factor , factorial , forget , fs olve , help , ifactor , indices , int , interface, isprime , lhs, matr ix , " }{TEXT 268 10 "matrixplot" }{TEXT 269 28 " , mul , nops , op , \+ plot , " }{TEXT 264 6 "plot3d" }{TEXT 265 3 " , " }{TEXT 270 9 "plotse tup" }{TEXT 271 42 " , print , product , readlib, rhs , RootOf" } {TEXT 259 1 "," }{TEXT 258 1 " " }{TEXT 273 4 "save" }{TEXT 272 72 " , seq , simplify , sort , sqrt , subs , subsop , sum , table , taylor , " }{TEXT 262 2 " " }{TEXT 263 39 "time , trace , type, writeto , what type" }}{PARA 0 "" 0 "" {TEXT -1 58 "Was ist die jeweilige Bedeutung? \+ Benutze im zweifelsfalle " }{TEXT 256 4 "help" }{TEXT -1 35 " , um die Erinnerung aufzufrischen." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "help() ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "c) Plotte die Kurve " } {TEXT 276 48 "x(s)=cos(s) , y(s) = BesselJ(0,s) , s=[0, 5*Pi]" } {TEXT -1 2 " !" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Loesung:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot( [cos(s),BesselJ(0,s),s =0..5*Pi] );" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "d) Wie vermeid et man die Singularitaet beim Plotten von " }{TEXT 277 10 "sin(x)/x^2 " }{TEXT -1 2 " ?" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Loesung:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Clipping, \+ z.B.:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot( sin(x)/x^2 , x=-3*Pi ..3*Pi, y=-1..1 );" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "e) Wie \+ erreicht man, dass die automatische Skalierung ausgeschaltet wird (so \+ dass z.B. Ellipsen als Ellipsen und nicht als Kreise erscheinen) ?" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([sin(phi),cos(phi)/2, phi=0..2*Pi],scaling=constrained);" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "f) Sei " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "f:=proc(x) \n if abs(x)<=1 then x \n elif x>=1 then exp(-(abs(x)-1)) \n else -exp(-(ab s(x)-1)) \n fi; \nend;" }}{PARA 0 "" 0 "" {TEXT -1 46 "Plotte diese st ueckweise definierte Funktion !" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " f(x);Platz;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Loesung:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Da " } {TEXT 280 4 "f(x)" }{TEXT -1 20 " mit symbolischem " }{TEXT 281 1 "x " }{TEXT -1 38 " nicht evaluiert werden kann, muss " }{TEXT 278 4 " f(x)" }{TEXT -1 2 " " }{TEXT 329 10 "verzoegert" }{TEXT -1 5 " an " }{TEXT 279 4 "plot" }{TEXT -1 20 " uebergeben werden:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot('f(x)',x=-5..5);" }}}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "g) Plotte die punktweise ueber ein em rechteckigen Gitter " }{TEXT 282 10 "x[i], y[j]" }{TEXT -1 9 " du rch " }{TEXT 283 38 "f(x[i],y[j]) == f[i,j] := sin( j/10 ) " }{TEXT -1 39 "/ i , i,j=1..50 definierte Funktion !" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Loesung :" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "with(plots); matrixplot(matrix(50,50,(i,j) -> sin(j/10)/i));" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "f) Wie spei chert man eine Graphik als Postscript in einer externen Datei " } {TEXT 284 7 "Bild.ps" }{TEXT -1 3 " ?