How Can I Solve 2nd-Order Coupled Nonlinear Differential Equations?

[eating]

Hi, folks. I'm new to this forum.

I joined because i am so desperate in solving a 2nd-order coupled nonlinear differential equation for the motion of a system similar to an inverted pendulum model. The equation, which i will call here EXACT governing eqn, is this

A(s) s" + B s' + C(s, s') = Dsin(t)

where s(t) = {u(t) v(t)}^T [2x1 vector].

The 2x2 matrix A(s) matrix given as

A(s) = [ h h^2 ]
[ h^2 h^2 + u^2]

where h is a constant and u=u(t).

My problem is that direct integration can not proceed because A(s) does not have an inverse when u becomes zero.

The equation can be simplified when s(t) are small and the governing equation reduces to third-order ODE in terms of v, i will call an APPROXIMATE governing eqn. u(t) can be solved as u = f(v, v').

In more than 6 months, i have tried several methods:

1. Method switching.
Switch from solving the EXACT to APPROXIMATE equation, when u(t)~0 or becomes
less than some numerical value say e, then switch bact to EXACT soln when u(t)
becomes large. This method, however, caused significant discontinuities at e; and
sometimes solution diverges when amplitude D is large.

2. Least squares.
I added a new equation to minimize the acceleration s"(t), i.e., added

W(u) * s'(t+dt) = W(u) * s'(t), W(u)=1 for simplicity here

and solve a system of 6-first order ODEs in terms of s(t+dt) and s'(t+dt) using
principle of least squares.

This EXACT solution is identical to the APPROXIMATE solution when amplitude D is
small, which is has to be in this cases. But when amplitude D is large, the additional
equation (minimizing acceleration s") may be too strong to cause significant changes
in the governing equation, thereby altering the expected motion.


ARE THERE WAYS TO SOLVE THIS TYPE OF PROBLEM?

I WOULD APPRECIATE ANY COMMENTS. THANK YOU.

 

[jvicens]

Hello and welcome to the forum! I understand your frustration with trying to solve this difficult differential equation. I have encountered similar challenges in my research.

First of all, I would suggest looking into numerical methods for solving differential equations. These methods use numerical approximations to solve equations that cannot be solved analytically. Some commonly used methods include Euler's method, Runge-Kutta methods, and the finite difference method. These methods may be able to handle the nonlinearity and coupling in your equation.

Another approach you could try is using perturbation methods. This involves expanding the solution in a power series and solving for each term. This can be useful for small parameters, such as your amplitude D, and can give you an approximate solution.

You could also try using a computer program or software specifically designed for solving differential equations. These programs often have built-in algorithms and methods for handling difficult equations.

Lastly, I would suggest reaching out to other scientists or researchers in your field who may have encountered similar equations or problems. Collaboration and exchanging ideas can often lead to new approaches and solutions.

I hope these suggestions are helpful to you. Good luck with your research!