sowie der Grafiken s3 *.ps serie3.ps, serie3.pdf

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1 sowie der Grafiken s3 *.ps serie3.ps, serie3.pdf % serie3.m % scrip m-file % gdgl: Richungsfelder, Loesungen und Runge-Kua-Verfahren fuer AWP diary serie3. echo on clear all clc clf forma long % serie3.pro % echo off % SERIE3 % %. disp(. ) % =, ()=, <= % --> f.m figure() dfield( f,:.:,-:.:3); % auch dfield(@f,:.:,-:.:3); ile( Richungsfeld zu = in [,][-,3] ); label( ); ylabel( ); plo([ ],[ep() ep()], r- ); prin s3_.ps -dpsc 3.5 Richungsfeld zu = in [,][,3] Richungsfeld zu = in [,][,3]

2 = ; N = ; % Funkionsdefiniionen und Aufrufe = :.5:; % one-line-funcion, Einzeiler, funfun =@() ep() () % inline funcion objec =inline( ep(), ) = Inline funcion: () = ep() () % funcion m-file % % fe.m ()=, scalar argumen % funcion y = f() % = ; % y = *ep(); fe(), fe() % fev.m ()=, array argumen % funcion y = f() % = ; % y = *ep(); fev() % eplici definiion, epression = :.5:; = *ep(); [e,e] = euler( f,,,,n); % auch [e,e] = euler(@f,,,,n); [e,e] = euler( f,,,,); [e,e] = euler( f,,,,); [e,e] = euler( f,,,,); figure() plo(,, b-,e,e, k-,e,e, m-,e,e, r-,e,e, c-,,, b- ); ile( Richungsfeld zu = in [,][,3], () und PZV zum AWP mi ()= ); ais([ 3]); label( ); e(.53,.86, () ); legend( (), PZV, N=, N=, N=, N=,... locaion, souheas ) plo(,, ko ); dfield( f,:.:,:.:3); prin s3_.ps -dpsc 3 Richungsfeld zu = in [,][,3], () und PZV zum AWP mi ()=.5 ().5.5 () PZV, N= N= N= N=

3 reshape([e,e,*ep(e)],,3) disp( PZV mi N=, () ) e e = Columns hrough Columns 5 hrough Columns 9 hrough.8.9. e e = Columns hrough Columns 5 hrough Columns 9 hrough reshape([e,e,*ep(e)],,3) disp( PZV mi N=, () ) reshape([e,e,*ep(e)],,3); disp( PZV mi N=, () ) reshape([e,e,*ep(e)],,3); disp(. c) ) % % =*sqr(), ()=, <=<= % --> f7.m figure(3) dfield( f7,:.:,:.:); ile( Richungsfeld zu =*sqr() in [,][,] ); label( ); ylabel( ); prin s3_3.ps -dpsc.5 Richungsfeld zu =*sqr() in [,][,] = ; = ; = ; au= ; = :.:; = (-au).^; [e,e] = euler( f7,,,,); [e,e] = euler( f7,,,,5); [c,c] = collaz( f7,,,,); [c,c] = collaz( f7,,,,5); [h,h] = heun( f7,,,,); % auch [e,e] = euler(@f7,,,,)

4 [k,k] = krkv( f7,,,,5); disp( PZV ); disp(e ); disp( MPZV, Collaz ); disp(c ); disp( HV ); disp(h ); disp( KRKV ); disp(k ); forma shor e = *e; disp( Tabellenausgabe ) disp( RKV: PZV, HV, MPZV, KRKV mi N=, (), alle Null ) reshape([e,e,h,c,k,e],,6) forma long % =*sqr(), ()=/9, <=<= =/9; = :.:; = (+/3).^; _pseudo = (-/3).^; [e,e] = euler( f7,,,,); [e,e] = euler( f7,,,,5); [c,c] = collaz( f7,,,,); [c,c] = collaz( f7,,,,5); [h,h] = heun( f7,,,,); [h,h] = heun( f7,,,,5); [k,k] = krkv( f7,,,,); [k,k] = krkv( f7,,,,5); forma shor e = (e+/3).^; disp( Tabellenausgabe ) disp( RKV: PZV, HV, MPZV, KRKV mi N=5, () ) reshape([e,e,h,c,k,e],6,6) forma long forma shor e = (e+/3).^; disp( Tabellenausgabe ) disp( RKV: PZV, HV, MPZV, KRKV mi N=, () ) reshape([e,e,h,c,k,e],,6) forma long