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Loesung:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot setup(plotdevice=postscript,plotoutput=`Bild.ps`);\nplot(sin(x),x=-Pi. .Pi);" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "g) Wie lenkt man den \+ Graphik-Output wieder in das Worksheet zurueck?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(sin( x),x=-10..10);" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Loesung:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot setup(plotdevice=inline);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(s in(x),x=-10..10);" }}}}}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 91 " ----- schneide das Worksheet hier ab und reiche die Loesung des unteren Tei ls ein ! -----" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 451 "Die folgende A ufgabe soll geloest und zur Korrektur abgeliefert werden (ihre erfolgr eiche Bearbeitung zaehlt als Teilkriterium fuer die Vergabe des Prakti kumsscheins zu diesem Kurs). Ergaenze dazu den Rest dieses Worksheets \+ mit den MAPLE-Befehlen, die die Loesung der Aufgabe liefern. Trage Nam en und (falls vorhanden) die flcaXX-Nummer ein. Raeume auf: entferne eventuelle Ausgaben aus dem Worksheet. Speichere das Worksheet, etwa \+ unter dem Namen " }{TEXT 285 11 "Loesung.mws" }{TEXT -1 53 " , dann \+ reiche diese Datei per electronic mail ein: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 287 48 "elm flca00@rz.uni-frankfurt.de < Loes ung.mws " }}{PARA 0 "" 0 "" {TEXT 292 56 "Die Abgabe hat bis zum Freit ag, den 6.6.97, zu erfolgen" }{TEXT -1 29 ". Nach diesem Datum wird i n " }{TEXT 288 24 "/home/fb12/kurse/flca00 " }{TEXT -1 15 "das Worksh eet " }{TEXT 286 19 "Musterloesung12.mws" }{TEXT -1 130 " lesbar gem acht. Korrigierte Versionen der eingesandten Loesungen werden individu ell per electronic mail zurueckgeschickt werden." }}{PARA 0 "" 0 "" {TEXT 289 10 "Trage ein:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "Name und Vorname: " }{TEXT 293 19 "Mustermann, Hermann" }} {PARA 0 "" 0 "" {TEXT -1 25 "flcaXX-Nummer : " }{TEXT 290 6 " flca??" }}{PARA 0 "" 0 "" {TEXT -1 33 "oder email : \+ " }{TEXT 291 26 "flca??@rz.uni-frankfurt.de" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Es geht im folge nden um eine numerische Loesung der Poissongleichung " }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "Delta u = Diff(u,x,x)+Diff(u,y ,y) *`=`* f(x,y)" "/*&%&DeltaG\"\"\"%\"uGF%,&-%%DiffG6%F&%\"xGF+F%*(-F )6%F&%\"yGF/F%%\"=GF%-%\"fG6$F+F/F%F%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "mit vorgegebenem " }{TEXT 294 6 "f(x,y)" }{TEXT -1 29 " ueber dem Einheitsquadrat " }{XPPEDIT 18 0 "Omega=\{ (x,y), 0<= x,y<=1" "/%&OmegaG<&%\"xG%\"yG1\"\"!F%1F&\"\"\"" }{TEXT -1 26 " . Hie rbei sind Randdaten" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 296 77 " u(x,0) = unten(x) , u(x,1) = oben(x) , u(0,y) = links(y) , u(1,y) = re chts(y)" }}{PARA 0 "" 0 "" {TEXT -1 18 "auf dem Rand von " }{XPPEDIT 18 0 "Omega " "I&OmegaG6\"" }{TEXT -1 24 " durch die Funktionen " } {TEXT 295 23 "unten,oben,links,rechts" }{TEXT -1 12 " vorgegeben." