Hello altogether,

I want to plot the numerical result of an ODE, which seems to be pretty simple at first sight, but the difficulty is that the boundaries are depending on the solution.

The following pseudo-code describes what I want to have, but it doesn't work. This code fills the RAM pretty fast and you will have to kill the process.

Free-boundary_prob.mw

Is it possible to calculate a solution to this problem numerically (or even analytically) and if yes, how?

Since I am new here, I am sorry for any bad-to-read maple code or any noob errors I have made. I would be very thankful, for any response and help.

Greetings

butterflyfart

sys := [(diff(c(t), t))*(diff(a(t), t))*(diff(b(t), t))+(diff(c(t), t))*t^2, (diff(c(t), t))*(diff(a(t), t))*t+(diff(c(t), t))*t*(diff(b(t), t)), (diff(c(t), t))*(diff(a(t), t))*(diff(b(t), t))+(diff(c(t), t))*(diff(a(t), t))*t+(diff(c(t), t))*t^2]
DEplot(sys, [a(t), b(t), c(t)], t = 0 .. 2, a = -15 .. 15, b = -15 .. 15, c = -15 .. 15, color = magnitude, title = `Stable Limit Cycles`, arrows = curve, dirfield = 800, axes = none);
odeplot(sys, [t, a(t), b(t), c(t)], -4 .. 4, color = orange);

with(RegularChains);
with(ConstructibleSetTools);
source1 := PolynomialRing([a, b, c, d]);
target1 := PolynomialRing([e1, e2, e3, e4]);
source1list := [eq2a, eq3a, eq4a, 0];
target1list := [eq2b, eq3b, eq4b, eq5b];
cs := PolynomialMapPreimage(target1list, source1list, source1, target1)

return
cs := constructible_set

then i type
Elements(constructible_set)

it return 

Elements(constructible_set)

 since source has less number of variables, in order to use this function

i add a dummy variable d to this and use 0 for the fourth equation

dsol := dsolve([eq2=diff(s(t),t), eq3=diff(s(t),t), eq4=diff(s(t),t)], [a(t),b(t),c(t)]);
eval(subs(t(t)=t,subs(_C1(t) = _C1, subs(_C2(t) = _C2, subs(_C3(t) = _C3, sol1)))));

after subs t(t) = t , it still  have t(t)

Dear Collagues

I wrote a code to solve a system of ode numerically. When the equations have been solved, I want to find the value of MTE which is in integral form. I check and see that the value of MTE is calculated wrong. I dont know why?

I calculated the values of MTE twice (another variable is masst below res1). Two different answers!!

I would be most grateful if you could help me.

Thank you

Here is my code

restart:
EPSILONE:=1:
#Digits:=10:

a[mu1]:=39.11:
b[mu1]:=533.9:
a[k1]:=7.47:
b[k1]:=0:

rhop:=5180:
rhobf:=998.2:
#gama1:=0.2;
mu1[bf]:=9.93/10000:
k1[bf]:=0.597:

rhost(eta):=1-phi(eta)+phi(eta)*rhop/rhobf;
#mu:=unapply(mu1[bf]/(1-phi(eta))^2.5,eta);
mu:=unapply(mu1[bf]*(1+a[mu1]*phi(eta)+b[mu1]*phi(eta)^2),eta);
#mu1[bf]:=9.93/10000:
k:=unapply(k1[bf]*(1+a[k1]*phi(eta)+b[k1]*phi(eta)^2),eta);
mu(eta)/mu1[bf];


#eq1:=diff((mu(eta)/mu1[bf])*diff(u(eta),eta),eta)+rhost(eta)-Ha^2*u(eta);
eq1:=(a[mu1]*(diff(phi(eta), eta))+2*b[mu1]*phi(eta)*(diff(phi(eta), eta)))*(diff(u(eta), eta))+(1+a[mu1]*phi(eta)+b[mu1]*phi(eta)^2)*(diff(u(eta), eta, eta))+rhost(eta)-Ha^2*u(eta);
eq2:=diff((k(eta)/k1[bf])*diff(T(eta),eta),eta);
eq3:=diff(phi(eta),eta)+phi(eta)/(N_bt*(1+gama1*T(eta))^2)*diff(T(eta),eta):
eq2:=subs(phi(0)=phi0,eq2);
eq3:=subs(phi(0)=phi0,eq3);



eval(dsolve({eq3,phi(1)=phiv},phi(eta)),(T)(1)=1);
Phi:=normal(combine(%));
Teq:=isolate(eval(eq2,Phi),diff(T(eta),eta));
ueq1:=eval(eq1,Phi)=0:
ueq2:=subs(Teq,ueq1):


