adel-00

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12 years, 12 days

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These are questions asked by adel-00

Hello experts..

The following is the IVP:

restart:Digits:=14:t0:=0.0:tN:=5000.0: N1:=5000;th:=evalf((tN-t0)/N1):

dsys1 :=diff(y(t),t)=y(t)*((1-y(t)/3-epsilon)-0.8*y(t)/(y(t)^2+0.5^2));

var:={y(t)}:ini1:=y(0)=0.5:

dsol1 :=dsolve({dsys1,ini1},var,numeric, output=listprocedure, abserr=1e-9, relerr=1e-8,range=0..1);

dsolu:=subs(dsol1,y(t)):

t1:=array(0..N1,[]):u1:=array(0..N1,[]):pt1:=array(0..N1,[]):

for i from 0 to N1 do t1[i]:=evalf(th*i):u1[i]:=evalf(dsolu(t1[i]));pt1[i]:=[t1[i],u1[i]]:

od:

mytab1:=eval([seq(pt1[i],i=0..N1)]):

the above code is to plot y(t) against the time t for fixed epsilon

Now the question is how to plot epsilon against the time???

I do appriciated any comments

 

 

Hi every one,

Q1:

I tried to get the max $ min of a following function:

 

l:=1:alpha:=1:b:=100:k:=20:

eq1 := (alpha+(l+alpha)*u+alpha*k*u^2)*a =
u*(alpha+(l+alpha)*u+alpha*k*u^2)*(1+l*alpha*b/((alpha+(l+alpha)*u+alpha*k*u^2))):

I did this code but it seems it didnt work for this equation

maximize(eq, u=1..12, location);

minimize(eq, u=1..12, location);

Also, I think about solving the cubic i feel i'm so close to the solve but couldn't

factor((rhs-lhs)(eq1));

eq:=collect(%,u);

Q:=(a,u)->eq;sol:=evalf(solve(Q(a,u),u)): S:=array([],1..3): S[1]:=sol[1]:S[2]:=sol[2]:S[3]:=sol[3]:

Q2:

the same thing wanted to get the maximum and the minimum of the function v

here the code

restart;
eq1:=(alpha+(l+alpha)*u+alpha*k*u^2)*a=
u*(alpha+(l+alpha)*u+alpha*k*u^2)*(1+l*alpha*b/((alpha+(l+alpha)*u+alpha*k*u^2)));
eq2:=v=alpha*b*(1+u+k*u^2)/(alpha+(l+alpha)*u+alpha*k*u^2);
factor((rhs-lhs)(eq1));
eq1:=collect(%,u);
params:={l=10,alpha=0.5,b=100,k=20};
U:=[solve(eval(eq1,params),u)]; #3 solutions for u
#plots:-complexplot(U,a=0..20,style=point); #plot in the complex u-plane
vua:=eval(solve(eq2,v),params): #v expressed in terms of u and a
V:=eval~(vua,u=~U): #the 3 solutions for v in terms of a

## PLOT the function V
plot(V,a=0..75,v=0..100,color=black,labels=[a,v],axes=boxed,numpoints=90,linestyle=1,font=[1,1,18],thickness=2,tickmarks=[4,4],view=[0..65,25..100]);

I do appricaited any advises

Hi;

Trying to find the maximum values of the implicit function.. I tree the following commands.

 

l:=1:alpha:=1:b:=100:k:=20:

(alpha+(l+alpha)*u+alpha*k*u^2)*a=u*(alpha+(l+alpha)*u+alpha*k*u^2)*(1+l*alpha*b/((alpha+(l+alpha)*u+alpha*k*u^2)));

with(plots):

implicitplot((alpha+(l+alpha)*u+alpha*k*u^2)*a=u*(alpha+(l+alpha)*u+alpha*k*u^2)*(1+l*alpha*b/((alpha+(l+alpha)*u+alpha*k*u^2))),a=0..100,u=0..20,numpoints=9000000,color=black,axes=boxed,font=[1,1,18],thickness=2,tickmarks=[3,3],view=[0..12,0..12],labels=[a,u]);

maximize((alpha+(l+alpha)*u+alpha*k*u^2)*a=u*(alpha+(l+alpha)*u+alpha*k*u^2)*(1+l*alpha*b/((alpha+(l+alpha)*u+alpha*k*u^2))), N=0.5..1, location);

Any help I'll appriciated 

 

I tried to get the maximum and minimum values of the following function. From the plot I get them but its not accurate. Please advise me to get them accurate.