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Ein moeglich es numerisches Loesungsverfahren ist das folgende: ueberziehe das Gebi et mit einem quadratischen Netz von Punkten " }{XPPEDIT 18 0 "x[i]=i/N , y[j]=j/N, i,j=0..N" "6&/&%\"xG6#%\"iG*&F'\"\"\"%\"NG!\"\"/&%\"yG6#% \"jG*&F0F)F*F+F'/F0;\"\"!F*" }{TEXT -1 7 " . Sei " }{XPPEDIT 18 0 "u[i ,j]=u(x[i],y[j]])" "7#/&%\"uG6$%\"iG%\"jG-F%6$&%\"xG6#F'&%\"yG6#F(" } {TEXT -1 84 " die Matrix der Loesung auf den Gitterpunkten. Man ersetz t nun den Laplace-Operator " }{XPPEDIT 18 0 "Delta u" "*&%&DeltaG\"\" \"%\"uGF$" }{TEXT -1 62 " durch eine Differenzenapproximation, d.h., \+ man approximiert " }{XPPEDIT 18 0 "Delta u (x[i],y[j])" "*&%&DeltaG\" \"\"-%\"uG6$&%\"xG6#%\"iG&%\"yG6#%\"jGF$" }{TEXT -1 48 " durch eine g eeignete Differenz von Werten von " }{TEXT 297 1 "u" }{TEXT -1 28 " i n der Naehe des Punktes " }{XPPEDIT 18 0 "``(x[j],y[j])" "-%!G6$&%\"x G6#%\"jG&%\"yG6#F(" }{TEXT -1 27 " , z.B. mittels der Werte " } {XPPEDIT 18 0 "u[i,j], u[i-1,j],u[i+1,j],u[i,j-1],u[i,j+1]" "6'&%\"uG6 $%\"iG%\"jG&F$6$,&F&\"\"\"\"\"\"!\"\"F'&F$6$,&F&F+\"\"\"F+F'&F$6$F&,&F 'F+\"\"\"F-&F$6$F&,&F'F+\"\"\"F+" }{TEXT -1 151 " . Setzt man diese Ap proximation in die Differentialgleichung ein, so entsteht ein endliche s (in diesem Fall lineares) Gleichungssystem fuer die Werte " } {XPPEDIT 18 0 "u[i,j]" "&%\"uG6$%\"iG%\"jG" }{TEXT -1 123 " dessen Loe sung die numerische Loesung des Randwertproblems darstellt (beachte, d ass durch die Randdaten einige der Werte " }{XPPEDIT 18 0 "u[i,j] " " &%\"uG6$%\"iG%\"jG" }{TEXT -1 26 " bereits vorgegeben sind)." }}} {EXCHG {PARA 258 "" 0 "" {TEXT 334 89 "Zunaechst ist eine geeignete Di fferenzenapproximation des Laplace-Operators zu bestimmen." }{TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 14 "Aufga be 3.a): " }{TEXT -1 10 " Zeige: " }{XPPEDIT 18 0 "Delta[5]*u(x,y)=D elta*u(x,y)+O(h^2)" "/*&&%&DeltaG6#\"\"&\"\"\"-%\"uG6$%\"xG%\"yGF(,&*& F%F(-F*6$F,F-F(F(-%\"OG6#*$%\"hG\"\"#F(" }{TEXT -1 8 " , wobei" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Delta[5]*u(x,y) =(u(x,y+h)+u(x-h,y)-4*u(x,y)+u(x+h,y)+u (x,y-h))/h^2" "/*&&%&DeltaG6#\"\"&\"\"\"-%\"uG6$%\"xG%\"yGF(*&,,-F*6$F ,,&F-F(%\"hGF(F(-F*6$,&F,F(F3!\"\"F-F(*&\"\"%F(-F*6$F,F-F(F7-F*6$,&F,F (F3F(F-F(-F*6$F,,&F-F(F3F7F(F(*$F3\"\"#F7" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "die sogenannte 5-S tern-Diskretisierung von " }{XPPEDIT 18 0 "Delta " "I&DeltaG6\"" } {TEXT -1 6 " ist." }}{PARA 0 "" 0 "" {TEXT -1 30 "Anleitung: Multipli kation mit " }{TEXT 301 3 "h^2" }{TEXT -1 5 " und " }{TEXT 302 6 "tayl or" }{TEXT -1 36 " oder Laurent-Entwicklung mittels " }{TEXT 303 6 " series" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 12 "Anmerkung: " }{TEXT 313 18 "D[1,2,2,2](u)(x,y)" }{TEXT -1 25 " etc. ist synonym fu er " }{TEXT 314 20 "diff(u(x,y),x,y,y,y)" }{TEXT -1 5 " etc." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 12 "Aufgabe 3.b)" }{TEXT -1 29 ": Bestimme die Koeffizienten " }{TEXT 300 5 "a,b,c" }{TEXT -1 36 " des symmetrischen 9-Stern-Ansat zes " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Delta[9]*u(x,y) =h^(-2)* `(`" " /*&&%&DeltaG6#\"\"*\"\"\"-%\"uG6$%\"xG%\"yGF(*&)%\"hG,$\"\"#!\"\"F(%\" (GF(" }}{PARA 0 "" 0 "" {TEXT -1 24 " " } {XPPEDIT 18 0 "a*u(x-h,y+h) + b*u(x,y+h)+a*u(x+h,y+h) " ",(*&%\"aG\"\" \"-%\"uG6$,&%\"xGF%%\"hG!