5 disp( Tabellenausgabe, formaier ) fprinf(, (i) KRKV (i), N=5 e(i) \n ); fprinf(, PZV HV MPZV KRKV \n ); fprinf(, \n ); for i = : fprinf(, %4.f %9.6f %9.6f %9.6f %9.6f %9.6f \n,... e(i),e(i),h(i),c(i),k(i),e(i)); Tabellenausgabe, formaier (i) KRKV (i), N=5 e(i) PZV HV MPZV KRKV figure(4) plo(,, b-,e,e, m-,e,h, k-,e,c, r-,e,k, c-,,, b- ); ile( Richungsfeld zu =*sqr() in [,][,], () und RKV zum AWP mi N=5, ()=/9 ); ais([ ]); label( ); e(.53,.88, () ); legend( (), PZV, HV, MPZV, KRKV, locaion, souheas ) plo(,/9, ko ); dfield( f7,:.:,:.:); prin s3_4.ps -dpsc Richungsfeld zu =*sqr() in [,][,], () und RKV zum AWP mi N=5, ()=/ ().6.4 () PZV. HV MPZV KRKV e = (e+/3).^; disp( ) disp( Tabellenausgabe, formaier ) fprinf(, (i) KRKV (i), N= e(i) \n ); fprinf(, PZV HV MPZV KRKV \n ); fprinf(, \n ); for i = : fprinf(, %4.f %9.6f %9.6f %9.6f %9.6f %9.6f \n,... e(i),e(i),h(i),c(i),k(i),e(i)); Tabellenausgabe, formaier (i) KRKV (i), N= e(i) PZV HV MPZV KRKV

6 ile( Richungsfeld zu =*sqr() in [,][,], () und RKV zum AWP mi N=, ()=/9 ); ais([ ]); label( ); e(.53,.88, () ); legend( (), PZV, HV, MPZV, KRKV, locaion, souheas ) plo(,/9, ko ); dfield( f7,:.:,:.:); prin s3_5.ps -dpsc Richungsfeld zu =*sqr() in [,][,], () und RKV zum AWP mi N=, ()=/ ().6.4 () PZV. HV MPZV KRKV disp(. d) Bewegungsgleichung eines Koerpers im zaehen Medium ) % % =c+k, ()=, c>, k<, <= % --> f7.m global c k c=; k=-; figure(6) dfield( f7,:.:,-:.:); ile( Richungsfeld zu =- in [,][-,] ); label( ); ylabel( ); plo([ ],[ ], r- ); prin s3_6.ps -dpsc.5 Richungsfeld zu = in [,][,]

7 = ; = :.5:; = ; = -ep(-); = +ep(-); m = -*ep(-); figure(7) dfield( f7,:.:,-:.:); ile( Richungsfeld zu =- in [,][-,] mi Loesungen zu AB ); label( ); ylabel( ); plo([ ],[ ], r-,[,,,],[-,,,], ko ); plo(,, b-,,, k-,,, m-,,m, c- ); prin s3_7.ps -dpsc.5 Richungsfeld zu = in [,][,] mi Loesungen zu AB % PZV = ; = 4; N = 6; = :.5:; = -ep(-); [e6,e6] = euler( f7,,,,n); % auch [e6,e6] = euler(@f7,,,,n); [e8,e8] = euler( f7,,,,8); [e4,e4] = euler( f7,,,,4); [e3,e3] = euler( f7,,,,3); [e,e] = euler( f7,,,,); figure(8) plo(,, b-,e,e, g-,e3,e3, k-,e4,e4, c-,e8,e8, m-,e6,e6, k- ); legend( (), PZV, N=, N=3, N=4, N=8, N=6,... locaion, souheas ) e(.5,.7, () ); ile( Richungsfeld zu =- in [,4][,], () und PZV zum AWP mi ()= ); ais([ ]); label( ); plo([ ],[ ], r:,,, ko ); dfield( f7,:.5:,:.:); prin s3_8.ps -dpsc Richungsfeld zu = in [,4][,], () und PZV zum AWP mi ()= ().4 () PZV, N= N=3. N=4 N=8 N=

