Solve Highly Nonlinear ODE: Fourier Series Analysis

[tburke2]

I have an equation of motion given by
$$f(z(t),t) = \frac{d^2z}{dt^2} + A\frac{dz}{dt} + B $$
where
$$f(z(t),t) = [(\frac{C}{z^2+C^2})^2-(\frac{D}{z^2+D^2})^4]^2(1+cos(wt))$$
and $A,B,C,D,$ and $w$ are constants

Is it possible to solve this for $z(t)$? I have been solving it numerically using Matlab's ODE solver but as this model is used to fit a set of experimental data (constants $A$ and $B$ are varied until a reasonably small error is achieved between this and the experimental data) it would greatly reduce computational time if a solution or even a close approximation can be found.

I know from numerically solving $z(t)$ that it is periodic so there must be a Fourier series that can be used to find a solution. I have basic knowledge of Fourier analysis and since $f$ is a function of $z$ and $t$, and $z$ is dependent on $t$, I'm unsure how to do this. If someone could point me in the right direction it would be much appreciated.

 

[mighty2000]

I would suggest looking into the method of separation of variables to solve this equation. This method involves assuming that the solution can be written as a product of two functions, one depending only on $z$ and the other depending only on $t$. This will allow you to separate the variables and solve for each one separately.

In this case, you can assume that $z(t) = X(t)Y(z)$ and then plug this into the equation of motion. This will result in two separate equations, one for $X(t)$ and one for $Y(z)$. You can then solve each equation separately using standard techniques such as integration or power series.

Once you have the solutions for $X(t)$ and $Y(z)$, you can combine them to get the solution for $z(t)$. This will give you an analytical solution for $z(t)$ which will be much faster and more accurate than using numerical methods.

Additionally, since you mentioned that the solution is periodic, you can also try using the method of Fourier series. In this method, you can express the solution as a sum of sine and cosine functions with different frequencies and amplitudes. By finding the coefficients of these functions, you can get an approximation of the solution for $z(t)$.

I hope this helps and good luck with your research!