hi dears. i cant solve this question by Analytical method such variational itteration method (VIM) or homotopy perturbation method (HPM), homotopy analysis method(HAM)
how to solve by vim or DTM?
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QM perturbation theory: too many levels of recursion
Hello everyone,
I've been trying to do some perturbation theory and ran into some problems I don't quite understand. I implemented the Hamiltonian of the Bose-Hubbard model and treated the hopping as a perturbation. Calculating the second order energy shift is easily accomplished, but when I'm only interested in one of the two occuring terms, I run into problems. The calculation takes minutes to finally fail, giving me an "too many levels of recursion"-error. I need to be able to just pick a few terms for some calculations, I'm doing, and can't figure out what I might be doing wrong. Here is the source code (download is below):
Download too_many_levels_of_recursion.mw
It would be great, if someone could point out the mistake, I'm making. I copy/pasted the last line, so there shouldn't be any typos.
Thanks in advance,
Sören
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Quantum perturbation theory: energy denominators aren't evaluated properly
Hi!
I seem to run into problems with (quantum) perturbation theory. In the following minimal working example, an energy denominator, as occuring in perturbation theory is not evaluated if I previously assume that the quantum number is a positive integer. It's supposed to return an error, as the energy denominator is 0.
Download energy_denominator.mw
Best regards,
Sören
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Homotopy perturbation methods
HPM_4.mwhi, I am using homotopy perturbation technique but there is arising an error in comaring coeffecient of p^0, p^1,.... plz help me
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Problem with HPM
Hi everyone.
I'm going to solve a problem with HPM in Maple. I wrote some initial codes but now I'm confused becouse of P^0 coefficients in A1 and B1. I mean I can't reach to f0 and g0.
I upload that file. these are codes that i typed. could you please help me how can I reach to them(f0 & g0)?
http://www.filehosting.org/file/details/573095/Maple%20Project+.mw
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Metric Perturbation
Hi All,
I'm using the Physics package, which enables GR calculations, ie defining metrics and tensor algebra.
Was just curious if it were possible to add a perturbation to the metric when calculating Ricci and Christoffels.
I would like something like
g_[] = g1_[mu,nu] + h[mu,nu]
And then do a calculation like,
Ricci[].
I know this would be possible if I define everything and re-write the calculations for calculating Ricci, i.e
Define(g1[mu,nu], h[mu,nu]);
and the proceed with GR calculations to find Ricci, however was hoping there was an easier way to do this.
Any help is appreciated.
Thanks guys.
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How do I solve a system of singular differential equation in Maple
Please help me to solve the system of 1st order singular O.D.E (see uploaded file)....New_Microsoft_Office_Word_Document.docx
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how to solve by vim or DTM?
hi dears. i cant solve this question by Analytical method such variational itteration method (VIM) or homotopy perturbation method (HPM), homotopy analysis method(HAM)
↧
perturbation theorie
Hi all, I am having the follwing DE:
restart:
(diff(z(x), x, x))-z(x) - cos(2*x)/(1+delta*z(x)) = 0:
With initial conditions: z(-pi/4)=z(pi/4)=0 and |delta|<<1
I showed by hand by using perturbation theory the second order approximation. The hint was: you can use Maple to check your answer.
Is there somebody who can help me with this?
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How do I use Maple to Solve a Differential Equation in Second-Order Perturbation Theory?
I am working with the following differential equation:
$\frac{d^2z}{dx^2}+z=\frac{\cos 2x}{1+\epsilon z},\:\:\:z(-\pi/4)=z(\pi/4)=0$
where modulus of $\epsilon$ is much less than $1$. The task is then to use perturbation theory (with Maple, if necessary) to show that the second-order approximation to the solution to this DE is:
$z=-\frac{1}{3}\cos 2x +\epsilon\bigg(\frac{1}{6}-\frac{8\sqrt{2}}{45}\cos x - \frac{1}{90}\cos 4x \bigg) + \epsilon^2 \bigg(\frac{2\sqrt{2}x}{45}\sin x - \frac{\sqrt{2}}{90}(\pi + 1)\cos x + \frac{7}{720} \cos 2x - \frac{\sqrt{2}}{90}\cos 3x - \frac{1}{1050}\cos 6x \bigg).$
I will then likely have to use Maple to determine the third-order term $\delta^{3}z_{3}(x)$ and evaluate $z_{3}(x)$ at $x=0$ and $x-\pi/8$.
My starting point is to use the theory for a regular perturbation (since the modulus of $\epsilon$ is much less than $1$). For the unperturbed equation, I could set $\epsilon=0$ as that would give a simple differential equation which should be solvable. I can then see that $1/{1+\epsilon z}$ can be expanded to second-order in $\epsilon$ as $1 - \epsilon z + \epsilon^2 z^2 + O(\epsilon^3), which looks promising. Could someone advise how I put this together? Do I then have to multiply the unperturbed solution by the expansion in $\epsilon$?
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asking for training exercise
Dear sir / madam
Hi
I'm a student of mechanical engineering and looking for some HPM ( homotopy perturbation method) maple codes with the related article to practice more in this field.
can I ask you for help?
unfortunately, I couldn't find any articles with maple solution about HPM.
sincerely
Aidin
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Is there any plot that can show perturbation of eigenvectors
If decimal place is 1 such as 123.1 as input matrix and output matrix is eigenvector
is there a easy way to see changes of eigenvector in graph ?
because 3 x 3 matrix has 9 elements
and eigenvector each vector may not fix in place,
how do mathematician observe the perturbation in this case?
would like to find which changes can cause different in eigenvectors
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How expand in perturbation series in maple?