8 % =c+k^, ()=, c>, k<, <= % --> f8.m global c k c=; k=-; figure(9) dfield( f8,:.:,-:.:); ile( Richungsfeld zu =-^ in [,][-,] ); label( ); ylabel( ); plo([ ],[ ], r-,[ ],[- -], r- ); prin s3_9.ps -dpsc.5 Richungsfeld zu = in [,][,] = ; = ; = ; = :.5:; = ; = anh(); = anh(+aanh(.)); p = anh(+aanh(.5)); m = anh(+aanh(-.5)); m34 = anh(+aanh(-.75)); m = -; figure() dfield( f8,:.:,-:.:); ile( Richungsfeld zu =-^ in [,][-,] mi Loesungen zu AB ); label( ); ylabel( ); plo([ ],[ ], r-,[ ],[- -], r-,[,,,,,,],[-,-/,,/,,,-3/4], ko ); plo(,, b-,,m, b-,,, k-,,, m-,,p, g-,,m, g:,...,m34, b- ); prin s3_.ps -dpsc.5 Richungsfeld zu = in [,][,] mi Loesungen zu AB

9 = 4; N = 6; = :.5:; = anh(); [e6,e6] = euler( f8,,,,n); [e8,e8] = euler( f8,,,,8); [e4,e4] = euler( f8,,,,4); [e3,e3] = euler( f8,,,,3); [e,e] = euler( f8,,,,); figure() plo(,, b-,e,e, g-,e3,e3, k-,e4,e4, c-,e8,e8, m-,e6,e6, k- ); legend( (), PZV, N=, N=3, N=4, N=8, N=6,... locaion, norheas ) e(.,.7, () ); ile( Richungsfeld zu =-^ in [,4][-,3], () und PZV zum AWP mi ()= ); ais([ - 3]); label( ); plo([ ],[ ], r:,[ ],[- -], r:,,, ko ); dfield( f8,:.5:,-:.:3); prin s3_.ps -dpsc 3.5 Richungsfeld zu = in [,4][,3], () und PZV zum AWP mi ()= () PZV, N= N=3 N=4 N=8 N=6.5.5 () disp(. f) Bewegungsgleichung fuer den freien Fall ohne Reibung, 3 Varianen ) % % =-g, ()=re+h, ()=, <= % --> f6s.m als Sysem % funcion y = fs(,) % g = 9.86; % re = 6.375e6; % y = zeros(size()); % y() = (); % y() = -g; g = 9.86; re = 6.375e6; % H = e3, e4, e5 % eake Loesungen _allg -/*g*.^+c*+c -/*g*.^+re+h -g* % Fallhoehe, Fallzei und Geschwindigkeien H = e3; (,H); F3 = *sqr(H) F3 = v_anf = s() v_anf = v_end = s(f3) v_end = e+ H = e4; (,H); F4 = *sqr(H) F4 = v_anf = s() v_anf =

10 e+ H = e5; (,H); F5 = *sqr(H) F5 = e+ v_anf = s() v_anf = v_end = s(f5) v_end = e+3 % Grafik zum freien Fall 3 = :.5:F3*.; 3 = (3,e3); 4 = :.5:F4*.; 4 = (4,e4); 5 = :.5:F5*.; 5 = (5,e5); figure() plo([ F5],[rE re], k- ); ile( Fallhoehen (), <=, bei H=e3,e4,e5 ); label( ); ylabel( ); e(6,re-3e3, Erdoberflaeche ); e(3,6.379e6, H=e3 ); e(3,6.388e6, H=e4 ); e(3,6.477e6, H=e5 ); plo([,,],[re+e3,re+e4,re+e5], ko ); plo(3,3, r-,4,4, b-,5,5, g- ); prin s3_.ps -dpsc 6.48 Fallhoehen (), <=, bei H=e3,e4,e5 6 H=e H=e H=e3 Erdoberflaeche % PZV = ; = 5; N = 5; H = e3; = [re+h,] ; [e35,e35] = euler( f6s,,,,n); [e33,e33] = euler( f6s,,,,*n); disp( Naeherungsloesung des PZV, N=5,3, fuer <=<=5 ); e35(:,) 6376 e35(:,n+).e+6 * e33(:,) 6376 e33(:,*n+).e+6 *