 

F:=0.85:B:=0.5:

K:=N->(N*(1+F*N/(N^2+B^2-F*N)));

 

implicitplot(((N^2+B^2-F*N)*K=N*(N^2+B^2-F*N+F*N),K=0..10,N=0..10,view=[0..5,0..4],numpoints=90000,axes=boxed,thickness=2,color=black,font=[1,1,20],tickmarks=[3, 3],linestyle=1));

 

Hi,

1-Triying to plot a function divided by its maximum value,sometimes it works with some parameters that means, the max.value of the plot is 1.

But when i change the data the max. value in the plot in graeter than 1 which is wrong!! should be 1.

dont know why??

2- Changing different data in the parameters, the programme takes long long time then i stop it?

 

please help me with these two problems.


restart:
>
------------------------- Defining the nature of the variables used ----------------------
assume(T,real):Digits:=25:n:=1:tau:=Pi:
theta:=0:phi:=0:
lambda:=n;Omega:=1:Gamma:=0.01:
--------------------- Input---------------------------------
1

J1

term1:=(exp((Gamma+I*d)*tau)-1)/(2*(Gamma+I*d)):
Ak1:=d->(exp((Gamma+I*(d+0.5*Omega-2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d+0.5*Omega-2*lambda*k/Gamma))+(exp((Gamma+I*(d-0.5*Omega+2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d-0.5*Omega+2*lambda*k/Gamma)):
Ak2:=d->(exp((Gamma+I*(d-0.5*Omega-2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d-0.5*Omega-2*lambda*k/Gamma))+(exp((Gamma+I*(d+0.5*Omega+2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d+0.5*Omega+2*lambda*k/Gamma)):
term2:=(evalf(-0.25*sum(BesselJ(k,Omega*Gamma/(4*n))*Ak1(d)+BesselJ(k,-(Omega*Gamma)/(4*n))*Ak2(d),k=0..50))):
J1:=(term1+term2):
J1mod:=(Re(J1))^2+(Im(J1))^2:
###### J2#########################
Ak1:=d->(exp((Gamma+I*(d+0.5*Omega-2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d+0.5*Omega-2*lambda*k/Gamma))-(exp((Gamma+I*(d-0.5*Omega+2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d-0.5*Omega+2*lambda*k/Gamma)):
Ak2:=d->(exp((Gamma+I*(d-0.5*Omega-2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d-0.5*Omega-2*lambda*k/Gamma))-(exp((Gamma+I*(d+0.5*Omega+2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d+0.5*Omega+2*lambda*k/Gamma)):

J2:=(evalf(-0.25*sum(BesselJ(k,Omega*Gamma/(4*n))*Ak1(d)+BesselJ(k,-Omega*Gamma/(4*n))*Ak2(d),k=0..100))):
######################

J2mod:=(Re(J2))^2+(Im(J2))^2:
J3 same as J1differ in sign
term1:=(exp((Gamma+I*d)*tau)-1)/(2*(Gamma+I*d)):
Ak1:=d->(exp((Gamma+I*(d+0.5*Omega-2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d+0.5*Omega-2*lambda*k/Gamma))+(exp((Gamma+I*(d-0.5*Omega+2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d-0.5*Omega+2*lambda*k/Gamma)):
Ak2:=d->(exp((Gamma+I*(d-0.5*Omega-2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d-0.5*Omega-2*lambda*k/Gamma))+(exp((Gamma+I*(d+0.5*Omega+2*lambda*k/Gamma))*tau)-1)/(Gamma+I*(d+0.5*Omega+2*lambda*k/Gamma)):
term2:=(evalf(0.25*sum(BesselJ(k,Omega*Gamma/(4*n))*Ak1(d)+BesselJ(k,-Omega*Gamma/(4*n))*Ak2(d),k=0..100))):
J3:=term1+term2:
J3mod:=(Re(J3))^2+(Im(J3))^2:
J4 same as J2 but -0.25-->2


J4:=-2*J2:
######################

J4mod:=(Re(J4))^2+(Im(J4))^2:

calculate the spectrum

 

Spec:=d->(exp(-2*Gamma*tau)*(J1mod*cos(theta/2)^2+J2mod+J3mod*sin(theta/2)^2-0.5*Re(J3*J4*sin(theta)*exp(I*phi))+0.5*Re(J1*J4*sin(theta)*exp(-I*phi)))):

with(plots):

tit:=sprintf("l=%g,W=%g,G=%g",lambda,Omega,Gamma):
Smax1:=max(seq(evalf(Spec(d)),d=-100..100)):
plot(evalf(Spec(d)/Smax1),d=-15..15,axes=boxed,title=tit,color=black,font=[2,3,18],thickness=2,tickmarks=[3,3],titlefont=[SYMBOL,14],font=[1,1,18],linestyle=1);

 

 

 

 

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