## Phase space plot ode ...

Asked by:

Hi

Any help will be appreciated

I have a continuous time dynamical system

x in R ( set of real number)

t  a positive real time

and the function f_t(x)=x-t

How can we plot or sketch their behaviour in the phase space and in the extended phase space

Many thanks

## LSSolve on a large ODE system...

Asked by:

hello i have the following set of ode's:

ode_sub := diff(S(t), t) = -k1*S(t)-S(t)/T1_s;
ode_P1 := diff(P1(t), t) = k1*S(t)-k2*(P1(t)-P2(t)/keq)-P1(t)/T1_p1;
ode_P2 := diff(P2(t), t) = -k2*(-keq*P1(t)+P2(t))/keq-k4*P2(t)-P2(t)/T1_p2;
ode_P2e := diff(P2_e(t), t) = k4*P2(t)-P2_e(t)/T1_p2_e;

ode_system := ode_sub, ode_P1, ode_P2, ode_P2e;

with these parameters:
s0 := 10000;
k2 := 1000; T1_s := 14; T1_p2_e := 35; T1_p2 := T1_p1;

i want to find the unkown parameters : T1_p1, k1, keq and k4

my idea was this:

init:=S(0)=s0,P1(0)=0,P2(0)=0,P2_e(0)=0

dsolve({ode_system,init})

sol := combine(expand(%));
PS := subs(sol, [S(t), P1(t), P2(t), P2_e(t)]);

P1fu := unapply(PS[2],t);
Sfu := unapply(PS[1],t);
P2fu := unapply(PS[3],t);
P2e_fu := unapply(PS[4],t);
P2_total := unapply(P2fu+P2e_fu, t);

the following data is given:

T:=<0,2,4,6,8>

S:=<9999.99913146527,8328.870587730016,6937.009129218748,5777.745632133724,4812.209983843559>

P1:=<0.0,67.86790056712294,114.88787098501874,145.95438088662502,164.85650644237887>

P2_P2e:=<0.0,271.68492651947497,461.9130396605823,589.3710176125417,668.9967533337124> # data from P2(t)+P2_e(t)

making the rediduals:

RP1 := convert(P1-P1fu~(T), list);
RS := convert(S-Sfu~(T), list);
RP2_P2e := convert(P2_P2_e-P2_total~(T), list);

RPs := [op(RS), op(RP2_P2_e), op(RP1)]

res := Optimization:-LSSolve(RPs, k1 = 0 .. 1, keq = 0 .. 10, k4 = 0 .. 1, T1_p1 = 0 .. 100)

i dont know wheter or not the last step work to get the parameters becuase it takes to long to compute. is there a smarter way to obtain the parameters of the ode's? a numeric approch ?

i tried with dsolve({ode_sysytem,init},numeric,'parameters'=[k1,keq,k4,T1_p1]) however it doesnt seem to get my anywhere since i need to know the parameters to use this (i think)

hope someone can help:)

## solve second order ode ...

Asked by:

Hi

We solve the following ode in the interval (0,Pi):

diff(u(x),x,x)+u(x)=f(x);

bcs=u(0)=0, u(Pi)=0;

Stating one example of many conditions for such equation to have a
valid solution

Many thanks

## plot an ode and a function in the same plot...

Asked by:

Hi

I have an ODE which is based on a seperate function, and I would like to make a plot with the information

dsolve([diff(X(W), W) = (0.536000000000000e-3*(1-X(W)))*(1+X(W)), X(0) = 0], numeric)

and

C_A:= C_A0*(1-X(W))*(1+X(W))

which has been used as part of the ODE.

I would really like to plot C_A as a function of W. I have no problem plotting X as a function to W using odeplot. Ideally I would like to plot C_A and X vs W in the same plot.

Regards

## solve ode srm motor...

Asked by:

How I can solve it for P?

P=i^(2)r+(&DifferentialD;)/(&DifferentialD; t) (1/(2)l(tetha)i^(2))+1/(2)i^(2)(&DifferentialD;l(tetha))/(&DifferentialD; theta) w

attach i(t) "corrente", l(t) "induttanza", theta "angular", w "rotary speed"graph of funtions

thanks

## How do I solve an ODE with initial value condition...

Asked by:

the question is

ODE5:= y(x)*diff(y(x),x,x) + (diff(y(x),x)^2=0

dsolve({ODE5,y(0)=4,D(y)(0)=7},y(x))

and my answer appears to be an integration! which is wrong

the correct answer : 2*(4+14*x)^(1/2)

Could someone tell me what did I do wrong? And how could I get to this result?

Thanks a lot!

## Need Help to Plot BVP graph...

Asked by:

Please i need help to plot the graph of f'' against episoln using the below BVP

HELP.mw

## To find the numerical solutions of system of nonli...

Asked by:

Dears

Hope everything fine with you. I want to solve the attached problem by numarically and want to plot it but failed. Please see the attachement and correct it. I am waiting your positive respone.

System_of_ODEs.mw

With my best regards and sincerely.

Muhammad Usman

School of Mathematical Sciences
Peking University, Beijing, China

## how can i solve the following odes...

Asked by:

i don't know much about maple, i need to solve the following odes system... I study a little on the help page of maple about numeric[midrich] that takes bvp and deal singularity as well but dint know how to used in the following system

odes.mw

## ODE with boundary conditions involving limits...

Asked by:

Respected member!
Please help me to find the solution of attached problem, I am a new user so pleaes forgive any mistakes.