11 figure(3) plo([ ],[re re], k- ); ile( Fallhoehe (), <=<=5,H=e3, und PZV, N=5,3 ); label( ); ylabel( ); e(6,re-.5e, Erdoberflaeche ); e(.4,re+h+.8e, H=e3 ); ais([ 5 re-e re+h+e]); plo(,re+h, ko ); plo(3,3, r-,e35,e35(,:), r:o,e33,e33(,:), r:. ); prin s3_3.ps -dpsc Fallhoehe (), <=<=5,H=e3, und PZV, N=5, H=e Erdoberflaeche % PZV mi groeberer Schriweie rechnen, dami Unerschiede erkennbar sind = ; = 6; N = 4; H = e3; = [re+h,] ; [e35,e35] = euler( f6s,,,,n); [e33,e33] = euler( f6s,,,,*n); = 48; N = ; H = e4; = [re+h,] ; [e45,e45] = euler( f6s,,,,n); [e43,e43] = euler( f6s,,,,*n); = 44; N = 36; H = e5; = [re+h,] ; [e55,e55] = euler( f6s,,,,n); [e53,e53] = euler( f6s,,,,*n); figure(4) plo([ 45],[rE re], k- ); ile( Fallhoehen (), <=, H=e3,e4,e5, und PZV, N,N, N=4,,36 ); label( ); ylabel( ); e(6,re-e3, Erdoberflaeche ); e(,re+e3+3e3, H=e3 ); e(,re+e4+3e3, H=e4 ); e(,re+e5+3e3, H=e5 ); ais([ 46 re-e4 re+e5+5e3]); plo([,,],[re+e3,re+e4,re+e5], ko ); plo(3,3, r-,e35,e35(,:), r:o,e33,e33(,:), r:.,... 4,4, b-,e45,e45(,:), b:o,e43,e43(,:), b:.,... 5,5, g-,e55,e55(,:), g:o,e53,e53(,:), g:. ); prin s3_4.ps -dpsc

12 H=e4 H=e3 Erdoberflaeche % % =-3g+g/rE, ()=re+h, ()=, <= % --> f6s.m als Sysem % funcion y = fs(,) % g = 9.86; % re = 6.375e6; % y = zeros(size()); % y() = (); % y() = g*(-3+/re*()); g = 9.86; re = 6.375e6; % H = e3, e4, e5 % eake Loesungen _allg 3*rE/+C*ep(sqr(*g/rE*))+C*ep(-sqr(*g/rE*)) 3*rE/+(-rE/4+H/)*(ep(sqr(*g/rE)*)+ep(-sqr(*g/rE)*)) (-re/4+h/)*sqr(*g/re)*(ep(sqr(*g/re)*)-ep(-sqr(*g/re)*)) % Fallhoehe, Fallzei und Geschwindigkeien H = e3; (,H); F3 = *log((3875+sqr(6375*H-H^))/(3875-H)) F3 = v_anf = s(,h) v_anf = v_end = s(f3,h) v_end = e+ H = e4; (,H); F4 = *log((3875+sqr(6375*H-H^))/(3875-H)) F4 = v_anf = s(,h) v_anf = v_end = s(f4,h) v_end = e+ H = e5; (,H); F5 = *log((3875+sqr(6375*H-H^))/(3875-H)) F5 = e+ v_anf = s(,h) v_anf = v_end = s(f5,h) v_end = e+3

13 3 = (3,e3); 4 = :.5:F4*.; 4 = (4,e4); 5 = :.5:F5*.; 5 = (5,e5); figure(5) plo([ F5],[rE re], k- ); ile( Fallhoehen (), <=, bei H=e3,e4,e5 ); label( ); ylabel( ); e(6,re-3e3, Erdoberflaeche ); e(3,6.379e6, H=e3 ); e(3,6.388e6, H=e4 ); e(3,6.477e6, H=e5 ); ais([ 46 re-e4 re+e5+5e3]); plo([,,],[re+e3,re+e4,re+e5], ko ); plo(3,3, r-,4,4, b-,5,5, g- ); prin s3_5.ps -dpsc 6.48 Fallhoehen (), <=, bei H=e3,e4,e5 6 H=e H=e4 H=e Erdoberflaeche % PZV = ; = 5; N = 5; H = e3; = [re+h,] ; [e35,e35] = euler( f6s,,,,n); [e33,e33] = euler( f6s,,,,*n); disp( Naeherungsloesung des PZV, N=5,3, fuer <=<=5 ); e35(:,) 6376 e35(:,n+).e+6 * e33(:,) 6376 e33(:,*n+).e+6 * figure(6) plo([ ],[re re], k- ); ile( Fallhoehe (), <=<=5,H=e3, und PZV, N=5,3 ); label( ); ylabel( ); e(6,re-.5e, Erdoberflaeche ); e(.4,re+h+.8e, H=e3 ); ais([ 5 re-e re+h+e]); plo(,re+h, ko ); plo(3,3, r-,e35,e35(,:), r:o,e33,e33(,:), r:. ); prin s3_6.ps -dpsc

