<Text-field style="Heading 1" layout="Heading 1">Tanksystem</Text-field>

with(LinearAlgebra): with(plots):

Der er givet et system best\303\245ende af to omr\303\270rte tanke. x(t) og y(t) er koncentrationerne af et bestemt stof A i de to tanke. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JUYuLUYjNiMtRjs2JFEiMkYnRj5GQA== er volumenstr\303\270mningshastigheder. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JUYuLUYjNiMtRjs2JFEiMkYnRj5GQA== er rumfangene af de to tanke. Der er intet stof A i den til tank 1 indkommende str\303\270m.

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Massebevarelse giver for tank 1:

Diff(V[1]*x(t),t)=F[1]*y(t)-F[1]*x(t);

Massebevarelse giver for tank 2:

Diff(V[2]*y(t),t)=-F[1]*y(t)+F[1]*x(t)-F[2]*y(t);

Med 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 f\303\245s s\303\245 systemet

sys:={diff(x(t),t)=r1*y(t)-r1*x(t),diff(y(t),t)=r2*x(t)-(r2+r3)*y(t)};

Med A defineret som

A:=Matrix([[-r1,r1],[r2,-r2-r3]]);

P\303\245 vektorform kan systemet skrives som f\303\270lger (f\303\270rst kun til udstillingsbrug):

<diff(x(t),t),diff(y(t),t)> = A &. <x(t),y(t)>;

eller ganget ud:

<diff(x(t),t),diff(y(t),t)> = A.<x(t),y(t)>;

Taleksempler. r-v\303\246rdierne er valgt simple for overskuelighedens skyld:

r1:=4: r2:=1: r3:=3:

sys;

Fuldst\303\246ndig l\303\270sning ved dsolve:

dsolve(sys);

L\303\270sningen opskrevet som en vektor:

FULD1:=subs(%,<x(t),y(t)>);

"H\303\245ndkraft":

EV,P:=Eigenvectors(A);

Den fuldst\303\246ndige l\303\270sning

L:=c1*exp(EV[1]*t)&*Column(P,1)+c2*exp(EV[2]*t)&*Column(P,2):

L=eval(L,`&*`=`*`);

Det er det samme som

P.<c1*exp(EV[1]*t),c2*exp(EV[2]*t)>;

De v\303\246rdier af konstanterne c1 og c2, der svarer til l\303\270sningen, der opfylder begyndelsesbetingelsen x(0) = 8, y(0) = 2 findes ved inds\303\246ttelse af t = 0 i den fuldst\303\246ndige l\303\270sning. Dvs. at c1 og c2 skal opfylde ligningen

P.<c1,c2> = <8,2>;

Vi l\303\270ser dette ligningssystem:

LinearSolve(P,<8,2>);

Dermed er l\303\270sningen

subs(c1=%[1],c2=%[2],L): %=eval(%,`&*`=`*`);

Den l\303\270sning, der opfylder begyndelsesbetingelsen x(0) = 8, y(0) = 2 kan findes direkte i Maple s\303\245ledes:

dsolve(sys union {x(0)=8,y(0)=2});

L\303\270sningen som vektor:

XY:=subs(%,<x(t),y(t)>);

plot(XY,t=0..3,color=[red,blue],legend=["Koncentrationen i tank 1, x","Koncentrationen i tank 2, y"]);

FASEPLANSBILLEDE animeret:

animate(plot,[[XY[1],XY[2],t=0..T],labels=[x,y],thickness=3],'T'=0..3,frames=50);

Vi tegner mange l\303\270sninger svarende til forskellige startv\303\246rdier (ogs\303\245 urealistiske - negative koncentrationer!):

with(DEtools):

DEplot(sys,[x(t),y(t)],t=0..5,[seq([x(0)=0,y(0)=k],k=-5..5),seq([x(0)=k,y(0)=0],k=-5..5)],stepsize=.05,arrows=none,linecolor=red);

En animeret version:

animate(DEplot,[sys,[x(t),y(t)],t=0..T,[seq([x(0)=0,y(0)=k],k=1..5),seq([x(0)=k,y(0)=0],k=1..5)],stepsize=.05,arrows=none,linecolor=red],T=[.01,.05,.1,.2,.3,.4,.5,.7,1,2,5]);