Non-linear second order ODE

In summary, you can solve the nonlinear ODE using a Fourier series to represent the driving function, but it will require a long calculation and the stiffness of the system will make it difficult to maintain accuracy.
  • #1
en51nm
1
0
Hi all,

I have a nonlinear ODE in the following form:

a x'' + b |x'|x' + c x' + d x = y

where x and y are functions of time and a,b,c and d are constants. As far as I can tell the only way to solve this is numerically, something I've managed to do successfully using a Rung-Kutta scheme. This however is a lengthy calculation, since the driving function (y) is oscillatory I am mostly interested in the phase and magnitude of the response I am looking for an alternative method.

I'm wondering if I can use a Fourier series to represent the driving function y and then calculate the resulting Fourier series for x (actually it is the x' term I am most interested in) using an FFT? But I'm not sure if this is possible due to the nonlinear term in x'.

Any help/suggestions would be greatly appreciated.
 
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  • #2
en51nm said:
Hi all,

I have a nonlinear ODE in the following form:

a x'' + b |x'|x' + c x' + d x = y

where x and y are functions of time and a,b,c and d are constants. As far as I can tell the only way to solve this is numerically, something I've managed to do successfully using a Rung-Kutta scheme. This however is a lengthy calculation, since the driving function (y) is oscillatory I am mostly interested in the phase and magnitude of the response I am looking for an alternative method.

I'm wondering if I can use a Fourier series to represent the driving function y and then calculate the resulting Fourier series for x (actually it is the x' term I am most interested in) using an FFT? But I'm not sure if this is possible due to the nonlinear term in x'.

Any help/suggestions would be greatly appreciated.

This is more of a question than an answer. If x and y are both functions of time (i.e., x(t) and y(t)), then can we not pick any x(t) to solve for y(t)? For example, if x(t)=t^3 then:

[itex] x'(t) = 3t^2, x''(t) = 6t [/itex]

Then let [itex]y(t) = t^4 + d t^3 + c t^2 + 6at + 9b [/itex], and you have an x(t) and y(t) such that [itex]a x''(t) + b |x'(t)|x'(t) + c x'(t) + d x(t) = y(t)[/itex], but there are in fact an infinite number of (x(t),y(t)) pairs, so this solution is somewhat trivial.

Might you mean x is a function of y? Then: [itex]a x''(y) + b |x'(y)|x'(y) + c x'(y) + d x(y) = y[/itex]

Or I'm just not understanding something. Let me know.
 
  • #3
en51nm said:
Hi all,

I have a nonlinear ODE in the following form:

a x'' + b |x'|x' + c x' + d x = y

where x and y are functions of time and a,b,c and d are constants.
A driven spring mass damper system with linear and quadratic damping? Ouch.

As far as I can tell the only way to solve this is numerically, something I've managed to do successfully using a Rung-Kutta scheme. This however is a lengthy calculation, since the driving function (y) is oscillatory I am mostly interested in the phase and magnitude of the response I am looking for an alternative method.

I'm wondering if I can use a Fourier series to represent the driving function y and then calculate the resulting Fourier series for x (actually it is the x' term I am most interested in) using an FFT? But I'm not sure if this is possible due to the nonlinear term in x'.
That quadratic drag (worse than quadratic, there's an absolute value) is going to make a Fourier decomposition tough.

The damping is going to make this a stiff system, and the quadratic damping is going to make the stiffness rather interesting.

The key problem with a fixed step size integrator such as basic Runge Kutta is that the stiffness mandates that the step size be very small. It's going to take a long time to solve and will lose accuracy in the process.

Have you tried using adaptive techniques such as Runge Kutta Fehlberg, Cash-Karp, adaptive Adams Bashforth Moulton, Gauss Jackson?

Have you tried using adaptive integration techniques such as Kaps-Rentrop, various descendants of the Gear technique that are aimed specifically at stiff problems?
 

1. What is a non-linear second order ODE?

A non-linear second order ODE (ordinary differential equation) is a mathematical equation that describes the relationship between a function and its derivatives up to the second order. Unlike linear ODEs, non-linear ODEs cannot be written in the form of a linear combination of the function and its derivatives.

2. What are some examples of non-linear second order ODEs?

Some examples of non-linear second order ODEs include the Van der Pol oscillator, the Duffing equation, and the Lotka-Volterra model. These equations are commonly used in physics, engineering, and biology to model complex systems with non-linear behavior.

3. How is a non-linear second order ODE solved?

Solving a non-linear second order ODE involves finding the general solution, which is a function that satisfies the equation for all possible values of the independent variable. This can be done analytically using techniques such as separation of variables, substitution, or series expansion. In some cases, numerical methods such as Euler's method or Runge-Kutta methods may be used to approximate the solution.

4. What are the applications of non-linear second order ODEs?

Non-linear second order ODEs have a wide range of applications in various fields of science and engineering. They can be used to model physical systems such as mechanical vibrations, electrical circuits, and chemical reactions. They are also used in population dynamics, control theory, and image processing.

5. What are the differences between linear and non-linear second order ODEs?

The main difference between linear and non-linear second order ODEs is that linear ODEs can be written in the form of a linear combination of the function and its derivatives, whereas non-linear ODEs cannot. This means that the behavior of linear ODEs is more predictable and easier to analyze, while non-linear ODEs can exhibit more complex and chaotic behavior. Additionally, the methods used to solve these equations are different, with linear ODEs often having analytical solutions, while non-linear ODEs may require numerical methods.

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