14 H=e Erdoberflaeche % PZV mi groeberer Schriweie rechnen, dami Unerschiede erkennbar sind = ; = 6; N = 4; H = e3; = [re+h,] ; [e35,e35] = euler( f6s,,,,n); [e33,e33] = euler( f6s,,,,*n); = 48; N = ; H = e4; = [re+h,] ; [e45,e45] = euler( f6s,,,,n); [e43,e43] = euler( f6s,,,,*n); = 44; N = 36; H = e5; = [re+h,] ; [e55,e55] = euler( f6s,,,,n); [e53,e53] = euler( f6s,,,,*n); figure(7) plo([ 45],[rE re], k- ); ile( Fallhoehen (), <=, H=e3,e4,e5, und PZV, N,N, N=4,,36 ); label( ); ylabel( ); e(6,re-e3, Erdoberflaeche ); e(,re+e3+3e3, H=e3 ); e(,re+e4+3e3, H=e4 ); e(,re+e5+3e3, H=e5 ); ais([ 46 re-e4 re+e5+5e3]); plo([,,],[re+e3,re+e4,re+e5], ko ); plo(3,3, r-,e35,e35(,:), r:o,e33,e33(,:), r:.,... 4,4, b-,e45,e45(,:), b:o,e43,e43(,:), b:.,... 5,5, g-,e55,e55(,:), g:o,e53,e53(,:), g:. ); prin s3_7.ps -dpsc 6.48 Fallhoehen (), <=, H=e3,e4,e5, und PZV, N,N, N=4,,36 6 H=e H=e4 H=e3 Erdoberflaeche

15 % --> f63s.m als Sysem % funcion y = fs(,) % g = 9.86; % re = 6.375e6; % y = zeros(size()); % y() = (); % y() = -g*re^/(().^); % keine eplizie Loesung g = 9.86; re = 6.375e6; H = e3; % e4,e5 % KRKV und ode45 mi hoher Genauigkei als eake Loesung = ; = 5; H = e3; = [re+h,] ; N = ; [k4,k4] = krkv( f63s,,,,n); op = odese( AbsTol,e-3); [45,45] = ode45(@f63s,[ ],,op); N = ma(size(45)) % N+ N = 53 ha = 45()-45() % Schriweien am Anfang und Ende ha = he = 45(N-)-45(N-) he = figure(8) plo([ ],[re re], k- ); ile( Fallhoehen (), <=<=5,H=e3, gemaess KRKV(N=) und ode45(e-3) ); label( ); ylabel( ); e(6,re-.5e, Erdoberflaeche ); e(.4,re+h+.8e, H=e3 ); ais([ 5 re-e re+h+e]); plo(,re+h, ko ); plo(k4,k4(,:), k-,45,45(:,), r- ); prin s3_8.ps -dpsc abs(k4(,n+)-45(n,)) % Absand am Inervallende e Fallhoehen (), <=<=5,H=e3, gemaess KRKV(N=) und ode45(e 3) H=e Erdoberflaeche N = ; [k4,k4] = krkv( f63s,,,,n); op = odese( AbsTol,e-6); [45,45] = ode45(@f63s,[ ],,op); N = ma(size(45)) % N+ N = 69 ha = 45()-45() % Schriweien am Anfang und Ende ha = e-6 he = 45(N-)-45(N-) he =

