Looking for a serie solution for a nonlinear ODE system

In summary, the conversation discusses the difficulties in finding a solution for a specific ODE system related to projectile motion. Various methods have been tried, including Picard iteration and power series, but a more accurate solution may be obtained using numerical methods or a computer algebra system. It is also suggested to experiment with different approaches, such as a Taylor series expansion, to find a solution for \artanh q.
  • #1
Theia
122
1
Hi!

\(\displaystyle \begin{cases} \dot{q} = a \left( 1 - q^2 \right) \\ \dot{a} = - \alpha - a^2 q\end{cases} \qquad \alpha \in (0, 1 ) \)

I've looked into this ODE system about 7 months now, but I've not got anything promising how to write down the solution. I'm mostly interested in \(\displaystyle q\)-serie. (To those of you who are now thinking why I'm doing something like this: I'm too much interested in physics and projectile motion. This ODE system comes from quadratic air resistance when describing projectile motion. See e.g. this paper where to start.)

Methods tried thus far: Picard iteration, regular power serie method.

Picard iteration gives first powers very nicely, but it also shows how complicated the solution is. Also, I know that there are a lot of cool math hiding in the system. For example one can form a Lagrange equation (and Hamiltonian) to generate the system... In this case

\(\displaystyle \mathcal{L} = \frac{-1}{2} \frac{\dot{q}^2}{\left( 1 - q^2 \right)^3} + \frac{\alpha}{2} \left( \frac{q}{1 - q^2} + \artanh q \right)\)

Methods speculated: Perhaps it would help, if there was a way to write \(\displaystyle \artanh q\) in some kind of serie (definitely not like a power serie, but more in the spirit of Min-Max-approximation)...

Any thoughts or comments what to try?
 
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  • #2


Hi there!

It seems like you have already tried some common methods for solving this ODE system. Have you considered trying numerical methods, such as Runge-Kutta or Euler's method? These methods may give you a more accurate solution compared to Picard iteration or power series methods.

Another approach you could try is to use a computer algebra system, such as Mathematica or Maple, to find a symbolic solution. These systems have built-in functions for solving differential equations and may be able to handle more complicated systems like the one you are working with.

As for writing \artanh q in a series form, you could try using a Taylor series expansion. However, keep in mind that this may not be a very useful approach since the series may not converge for all values of q.

Overall, I would recommend exploring different numerical and symbolic methods to see which one gives you the best solution for your specific problem. Good luck!
 

FAQ: Looking for a serie solution for a nonlinear ODE system

1. What is a series solution for a nonlinear ODE system?

A series solution for a nonlinear ODE system is a method of solving a system of ordinary differential equations (ODEs) by representing the solution as a series of terms. This method is often used when other analytical or numerical methods are not feasible.

2. How is a series solution different from other methods of solving ODE systems?

A series solution is different from other methods of solving ODE systems in that it involves representing the solution as a series of terms, rather than finding an exact or approximate solution using algebraic or numerical techniques. This can be useful when dealing with highly nonlinear systems that cannot be solved using traditional methods.

3. What are the advantages of using a series solution for a nonlinear ODE system?

One advantage of using a series solution for a nonlinear ODE system is that it can provide a more accurate solution compared to other methods, especially when dealing with highly nonlinear systems. Additionally, the series solution can be used to approximate solutions for a wide range of initial conditions.

4. What are some limitations of using a series solution for a nonlinear ODE system?

One limitation of using a series solution for a nonlinear ODE system is that it can be time-consuming and computationally intensive. Additionally, the series may not converge for certain initial conditions, making it difficult to find a solution. It also may not provide an exact solution, but rather an approximation.

5. Are there any real-world applications of using a series solution for a nonlinear ODE system?

Yes, there are many real-world applications of using a series solution for a nonlinear ODE system. Some examples include modeling chemical reactions, analyzing population growth in ecology, and predicting the behavior of electrical circuits. This method can also be used in physics, engineering, and other fields to understand and solve complex systems.

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