I have the following ODE perturbation problem which I want maple to solve for me:
q'(\tau)=f(p(eps*\tau)+eps*q(\tau),r(eps*\tau)+s(\tau))-f(p(eps*\tau,r(eps*\tau)+s(\tau))-f(p(eps*\tau),r(eps*\tau))
where q(\tau)=q_0(\tau)+eps*q_1(\tau)
p(eps*\tau)=p_0(eps*\tau)+eps*p_1(eps*\tau)
s(\tau)=s_0(\tau)+eps*s_1(\tau)
r(eps*\tau)=r_0(eps*\tau)+eps*r_1(eps*\tau)
I want maple to expand every function that depends on eps in its arguments by a Taylor series around eps=0, i.e h(eps)=h(0)+eps*h'(0)
and also expand the difference above the fs with an eps-expansion around eps=0.
I did all this manually now I want to check if my calculations are correct, eventaully I want to equate same powers of eps of the RHS and LHS of the first ODE I wrote above.
Then how to use maple for this?
Thnaks.
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How do I use Homotopy Perturbation Method to find an analytical solution for Euler-Bernoulli Beam on Pasternak foundation?
Please, I need to use Maple to solve Euler-Bernoulli Beam on Pasternak Foundation using Homotopy Perturbation Method.
The governing equation is 
initial conditions are 
the boundary condition is 
The governing equation represents Euler-Bernoulli beam on a generalized Pasternak viscoelastic foundation under an
arbitrary distributed dynamic load. in which E, I , ρ ,A are the parameters of the beam, representing Young’s modulus of elasticity, moment of inertia, density and area of cross section, respectively. K,C and Gp are spring stiffness, damping coefficient, and shear coefficient of the foundation. Moreover, y(x, t) and F(x, t) are defined as the vertical deflection of the beam and the generic arbitrary dynamic loads, respectively, where the loads distribute along the x-axis and t is time.
I will appreciate anyone who can help me with a Maple solution.
Thank you.
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Simulation on perturbations
Consider a wave y=f(x,t) with amplitude modulation such that it becomes
y_new=(1+A)*f(x,t), where A=A(t) is a small perturbation.
How can I do the simulation to test the robustness of the system to see if the wave is stable / unstable under perturbation? As the time t goes to infinity, I want to see if the original wave is distorted or not under either random or prescribed perturbation.
P.S. For example, f(x,t) = (sech(0.5*x-0.1*t))^2.
Thank you for your help.
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Why the solution is not shown in a complete form?
As you can see in the code, I am trying to print the solution of the ODE using the Homotopy perturbation method for N = 10 ( f[0] till f[10]), but Maple is only printing f[0],f[1], and the rest of the terms are not printed out, what could cause this issue to take place?
```
restart;
N := 10:
F(Y) := add(p^i*f[i](Y), i = 0..N);
HPMEq := (1 - p)*diff(F(Y), Y $ 4) + p*(diff(F(Y), Y $ 4) + R*(diff(F(Y), Y $ 3)*F(Y)- diff(F(Y), Y $ 2)*diff(F(Y), Y $ 1))-G*diff(F(Y), Y $ 2));
sol:=[]:
for i from 0 to N do
sol:= [ sol[],
dsolve
( [ eval
( coeff(HPMEq, p, i) = 0,
sol
),
f[i](0) = -a,
f[i](1) = -b,
D(f[i])(0) = B * D(D(f[i]))(0),
D(f[i])(1) = -B * D(D(f[i]))(1)
]
)
];
end do:
sol;
```
Please kindly try to run this code in your Maple version and tell me if all terms are printed.
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Analytical solution of ODE?
Hi, I am tryig to solve ode with the perturbation method but getting an error. can anyone correct my fil and i also want to draw the graph of the involved parameter beta and how can I compare the result obtained through the perturbation method with the numerical result?.... beta is very very small like 0.1..0.9...
In my maple sheet the whole ode is equati=al q which is q=dp/dx and also want to involve its zero and first-order terms like dp0/dx, dp1/dx in the obtained result of u.
Respected Comunity waiting for your kind response
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Perturbation with anti-secularity is older than was thought
Dear all,
Recently we learned that the idea of "anti-secularity" in perturbation methods was known to Mathieu already by 1868, predating Lindstedt by several years. The Maple worksheet linked below recapitulates Mathieu's computations:
https://github.com/rcorless/MathieuPerturbationMethod
Nic Fillion and I wrote a more general introduction to perturbation methods using Maple and you can find that paper at
https://arxiv.org/abs/1609.01321
and the supporting Maple code in a workbook at
https://github.com/rcorless/Perturbation-Methods-in-Maple
For instance, one of the problems solved is the lengthening pendulum and when we do so taking proper account of anti-secularity (we use renormalization for that one, I seem to remember) we get an error curve that is bounded over time.
Hope that some of you find this useful.
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Is there any plot that can show perturbation of eigenvectors
If decimal place is 1 such as 123.1 as input matrix and output matrix is eigenvector
is there a easy way to see changes of eigenvector in graph ?
because 3 x 3 matrix has 9 elements
and eigenvector each vector may not fix in place,
how do mathematician observe the perturbation in this case?
would like to find which changes can cause different in eigenvectors
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Solution of Logistic equation in Perturbation theory
Hi everyone,
Please I need your help, if anyone has idea of using Perturbation Theory to Solve the following Logistic Fractional Equation and ploting with iteration. Thanks
Restart
u(t):=1/(m)(u(t)-(u^(2)(t))/(k));
NULL
uu(t):=(k*u_0)/((u_0+(u_0)*k)*(e)^(-t/(m)))
NU
LL
u(0) = u_0
NULL
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