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Sitzung" }{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Erinnerung an die 13. S itzung" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "a) Gehe zurueck zur 13. Sitzung, f alls diese noch nicht abgeschlossen war. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "b) Es sind in den ersten 11 Sitzungen die folgenden Opera toren und Funktionen aufgetaucht:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {TEXT 257 31 "$ , ` ` , ' ' , . , &*" }{TEXT 261 1 " " } {TEXT 260 111 ", add , alias , allvalues, array, assign, BesselJ, bino mial, cat , convert , copy , cost , debug , diff , Diff" }{TEXT 266 1 " " }{TEXT 267 27 ", display , eval , evalf , " }{TEXT 274 6 "evalhf " }{TEXT 275 49 " , evaln , evalm , expand , factor , factorial , " } {TEXT 270 3 "FFT" }{TEXT 271 38 " , forget , fsolve , help , ifactor , " }{TEXT 272 5 "iFFT " }{TEXT 273 63 ", indices , int , interface, is prime , lhs, matrix , matrixplot" }{TEXT 265 1 " " }{TEXT 268 33 ", mu l , nops , op , plot , plot3d" }{TEXT 264 1 " " }{TEXT 269 32 ", plots etup , print , product , " }{TEXT 276 4 "rand" }{TEXT 277 3 " , " } {TEXT 278 10 "randmatrix" }{TEXT 279 3 " , " }{TEXT 280 8 "readdata" } {TEXT 281 24 " , readlib, rhs , RootOf" }{TEXT 259 1 "," }{TEXT 258 77 " save , seq , simplify , sort , sqrt , subs , subsop , sum , table , taylor ," }{TEXT 262 2 " " }{TEXT 263 39 "time , trace , type, wri teto , whattype" }}{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 256 "" 0 "" {TEXT -1 91 " ----- schn eide das Worksheet hier ab und reiche die Loesung des unteren Teils ei n ! -----" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 451 "Die folgende Aufgab e soll geloest und zur Korrektur abgeliefert werden (ihre erfolgreiche Bearbeitung zaehlt als Teilkriterium fuer die Vergabe des Praktikumss cheins zu diesem Kurs). Ergaenze dazu den Rest dieses Worksheets mit d en MAPLE-Befehlen, die die Loesung der Aufgabe liefern. Trage Namen un d (falls vorhanden) die flcaXX-Nummer ein. Raeume auf: entferne even tuelle Ausgaben aus dem Worksheet. Speichere das Worksheet, etwa unter dem Namen " }{TEXT 282 11 "Loesung.mws" }{TEXT -1 53 " , dann reich e diese Datei per electronic mail ein: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 284 48 "elm flca00@rz.uni-frankfurt.de < Loesung.mws \+ " }}{PARA 0 "" 0 "" {TEXT 289 35 "Abgabe : bis Freitag, den 20.6.97 " }{TEXT -1 29 ". Nach diesem Datum wird in " }{TEXT 285 24 "/home/fb 12/kurse/flca00 " }{TEXT -1 15 "das Worksheet " }{TEXT 283 19 "Muster loesung14.mws" }{TEXT -1 130 " lesbar gemacht. Korrigierte Versionen \+ der eingesandten Loesungen werden individuell per electronic mail zuru eckgeschickt werden." }}{PARA 0 "" 0 "" {TEXT 286 10 "Trage ein:" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 307 18 "Name und Vorname :" } {TEXT -1 6 " " }{TEXT 290 19 "Mustermann, Hermann" }}{PARA 0 "" 0 "" {TEXT 308 18 "flcaXX-Nummer :" }{TEXT -1 7 " " }{TEXT 287 6 "flca??" }}{PARA 0 "" 0 "" {TEXT 309 18 "oder email :" } {TEXT -1 7 " " }{TEXT 288 26 "flca??