Agama1:=<<0.1>,<0.2>,<0.3>>:
ANBT:=<<0.2>,<1>,<10>>:
Aphiv:=<<0.02>,<0.06>,<0.1>>:
lambda:=0;
Ha:=0;
#NBT:=0.5:
N_bt:=cc*NBT+(1-cc)*10;
#phiv:=0.02:
 
kratio:=k1[p]/k1[bf];




eq1;
eq2;
eq3;

res1 := dsolve(subs(NBT=1,gama1=0.3,phiv=0.06,{eq1,eq2,eq3,u(0)=lambda*D(u)(0),D(u)(1)=0,T(0)=0,phi(1)=phiv,T(1)=1}), numeric,method=bvp[midrich],output=listprocedure,continuation=cc):
G0,G1,G2:=op(subs(subs(res1),[phi(eta),u(eta),diff(T(eta),eta)])):

masst:=evalf(int((1-G0(eta)+G0(eta)*rhop/rhobf)*G1(eta), eta = 0..1));
heatt:=(1+a[k1]*G0(0)+b[k1]*G0(0)^2)*G2(0);



for iNBT from 1 to 3 do

for iphi from 1 to 3 do

for igamma from 1 to 3 do

res := dsolve(subs(NBT=ANBT(iNBT),gama1=Agama1(igamma),phiv=Aphiv(iphi),{eq1,eq2,eq3,u(0)=lambda*D(u)(0),D(u)(1)=0,T(0)=0,phi(1)=phiv,T(1)=1}), numeric,method=bvp[midrich],output=listprocedure,continuation=cc):
F0,F1,F2:=op(subs(subs(res),[phi(eta),u(eta),diff(T(eta),eta)])):
MTE[iNBT,iphi,igamma]:=evalf(Int((1-F0(eta)+F0(eta)*rhop/rhobf)*F1(eta), eta = 0..1)):
HTC[iNBT,iphi,igamma]:=(1+a[k1]*F0(0)+b[k1]*F0(0)^2)*F2(0):

end do;

end do;

end do;

for iNBT from 1 to 3 do

for iphi from 1 to 3 do

for igamma from 1 to 3 do



print (convert(HTC[iNBT,iphi,igamma],float,9));

end do;

end do;

end do;
for iNBT from 1 to 3 do

for iphi from 1 to 3 do

for igamma from 1 to 3 do


print (convert(MTE[iNBT,iphi,igamma],float,9));

end do;

end do;

end do;

Amir

1.

with(RegularChains):
with(ConstructibleSetTools):
source1 := PolynomialRing([a,b,c]);
target1 := PolynomialRing([e1,e2,e3,e4]);
source1list := [eq2a, eq3a,eq4a];
target1list := [eq2b, eq3b, eq4b,eq5b];
cs := PolynomialMapPreimage(target1list, source1list, source1, target1);

Error, (in RegularChains:-ConstructibleSetTools:-PolynomialMapPreimage) number of map functions is different from number of variables in target space

in maple 15 do not have error, however now it has error for the code below

with(RegularChains): with(RegularChains): with(ConstructibleSetTools): 

source1 := PolynomialRing([e1, e2, e3]);
target1 := PolynomialRing([a, b, c]);
source1list := [-1*e1+2*e2+*e3, -1*e1+2*e2+*e3, -1*e1+2*e2+*e3];
target1list := [.....]; cs :=
PolynomialMapPreimage(target1list, source1list, source1, target1);

Error, invalid product/quotient

Hi guys,

i want to find d'alembertian of a function of Ricci scalar (f(R)) , but it seems something is wrong .

f(R).mw

thanks in advance

Dear all,

I need you help to finish some steps of this idea to approximate the roots of a given equation (polynom). Thanks in advance for your help. 

I have a sturm sequence, I would like to use Bisection method to approximation the roots using Sturm decomposition of my polynom. For example, my polynom is  P=x^6-4*x^3+x-2

s := sturmseq(x^6-4*x^3+x-2,x);

sturm(s,x,-2,2); # The number of roots in the interval (-2,2)

Here, i would like to find the roots in (-M,M) :

Bounding all roots in [-M,M] where M = max{1, sum^(n-1) |ai|/an}.

f0 = f, f1 = f', then use -remainder,

I know that  sturm(s,x,-M,M); gives the number of roots in (-M,M)  but is it possible to use the variation of sign like :

      gives a Sturm sequence for f.

      variation of sign, varsign(a0,a1,...,ar).