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Download mplprimes.mw

## How can I solve Linear ODEs system using matrix me...

Asked by:

$$\textbf{x}' = \begin{bmatrix} -4 & -2 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}+\begin{bmatrix} -t \\ -2t-1 \end{bmatrix},\textbf{x}(0)=\begin{bmatrix} 3 \\ -5 \end{bmatrix}$$

As I know firstly, when the matrix is denoted by $A$, we must compute $e^{At}$ by diagonalizing $A$: if $A=PDP^{-1}$ for a diagonal $D$ then $e^{At} = P e^{Dt} P^{-1}$ where $e^{Dt}$ is a diagonal matrix with $(e^{Dt})_{ii} = e^{D_{ii} t}$...

How can I write The Maple code? maple.stackexchange)

restart: with(LinearAlgebra):

A := Matrix(2,2,[-4,-2,3,1]);

....

## boundary conditions...

Asked by:

Dear sir in this problem should accept five boundaryconditions but it is not working for five boundary conditions and showing the following error please can you tell why it is like this ??

Error, (in dsolve/numeric/bvp/convertsys) too many boundary conditions: expected 4, got 5
Error, (in plots:-display) expecting plot structures but received: [fplt[1], fplt[2], fplt[3], fplt[4], fplt[5], fplt[6], fplt[7]]
Error, (in plots:-display) expecting plot structures but received: [tplt[1], tplt[2], tplt[3], tplt[4], tplt[5], tplt[6], tplt[7]]

and for the progam please check the following link

stretching_cylinder_new1.mw

## Does TWS command of Maple solves system of ODEs ?...

Asked by:

Dear all

I am trying to solve system of ODEs by TWS command for traveling wave solution, but an error is showing. When I enter sinlge ODE or PDE the command does not show any error. Why it is showing error for system of ODEs ?

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Download ODEs.mw

## Method for numerical solving BVP of ODE with disco...

Asked by:

I try to solve numerically a boundary VP for ODE with different order of discontinuity of right part.

Say, the following BVP is given:

y''(x)+y'(x)+y(x)=F(x)

y(0)=1, y(2)=1

Let's use piecewise right part

F  := piecewise(x<=1, -x, x>1, 2*x+(x-1)^2)

The function

piecewise(x<=1, 1-x, x>1, (x-1)^2)

as obviuos, satisfies the BVP exclung the point x=1, where its 1st and 2nd derivatives are discontinuos.

Numerical solution

N0:=6:
As:=dsolve([diff(y(x), x\$2)+diff(y(x), x)+y(x)=F,  y(0)=1, y(2)=1], y(x), type=numeric, output = Array([seq(2.0*k/N0, k=0..N0)]), 'maxmesh'=500, 'abserr'=1e-3):

provides the solution essentially different to exact one described above:

But if to use the right part

F := piecewise(x<=1, x^2+x+2, x>1, -x^2+x)

for which the function

piecewise(x<=1, 1-x+x^2, x>1, -1+3*x-x^2)

satisfies the BVP excluding x=1, where this function has discontinuity of 2nd derivative only, the corresponding numerical solution is very similar to this exact solution:

This reason of the difference between these two cases is clear. In the first case both 1st and 2nd derivatives are discontiuos, while in the second one -- 1st derivative is contiuos.

I wonder, if there are numerical methods, implemeted in Maple, for numerical solution of the first type BVP with non-smooth right part?

## Library, Shooting, ode...

Asked by:

what does it means and what it will do. Can some one help me for solving this

Shootlib := "C:/Shoot9"; libname := Shootlib, libname; with(Shoot);

while i m receiving the following message:

"Error, invalid input: with expects its 1st argument, pname, to be of type {module, package}, but received Shoot "

Full program is :

restart; Shootlib := "C:/Shoot9"; libname := Shootlib, libname; with(Shoot);
with(plots):
N1 := 1.0; N2 := 2.0; N3 := .5; Bt := 6; Re_m := N1*Bt; gamma1 := 1;
FNS := {f(eta), fp(eta), fpp(eta), g(eta), gp(eta), m(eta), mp(eta), n(eta), np(eta), fppp(eta)};
ODE := {diff(f(eta), eta) = fp(eta),
diff(fp(eta), eta) = fpp(eta),
diff(fpp(eta), eta) = fppp(eta),
diff(g(eta), eta) = gp(eta),
diff(gp(eta), eta) = N1*(2.*g(eta)+(eta-2.*f(eta))*gp(eta)+2.*g(eta)*fp(eta)+2.*N2*N3*(m(eta)*np(eta)-n(eta)*mp(eta))),
diff(m(eta), eta) = mp(eta),
diff(mp(eta), eta) = Re_m*(m(eta)+(eta-2.*f(eta))*mp(eta)+2.*m(eta)*fp(eta)),
diff(n(eta), eta) = np(eta),
diff(np(eta), eta) = Re_m*(2.*n(eta)+(eta-2.*f(eta))*np(eta)+2.*N2/N3*m(eta)*gp(eta)),
diff(fppp(eta), eta) = N1*(3.*fpp(eta)+(eta-2.*f(eta))*fppp(eta)-2.*N2*N2*m(eta)*(diff(mp(eta), eta)))
}:

blt := 1.0;
IC := { f(0) = 0,
fp(0) = 0,
fpp(0) = alpha1,
g(0) = 1,
gp(0) = beta1,
m(0) = 0,
mp(0) = beta2,
n(0) = 0,
np(0) = beta3,
fppp(0) = alpha2
};
BC := { f(blt) = .5,
fp(blt) = 0,
g(blt) = 0,
m(blt) = 1,
n(blt) = 1};
infolevel[shoot] := 1;

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