@rz.uni-frankfurt.de" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 " Es geht im folgenden darum, Stammfunktionen von Differentialpolynomen \+ zu bestimmen. Sei " }{TEXT 302 6 "u=u(x)" }{TEXT -1 52 " eine unbesti mmte Funktion in einer Variablen und " }{XPPEDIT 18 0 "expr=Polynom(u ,u[x],u[xx],`..`,u[``(k)])" "/%%exprG-%(PolynomG6'%\"uG&F'6#%\"xG&F'6# %#xxG%#..G&F'6#-%!G6#%\"kG" }{TEXT -1 26 " . Der Systemintegrierer " }{TEXT 291 3 "int" }{TEXT -1 64 " kann nur in den einfachsten Faellen Stammfunktionen bestimmen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "alias(u=u(x)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(u*diff(u,x),x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(diff(u,x,x),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Bereits" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "int( u*diff(u,x,x, x) ,x);" }}{PARA 0 "" 0 "" {TEXT -1 52 "wird nicht mehr integriert, ob wohl die Stammfunktion" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "stammfunk tion:=u*diff(u,x,x)-diff(u,x)^2/2;" }}{PARA 0 "" 0 "" {TEXT -1 20 "lei cht angebbar ist:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(stammfunk tion,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 10 "Aufgabe 4:" }{TEXT -1 35 " Implementiere einen Integrierer " }{TEXT 293 29 "Int:=proc( \+ expr , x ) ... end" }{TEXT -1 40 " , der in der Lage ist, alle Ausdrue cke " }{TEXT 294 4 "expr" }{TEXT -1 128 ", die die Ableitung eines Di fferentialpolynoms sind, zu integrieren (d.h., das entsprechende Diffe rentialpolynom zu finden) ! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 " Beispiel:\n Int( u*diff(u,x,x,x)+u^2 , x ) ---> " }{XPPEDIT 18 0 "u *diff(u,x,x)-diff(u,x)^2/2+Int(u^2,x)" ",(*&%\"uG\"\"\"-%%diffG6%F$%\" xGF)F%F%*&-F'6$F$F)\"\"#\"\"#!\"\"F/-%$IntG6$*$F$\"\"#F)F%" }{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Anleitung:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 303 10 "Anleitung:" }{TEXT -1 2 " " }{TEXT 295 3 "Int" }{TEXT -1 150 " ist linear: zerlege Summen, ziehe Konstante n (nicht von x abhaengende Faktoren) vor die Integration. Damit verble ibt das Problem, Monome der Form " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "expr := u[` `]^p[0]*u[x]^p[1]*`...`*u[``(k-1)]^p[k-1]* u[``(k)]^p[k]" ">%%exprG*,)&%\"uG6#%\"~G&%\"pG6#\"\"!\"\"\")&F'6#%\"xG &F+6#\"\"\"F.%$...GF.)&F'6#-%!G6#,&%\"kGF.\"\"\"!\"\"&F+6#,&F>F.\"\"\" F@F.)&F'6#-F;6#F>&F+6#F>F." }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " (mit " }{XPPEDIT 18 0 "u[``(k)]=`diff( u , x$k )`" "/&%\"uG6#- %!G6#%\"kG%0diff(~u~,~x$k~)G" }{TEXT -1 18 " ) zu behandeln. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Ein moegl icher Normalformalgorithmus fuer " }{TEXT 296 3 "Int" }{TEXT -1 153 " \+ arbeitet folgendermassen: die Idee ist, durch partielle Integration \+ die Ableitungsordnung (die hoechste vorkommende Ableitung im Monom) zu minimieren." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Finde dazu den Faktor " }{XPPEDIT 18 0 "u[``(k)]^p[k]" ")&%\"uG 6#-%!G6#%\"kG&%\"pG6#F)" }{TEXT -1 53 " mit der hoechsten Ableitung, \+ dann finde den Faktor " }{XPPEDIT 18 0 "u[``(k-1)]^p[k-1]" ")&%\"uG6#- %!G6#,&%\"kG\"\"\"\"\"\"!