16 figure(9) plo([ ],[re re], k- ); ile( Fallhoehen (), <=<=5,H=e3, gemaess KRKV(N=) und ode45(e-6) ); label( ); ylabel( ); e(6,re-.5e, Erdoberflaeche ); e(.4,re+h+.8e, H=e3 ); ais([ 5 re-e re+h+e]); plo(,re+h, ko ); plo(k4,k4(,:), k-,45,45(:,), r- ); prin s3_9.ps -dpsc abs(k4(,n+)-45(n,)) % Absand am Inervallende Fallhoehen (), <=<=5,H=e3, gemaess KRKV(N=) und ode45(e 6) H=e Erdoberflaeche % Grafik zum freien Fall = 5; H = e3; = [re+h,] ; N = ; [k43,k43] = krkv( f63s,,,,n); = 46; H = e4; = [re+h,] ; N = 3; [k44,k44] = krkv( f63s,,,,n); = 45; H = e5; = [re+h,] ; N = ; [k45,k45] = krkv( f63s,,,,n); figure() plo([ F5],[rE re], k- ); ile( Fallhoehen (), <=, bei H=e3,e4,e5 ); label( ); ylabel( ); e(6,re-3e3, Erdoberflaeche ); e(3,6.379e6, H=e3 ); e(3,6.388e6, H=e4 ); e(3,6.477e6, H=e5 ); ais([ 46 re-e4 re+e5+5e3]); plo([,,],[re+e3,re+e4,re+e5], ko ); plo(k43,k43, r-,k44,k44, b-,k45,k45, g- ); prin s3_.ps -dpsc

17 H=e4 H=e Erdoberflaeche % PZV = ; = 5; N = 5; H = e3; = [re+h,] ; [e35,e35] = euler( f63s,,,,n); [e33,e33] = euler( f63s,,,,*n); disp( Naeherungsloesung des PZV, N=5,3, fuer <=<=5 ); e35(:,) 6376 e35(:,n+).e+6 * e33(:,) 6376 e33(:,*n+).e+6 * figure() plo([ ],[re re], k- ); ile( Fallhoehe (), <=<=5,H=e3, und PZV, N=5,3 ); label( ); ylabel( ); e(6,re-.5e, Erdoberflaeche ); e(.4,re+h+.8e, H=e3 ); ais([ 5 re-e re+h+e]); plo(,re+h, ko ); plo(k43,k43, r-,e35,e35(,:), r:o,e33,e33(,:), r:. ); prin s3_.ps -dpsc Fallhoehe (), <=<=5,H=e3, und PZV, N=5, H=e Erdoberflaeche

18 = 6; N = 4; H = e3; = [re+h,] ; [e35,e35] = euler( f63s,,,,n); [e33,e33] = euler( f63s,,,,*n); = 48; N = ; H = e4; = [re+h,] ; [e45,e45] = euler( f63s,,,,n); [e43,e43] = euler( f63s,,,,*n); = 44; N = 36; H = e5; = [re+h,] ; [e55,e55] = euler( f63s,,,,n); [e53,e53] = euler( f63s,,,,*n); figure() plo([ 46],[rE re], k- ); ile( Fallhoehen (), <=, H=e3,e4,e5, und PZV, N,N, N=4,,36 ); label( ); ylabel( ); e(6,re-e3, Erdoberflaeche ); e(,re+e3+3e3, H=e3 ); e(,re+e4+3e3, H=e4 ); e(,re+e5+3e3, H=e5 ); ais([ 46 re-e4 re+e5+5e3]); plo([,,],[re+e3,re+e4,re+e5], ko ); plo(k43,k43, r-,e35,e35(,:), r:o,e33,e33(,:), r:.,... k44,k44, b-,e45,e45(,:), b:o,e43,e43(,:), b:.,... k45,k45, g-,e55,e55(,:), g:o,e53,e53(,:), g:. ); prin s3_.ps -dpsc 6.48 Fallhoehen (), <=, H=e3,e4,e5, und PZV, N,N, N=4,,36 6 H=e H=e4 H=e3 Erdoberflaeche

19 disp(. a) ) % =-^, ()=, <=<=, Ricai-DGL % --> f4.m figure(3) dfield( f4,:.:,-:.:); ile( Richungsfeld zu =-^ in [,][-,] ); label( ); ylabel( ); prin s3_3.ps -dpsc.5 Richungsfeld zu = in [,][,] = ; = ; = ; N = ; [em,em] = krkv( f4,,,-/,n); [e,e] = krkv( f4,,,,n); [e,e] = krkv( f4,,,/,n); [e,e] = krkv( f4,,,,n); [e,e] = krkv( f4,,,,n); figure(4) plo(em,em, c-,e,e, b-,e,e, k-,e,e, m-,e,e, r- ); ile( Richungsfeld zu =-^ in [,][-,], () und KRKV (N=) zum AWP mi AB ); ais([ - ]); label( ); legend( KRKV, ()=-/, ()=, ()=/,... ()=, ()=,... locaion, norheas ) plo([,,,,],[-/,,/,,], ko ); dfield( f4,:.:,-:.:); prin s3_4.ps -dpsc.5 Richungsfeld zu = in [,][,], () und KRKV (N=) zum AWP mi AB KRKV, ()= / ()= ()=/ ()= ()=