\"\",&%\"yGF%F+F%F%F%*&%\"bGF%-F'6$F*,&F.F%F+ F%F%F%*&F$F%-F'6$,&F*F%F+F%,&F.F%F+F%F%F%" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 19 " " }{XPPEDIT 18 0 "``+b*u(x-h,y) *` `-c*u(x,y)*` `+b*u(x+h,y) " ",*%!G\"\"\"*(%\"bGF$-% \"uG6$,&%\"xGF$%\"hG!\"\"%\"yGF$%)~~~~~~~~GF$F$*(%\"cGF$-F(6$F+F.F$F/F $F-*&F&F$-F(6$,&F+F$F,F$F.F$F$" }{TEXT -1 8 " " }}{PARA 0 "" 0 "" {TEXT -1 19 " " }{XPPEDIT 18 0 "``+a*u(x-h,y-h) + b*u(x,y-h)+a*u(x+h,y-h) " ",*%!G\"\"\"*&%\"aGF$-%\"uG6$,&%\"xGF$%\"hG !\"\",&%\"yGF$F,F-F$F$*&%\"bGF$-F(6$F+,&F/F$F,F-F$F$*&F&F$-F(6$,&F+F$F ,F$,&F/F$F,F-F$F$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 " \+ " }{XPPEDIT 18 0 "`)`" "I\")G6\"" }}{PARA 0 "" 0 "" {TEXT -1 42 "so, dass mit einer geeigneten Konstanten " } {XPPEDIT 18 0 "beta" "I%betaG6\"" }{TEXT -1 7 " gilt:" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "Delta[9] *u = Delta *u+ beta * h^2*Delta*Delta*u+O(h^4) " "/*&&%&DeltaG6#\"\"*\"\"\"%\"uGF(,(*&F%F( F)F(F(*,%%betaGF(*$%\"hG\"\"#F(F%F(F%F(F)F(F(-%\"OG6#*$F/\"\"%F(" }} {PARA 0 "" 0 "" {TEXT -1 11 "Hilfe: mit " }{TEXT 304 17 "collect(expr, D) " }{TEXT -1 43 "werden die Koeffizienten von verschiedenen " } {TEXT 306 2 "D-" }{TEXT -1 17 "Termen gesammelt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "Delta9:=(a*u(x-h,y+h)+b*u(x,y+h)+a*u(x+h,y+h)\n +b*u(x-h,y )-c*u(x,y )+b*u(x+h,y) \n +a*u(x-h,y-h)+b*u(x,y-h)+a*u(x+h, y-h)\n )/h^2 ; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "Fehler:= Delta9\n -( D[1,1](u)(x,y)+D[2,2](u)(x,y) )\n -beta*h^2* ( \+ D[1,1,1,1](u)(x,y)\n +2*D[1,1,2,2](u)(x,y)\n \+ + D[2,2,2,2](u)(x,y)\n );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Bestimme a,b,c,beta !" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 335 32 " Konstruktion eines Testproblems" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Es wird ein Testproble m gebastelt, an dem die Numerik ausprobiert werden soll. Sei" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "exakt:= -sin(x* Pi)*sin(y*Pi)/2/Pi^2;" }}{PARA 0 "" 0 "" {TEXT -1 35 "die exakte Loesu ng, fuer die sich " }{TEXT 305 3 "f:=" }{XPPEDIT 307 0 "Delta*u" "*&% &DeltaG\"\"\"%\"uGF$" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=diff(exa kt,x,x)+diff(exakt,y,y);" }}{PARA 0 "" 0 "" {TEXT -1 38 "ergibt. Die R andbedingungen, die von " }{TEXT 308 5 "exakt" }{TEXT -1 32 " erfuel lt werden, sind homogen:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(y=0,exakt):unten:=\";" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(y=1,exakt):oben:=\";" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(x=0,exakt):links:=\";" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(x=1,exakt):rechts:=\";" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 336 18 "Numerische Loesung" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 309 15 "Aufgabe 3.c): " }{TEXT -1 53 "Bestimme \+ eine numerische Loesung des Randwertproblems" }}{PARA 0 "" 0 "" {TEXT 311 4 " " }{XPPEDIT 18 0 "Delta*u = f(x,y)* `=`*sin(x*Pi)*sin(y*Pi " "/*&%&DeltaG\"\"\"%\"uGF%**-%\"fG6$%\"xG%\"yGF%%\"=GF%-%$sinG6#*&F+F %%#PiGF%F%-F/6#*&F,F%F2F%F%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "ueber dem Einheitsquadrat mit homogenen Randbedingungen. " }} {PARA 0 "" 0 "" {TEXT -1 12 "Setze dazu " }{TEXT 310 15 "N:=10 ; h:=1 /N;" }{TEXT -1 41 " und loese das lineare Gleichungssystem " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "Delta[5]*u(x[i],y[j])=sin (x[i]*Pi)*sin(y[j]*Pi)" "/*&&%&DeltaG6#\"\"&\"\"\"-%\"uG6$&%\"xG6#%\"i G&%\"yG6#%\"jGF(*&-%$sinG6#*&&F-6#F/F(%#PiGF(F(-F66#*&&F16#F3F(F;F(F( " }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x[i]=i/N, y[j]=j/N, i,j=1..N-1" "6 &/&%\"xG6#%\"iG*&F'\"\"\"%\"NG!