\"\"&%\"pG6#,&F*F+\"\"\"F-" }{TEXT -1 88 " mi t dem naechstniedrigeren Ableitungsgrad. Die restlichen Faktoren seien zum Ausdruck " }{XPPEDIT 18 0 "A=A(u,u[x],`...`,u[``(k-2)]" "/%\"AG- F#6&%\"uG&F&6#%\"xG%$...G&F&6#-%!G6#,&%\"kG\"\"\"\"\"#!\"\"" }{TEXT -1 17 " zusammengefasst." }}{PARA 0 "" 0 "" {TEXT 297 7 "Fall 1:" } {TEXT -1 2 " " }{XPPEDIT 18 0 "p[k]>1" "2\"\"\"&%\"pG6#%\"kG" }{TEXT -1 6 " oder " }{TEXT 301 3 "k=0" }{TEXT -1 13 " : liefere " }{TEXT 304 12 "Int(expr,x) " }{TEXT -1 21 " unevaluiert zurueck." }}{PARA 0 " " 0 "" {TEXT 298 7 "Fall 2:" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1=p[k]" " /\"\"\"&%\"pG6#%\"kG" }{TEXT -1 6 " und " }{XPPEDIT 18 0 "k <>0 " "0% \"kG\"\"!" }{TEXT -1 13 ": es gilt " }{XPPEDIT 18 0 "expr*` = `*A*` `*u[``(k-1)]^p[k-1]*` `*u[``(k)]= A/(p[k-1]+1" "/*0%%exprG\"\"\"%$~=~ GF%%\"AGF%%\"~GF%)&%\"uG6#-%!G6#,&%\"kGF%\"\"\"!\"\"&%\"pG6#,&F1F%\"\" \"F3F%F(F%&F+6#-F.6#F1F%*&F'F%,&&F56#,&F1F%\"\"\"F3F%\"\"\"F%F3" } {TEXT -1 2 " " }{XPPEDIT 18 0 "diff( u[(k-1)]^(p[k-1]+1),x)" "-%%diff G6$)&%\"uG6#,&%\"kG\"\"\"\"\"\"!\"\",&&%\"pG6#,&F*F+\"\"\"F-F+\"\"\"F+ %\"xG" }{TEXT -1 34 " . Durch partielle Integration " }{XPPEDIT 18 0 "` Int`(A*u[``(k-1)]^p[k-1]*u[``(k)],x )" "-%%~IntG6$*(%\"AG\"\"\")& %\"uG6#-%!G6#,&%\"kGF'\"\"\"!\"\"&%\"pG6#,&F0F'\"\"\"F2F'&F*6#-F-6#F0F '%\"xG" }}{PARA 0 "" 0 "" {TEXT -1 26 " ---> " } {XPPEDIT 18 0 "1/(p[k-1]+1)" "*&\"\"\"\"\"\",&&%\"pG6#,&%\"kGF$\"\"\"! \"\"F$\"\"\"F$F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "A*u[(k-1)]^(p[k-1]+1 )" "*&%\"AG\"\"\")&%\"uG6#,&%\"kGF$\"\"\"!\"\",&&%\"pG6#,&F*F$\"\"\"F, F$\"\"\"F$F$" }{TEXT -1 5 " - " }{XPPEDIT 18 0 "1/(p[k-1]+1)" "*&\" \"\"\"\"\",&&%\"pG6#,&%\"kGF$\"\"\"!\"\"F$\"\"\"F$F," }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "` Int`(A[x]*u[(k-1)]^(p[k-1]+1),x)" "-%%~IntG6$*&&% \"AG6#%\"xG\"\"\")&%\"uG6#,&%\"kGF*\"\"\"!\"\",&&%\"pG6#,&F0F*\"\"\"F2 F*\"\"\"F*F*F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 94 "wird d er Differentiationsgrad im verbleibenden Integrationsproblem verkleine rt (beachte, dass " }{XPPEDIT 18 0 "A[x]" "&%\"AG6#%\"xG" }{TEXT -1 51 " ein Differentialpolynom ist, indem hoechstens die " }{TEXT 299 5 "(k-1)" }{TEXT -1 18 ".te Ableitung von " }{TEXT 305 1 "u" }{TEXT -1 13 " auftaucht)." }}{PARA 0 "" 0 "" {TEXT -1 220 "Rekursiv wird so de r Differentiationsgrad verringert, bis Fall 1 auftritt. Die gesuchte S tammfunktion setzt sich zusammen aus den Randtermen der partiellen Int egrationen und den nicht evaluierbaren Integralen aus Fall 1." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 8 "Hinweis: " }{TEXT -1 72 " die in Kapitel 8.12.2 der Vorlesung/des Skriptes ange fuehrte Prozedur " }{TEXT 300 11 "diffOrdnung" }{TEXT -1 94 " kann f uer die Suche nach den hoechsten und zweithoechsten Ableitungstermen a ngepasst werden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "unprotect(Int);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Int:=proc(expr,x) ... end;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 310 13 "Einige Tests:" }}{PARA 0 "" 0 "" {TEXT -1 13 "Abkuerzu ngen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "alias(u=u(x),seq(u.i=diff( u(x),x$i),i=1..20));" }}{PARA 0 "" 0 "" {TEXT 314 8 "Achtung:" }{TEXT -1 33 " wenn in der Implementierung von " }{TEXT 312 3 "Int" }{TEXT -1 12 " der Name " }{TEXT 313 1 "u" }{TEXT -1 132 " der unbestimmt en Funktion explizit verwendet wurde (dies ist bei geschickter Impleme ntierung nicht noetig), so erzeugt das obige " }{TEXT 311 5 "alias" } {TEXT -1 16 " ein Problem !!!" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(u,x);Int(u^2,x);Int(diff(u,x,x),x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "expr:=u^5*diff(u^10,x$5); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Int(diff(expr,x),x);" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "expand(\"\"-\");" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "expr:=diff(u^5,x$3)*diff(u^2,x$5)^3;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "Int(diff(expr,x),x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "expand(\"\"-\");" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "Test:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Der Hintergrund des folgenden Tests ist die Korteweg-de Vries-Gleichung" }}{PARA 0 "" 0 " " {TEXT -1 7 " " }{XPPEDIT 18 0 "Diff(u,t)=diff(u,x,x,x)+6*u*dif f(u,x)*` ( = `*K[2]*` )`" "/-%%DiffG6$%\"uG%\"tG,&-%%diffG6&F&%\"xGF ,F,\"\"\"*.\"\"'F-F&F--F*6$F&F,F-%(~~(~=~~GF-&%\"KG6#\"\"#F-%#~)GF-F- " }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 105 "welche unendlich v iele ``Symmetrien'' besitzt (dies sind Differentialpolynome, die mit d er rechten Seite " }{XPPEDIT 18 0 "K[2]" "&%\"KG6#\"\"#" }{TEXT -1 44 " der Differentialgleichung ``kommutieren'')." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Abkuerzungen u=u(x), \+ u1=diff(u(x),x), u2=diff(u(x),x,x) etc.:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "alias(u=u(x),seq(u.i=diff(u(x),x$i),i=1..20));" }} {PARA 0 "" 0 "" {TEXT -1 44 "Betrachte die folgenden Differentialpolyn ome" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "K[1]:=dif f(u,x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "K[2]:=diff(u,x,x,x)+6*u* diff(u,x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "K[3]:=diff(u,x$5)+10 *u*diff(u,x$3)\n +20*diff(u,x)*diff(u,x,x)\n \+ +30*u^2*diff(u,x);" }}{PARA 0 "" 0 "" {TEXT -1 55 "Sie bilden d ie ersten Ausdruecke des rekursiven Schemas" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "N:=7:\nfor i to N-1 do \n K[i+1]:=expand( diff(K[i],x ,x)+4*u*K[i]+2*diff(u,x)*Int(K[i],x) );\nod;" }}{PARA 0 "" 0 "" {TEXT -1 158 "Man kann zeigen, dass hierdurch eine (unendliche) Hierarchie v on Differentialpolynomen erzeugt wird (d.h., alle Integrationen koenne n ausgefuehrt werden). Mit" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "komm utator:=proc(K1,K2) local eps;\n subs(u=u+eps*K1,K2)-subs(u=u+eps*K2 ,K1); \n diff(\",eps);\n subs(eps=0,\");\n expand(\");\nend; " } }{PARA 0 "" 0 "" {TEXT -1 74 "muessen die gefundenen Stammfunktionen s o sein, dass alle ``Kommutatoren''" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "for i to N do\n for j from i+1 to N do\n print(i,j,kommutator(K[ i],K[j]));\n od;\nod;" }}{PARA 0 "" 0 "" {TEXT -1 13 "verschwinden." } {MPLTEXT 1 0 0 "" }}}}}{MARK "14" 0 }{VIEWOPTS 1 1 0 3 2 1804 }