How expand in perturbation series in maple?

April 10, 2019, 9:58 pm

≫ Next: How do I use Homotopy Perturbation Method to find an analytical solution for Euler-Bernoulli Beam on Pasternak foundation?

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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?

May 23, 2019, 1:00 pm

≫ Next: Simulation on perturbations

≪ Previous: How expand in perturbation series in maple?

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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.

Simulation on perturbations

June 6, 2019, 12:50 pm

≫ Next: Why the solution is not shown in a complete form?

≪ Previous: How do I use Homotopy Perturbation Method to find an analytical solution for Euler-Bernoulli Beam on Pasternak foundation?

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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.

Why the solution is not shown in a complete form?

January 25, 2021, 8:00 am

≫ Next: Perturbation with anti-secularity is older than was thought

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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.

Perturbation with anti-secularity is older than was thought

September 20, 2021, 5:49 pm

≫ Next: Solution of Logistic equation in Perturbation theory

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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.

Solution of Logistic equation in Perturbation theory

July 30, 2023, 9:09 am

≪ Previous: Perturbation with anti-secularity is older than was thought

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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)))

NULL

u(0) = u_0

NULL