\"\"/&%\"yG6#%\"jG*&F0F)F*F+F'/F0;\"\" \",&F*F)\"\"\"F+" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 22 "Die \+ Unbekannten sind " }{XPPEDIT 18 0 "u[i,j] =u(x[i],y[j]" "/&%\"uG6$%\" iG%\"jG-F$6$&%\"xG6#F&&%\"yG6#F'" }{TEXT -1 10 " , wobei " }{XPPEDIT 18 0 "u[i,0],u[i,N],u[0,j],u[N,j]" "6&&%\"uG6$%\"iG\"\"!&F$6$F&%\"NG&F $6$F'%\"jG&F$6$F*F-" }{TEXT -1 42 " wegen der Randbedingungen verschw inden. " }}{PARA 0 "" 0 "" {TEXT -1 55 "Plotte die numerische Loesung \+ und bestimme den Fehler " }{XPPEDIT 18 0 "max( abs(u[i,j]-exakt(x[i], y[j]))" "-%$maxG6#-%$absG6#,&&%\"uG6$%\"iG%\"jG\"\"\"-%&exaktG6$&%\"xG 6#F,&%\"yG6#F-!\"\"" }{TEXT -1 2 " !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "N:=10;h:=1/N: u:=array(0..N,0..N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Delta5:= ( u[i,j+1]\n +u[i-1, j]-4*u[i,j] +u[i+1,j]\n +u[i,j-1]\n )/h^ 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Rechnung:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Visualisierung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plots[matrixplot](convert(u,matrix));" }}{PARA 0 "" 0 "" {TEXT -1 23 "Berechnung des Fehlers:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "exakt:=(i,j)->evalf(-sin(i*Pi/N)*sin(j*Pi/N)/2/Pi^2 ) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Fehler:=max(seq(seq(abs(u[i,j] -exakt(i,j)),i=0..N),j=0..N);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 14 "Aufgabe 3.d) :" }{TEXT -1 37 " Loese noch einmal das Testproblem " }{XPPEDIT 18 0 "Delta *u = f" "/*&%&DeltaG\"\"\"%\"uGF%%\"fG" }{TEXT -1 49 " aus Aufgabe 3.c), diesmal, indem die Gleichung " }{XPPEDIT 18 0 "Delta*u+beta*h^2*Delta*Delta*u=f+beta*h^2 Delta*f" "/,&*&%&Delta G\"\"\"%\"uGF&F&*,%%betaGF&*$%\"hG\"\"#F&F%F&F%F&F'F&F&,&%\"fGF&**F)F& *$F+\"\"#F&F%F&F.F&F&" }{TEXT -1 82 " diskretisiert wird. Ersetze die linke Seite durch die Differenzenapproximation " }{XPPEDIT 18 0 "Del ta[9]*u" "*&&%&DeltaG6#\"\"*\"\"\"%\"uGF'" }{TEXT -1 112 " aus Aufgab e 3.b) und verfahre numerisch wie in Aufgabe 3.c) ! Vergleiche die Fe hler aus den Teilen c) und d)!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r estart: N:=10;h:=1/N: u:=array(0..N,0..N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a:=1/6:b:=2/3:c:=10/3:beta:=1/12:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 154 "Delta9:=(a*u[i+1,j-1]+b*u[i+1,j]+a*u[i+1,j+1]\n \+ +b*u[i ,j-1]-c*u[i ,j]+b*u[i ,j+1] \n +a*u[i-1,j-1]+b* u[i-1,j]+a*u[i-1,j+1]\n )/h^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Rechnung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "exakt:=(i,j)->evalf(-sin(i*Pi/N)*sin(j*Pi/N)/2/Pi^2 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Fehler:=max(seq(seq(abs(u[i,j]-exakt(i,j)), i=0..N),j=0..N);" }}}}{MARK "14 0 1" 0 }{VIEWOPTS 1 1 0 3 2 1804 }