20 e(),e() [e,e] = krkv( f4,,,,9); e(9),e(9) [e,e] = krkv( f4,,,,95); e(95),e(95) [e,e] = krkv( f4,,,,); e(),e() e=e() e = % PZV, N=5,, [e5,e5] = euler( f4,,,,5) e5 = Columns hrough Columns 5 hrough 6.8. e5 = Columns hrough Columns 5 hrough [e,e] = euler( f4,,,,); [e,e] = euler( f4,,,,); err5 =abs(e-e5(6)) err5 = err=abs(e-e()) err = err=abs(e-e()) err = disp( Genauigkei des PZV is O(h), weil halber Fehler bei Halbierung von h )

21 % =-lambda, ()=, <=<=5 % --> f3.m global lambda lambda=; e ep(-) figure(5) dfield( f3,:.:5,-:.:); ile( Richungsfeld zu =- in [,5][-,] ); label( ); ylabel( ); prin s3_5.ps -dpsc.5 Richungsfeld zu = in [,5][,] = ; = 5; = ; N = ; [em,em] = krkv( f3,,,-/,n); [e,e] = krkv( f3,,,,n); [e,e] = krkv( f3,,,/,n); [e,e] = krkv( f3,,,,n); [e,e] = krkv( f3,,,,n); figure(6) plo(em,em, c-,e,e, b-,e,e, k-,e,e, m-,e,e, r- ); ile( Richungsfeld zu =- in [,5][-,], () und KRKV (N=) zum AWP mi AB ); ais([ - ]); label( ); legend( KRKV, ()=-/, ()=, ()=/,... ()=, ()=,... locaion, norheas ) plo([,,,,],[-/,,/,,], ko ); dfield( f3,:.:,-:.:); prin s3_6.ps -dpsc.5 Richungsfeld zu = in [,5][,], () und KRKV (N=) zum AWP mi AB KRKV, ()= / ()= ()=/ ()= ()=

22 e(),e() e5=ep(-5) e5 = % PZV, N=5,, [e5,e5] = euler( f3,,,,5) e5 = e5 = [e,e] = euler( f3,,,,); [e,e] = euler( f3,,,,); err5 =abs(e5-e5(6)) err5 = err=abs(e5-e()) err = err=abs(e5-e()) err = disp( Genauigkei des PZV is O(h), weil halber Fehler bei Halbierung von h ) % disp(. c) ) % =-, ()=, <=<= % --> f.m e *ep(-)+- figure(7) dfield( f,:.:,-:.:); ile( Richungsfeld zu =- in [,][-,] ); label( ); ylabel( ); prin s3_7.ps -dpsc.5 Richungsfeld zu = in [,][,] = ; = ; = ; N = ; [em,em] = krkv( f,,,-/,n); [e,e] = krkv( f,,,,n); [e,e] = krkv( f,,,/,n); [e,e] = krkv( f,,,,n); [e,e] = krkv( f,,,,n); figure(8) plo(em,em, c-,e,e, b-,e,e, k-,e,e, m-,e,e, r- ); ile( Richungsfeld zu =- in [,][-,], () und KRKV (N=) zum AWP mi AB ); ais([ - ]);

23 ()=, ()=,... locaion, norheas ) plo([,,,,],[-/,,/,,], ko ); dfield( f,:.:,-:.:); prin s3_8.ps -dpsc.5 Richungsfeld zu = in [,][,], () und KRKV (N=) zum AWP mi AB KRKV, ()= / ()= ()=/ ()= ()= % e()=ep(-) [e,e] = krkv( f,,,,); e(),e() e=*ep(-) e = e=e() e = % PZV, N=5,, [e5,e5] = euler( f,,,,5) e5 = Columns hrough Columns 5 hrough 6.8. e5 = Columns hrough Columns 5 hrough [e,e] = euler( f,,,,); [e,e] = euler( f,,,,); err5 =abs(e-e5(6)) err5 = err=abs(e-e()) err = err=abs(e-e()) err = disp( Genauigkei des PZV is O(h), weil halber Fehler bei Halbierung von h ) echo off diary off