      Thm: (Sturm) varsign(f0(alpha),...,fr(alpha)) - varsign(f0(beta),..., fr(beta))

      is the number of distinct roots of f in [alpha,beta].

then i would like Isolating roots of rational polynomials

Method: reduce, remove rational roots, divide and conquer in [-M,M],

      then use bisection  in disjoint closed intervals ctg one root each

 Bisection method :

Bisection

      Setup: f(a) < 0, f(b) > 0 (or conversely).

      Repeated subdivision of [a,b] guaranteed to get close to a root.

      Error analysis: for error eps, solve (b-a)/ 2^(n+1)  < tol for n. where tol is the tolerance

Thanks

in harmonic ocillator version, why it has error after it dsolve?

with(Physics[Vectors]):
with(SumTools):
Setup(mathematicalnotation = true):

Qs is Qss*cos(m*t+phi);

H := 1/(8*Pi*c^2)*Summation(Commutator(diff(Qs*cos(w*t+phi),t)*diff(conjugate(Qs*cos(w*t+phi)),t)+w^2*Qs*cos(w*t+phi)*conjugate(Qs*cos(w*t+phi))),ks=1..3)-1/c*(Summation(conjugate(Qs*cos(w*t+phi)),ks=1..3));

or

H := 1/(8*Pi*c^2)*Summation(Commutator(diff(Qs,t)*diff(conjugate(Qs),t)+w^2*Qs*conjugate(Qs)),ks=1..3);

#Qs = p, Ps := Pss*sin(m*t+phi) Ps = qs

i use

H := 1/(8*Pi*c^2)*Summation(Commutator(diff(Qs*cos(w*t+phi),t)*diff(conjugate(Qs*cos(w*t+phi)),t)+w^2*Qs*cos(w*t+phi)*conjugate(Qs*cos(w*t+phi))),ks=1..3)-1/c*(Summation(conjugate(Qs*cos(w*t+phi)),ks=1..3));

 HJ := subs(Qs= diff(f(q,P,t), q), H);

H:=subs( f(q,P,t) = f1(q) + f2(t), HJ);
sol:=dsolve({rhs(H)=E,lhs(H)=E});
S:=rhs(sol[1][1]+sol[1][2]);
p:=diff(S,q);
Q:=diff(S,E);

Hello

when i am trying to isolate for Rev in the following expression

the program doesnt respond.do you know why?

I have tried without sucess to run the

Equation manipulator.

I get the following error:

"

Error, (in Maplets:-Tools:-CleanElements) invalid input: op expects 1 or 2 arguments, but received 0

"

Do I have to load some package for it to run?

I have searched for a solution but can not find one.

Getting desperate.. anyone?

TIA

LWc

Hi

How  can we draw diagrams in tripartite coordinate system?

Hello,

How to Factorize the coefficients of this matrix :

> A:=Matrix([[x^3+x-2,x^2+x],[x,x+1]]);

factor(A) does not work !

Thank you,

Gérard.

I would like to ask your assistance in understanding the following Maple interaction which speaks for itself.

> sum(product(k+4-m, m=0..k-1)*z^k ,k=0..j);                                         
                    /    j                 \                                         
                    | --------'            |
                    |'  |  |               |  (j + 1)
                    |   |  |    (j + 5 - m)| z
                    |   |  |               |
                    |   |  |               |
                    \  m = 0               /               1
                    --------------------------------- - -------
                                 4 z - 1                4 z - 1

> series(simplify(subs(j=6, %)), z=0, 16);   
              2       3        4         5         6            7            8
1 + 4 z + 16 z  + 64 z  + 256 z  + 1024 z  + 4096 z  - 1646816 z  - 6587264 z  -

              9              10              11               12               13
    26349056 z  - 105396224 z   - 421584896 z   - 1686339584 z   - 6745358336 z

                    14                 15      16
     - 26981433344 z   - 107925733376 z   + O(z  )

> sum(product(k+4-m, m=0..k-1)*z^k ,k=0..6);
                     6          5         4        3       2
             151200 z  + 15120 z  + 1680 z  + 210 z  + 30 z  + 5 z + 1

> sum((k+4)!/4!*z^k ,k=0..6);
                     6          5         4        3       2
             151200 z  + 15120 z  + 1680 z  + 210 z  + 30 z  + 5 z + 1

This was with Maple 18 (X86 64 LINUX).

Best regards, Marko Riedel