24 % dfield.m % Richungsfeld der gdgl =f(,), (,) aus Gier % mi Normierung und Skalierung funcion dfield(f,v,yv) [,y] = meshgrid(v,yv); [n,m] = size(); mag = min((v(m)-v())/m,(yv(n)-yv())/n); % fuer Skalierung for i=:n for j=:m v = [,feval(f,(i,j),y(i,j))] ; v =.4*mag*v/norm(v); % Normierung und Skalierung line([(i,j)-v(),(i,j)+v()],[y(i,j)-v(),y(i,j)+v()]); if v(m)*v()<= % fuer Achseneinragung plo([,],[yv(),yv(n)], k- ); if yv(n)*yv()<= plo([v(),v(m)],[,], k- ); % % euler.m % Eplizies Euler-Verfahren, PZV funcion [,] = euler(f,,,,n) h = (-)/N; () = ; () = ; for i = :N (i+) = +h*i; (i+) = (i)+h*feval(f,(i),(i)); % collaz.m % Eplizies Collaz-Verfahren, MPZV fuer gdgl funcion [,] = collaz(f,,,,n) h = (-)/N; () = ; () = ; for i = :N (i+) = +h*i; k = feval(f,(i),(i)); k = feval(f,(i)+.5*h,(i)+.5*h*k); (i+) = (i)+h*k; % heun.m % Eplizies Heun-Verfahren, VPZV fuer gdgl funcion [,] = heun(f,,,,n) h = (-)/N; () = ; () = ; for i = :N (i+) = +i*h; k = feval(f,(i),(i)); k = feval(f,(i)+h,(i)+h*k); (i+) = (i)+.5*h*(k+k); % krkv.m % Eplizies Klassisches Runge-Kua-Verfahren, KRKV fuer gdgl funcion [,] = krkv(f,,,,n) h = (-)/N; () = ; () = ; for i = :N (i+) = +i*h; k = feval(f,(i),(i)); k = feval(f,(i)+.5*h,(i)+.5*h*k); k3 = feval(f,(i)+.5*h,(i)+.5*h*k); k4 = feval(f,(i)+h,(i)+h*k3); (i+) = (i)+h/6*(k+*k+*k3+k4);

25 % Eplizies Euler-Verfahren, PZV fuer SysgDGl funcion [,] = euler(f,,,,n) h = (-)/N; n = size(,); = zeros(n,n+); () = ; (:,) = ; for i = :N (i+) = +i*h; (:,i+) = (:,i)+h*feval(f,(i),(:,i)); % collaz.m % Eplizies Collaz-Verfahren, MPZV fuer SysgDGl funcion [,] = collaz(f,,,,n) h = (-)/N; n = size(,); = zeros(n,n+); () = ; (:,) = ; for i = :N (i+) = +i*h; k = feval(f,(i),(:,i)); k = feval(f,(i)+.5*h,(:,i)+.5*h*k); (:,i+) = (:,i)+h*k; % heun.m % Eplizies Heun-Verfahren, VPZV fuer SysgDGl funcion [,] = heun(f,,,,n) h = (-)/N; n = size(,); = zeros(n,n+); () = ; (:,) = ; for i = :N (i+) = +i*h; k = feval(f,(i),(:,i)); k = feval(f,(i)+h,(:,i)+h*k); (:,i+) = (:,i)+.5*h*(k+k); % krkv.m % Eplizies Klassisches Runge-Kua-Verfahren, KRKV fuer SysgDGl funcion [,] = krkv(f,,,,n) h = (-)/N; n = size(,); = zeros(n,n+); () = ; (:,) = ; for i = :N (i+) = +i*h; k = feval(f,(i),(:,i)); k = feval(f,(i)+.5*h,(:,i)+.5*h*k); k3 = feval(f,(i)+.5*h,(:,i)+.5*h*k); k4 = feval(f,(i)+h,(:,i)+h*k3); (:,i+) = (:,i)+h/6*(k+*k+*k3+k4);