Numerical methods for 2nd order non-linear ODE $\ddot y=f(x)$ where $f$ is unknown

Question

Say we have a simple 2nd order non-linear ODE $\ddot y=f(x)$. We don't know what $f$ is but have several known data points $(x_1,f(x_1)),...,(x_n,f(x_n))$. Could you help suggest numerical methods (esp. iterative algorithms) to numerically estimate $y$? Any suggestion is helpful, including classic ones and new methods in recent researches. Thanks!

Answer 1

I think this is problem can be reduced to an interpolation problem. I am pretty sure you are missing some initial conditions to solve the problem.

You can assume a polynomial as an interpolation function: \begin{equation} \hat{y}(x) = \sum^{n+1}_{k=0}a_k\,x^k \end{equation} If you differentiate the interpolation polynomial you get: \begin{equation} \mathrm{\tfrac{d^2}{dx^2}}\hat{y}(x) = \sum^{n+1}_{k=2}k\,(k+1)\,a_{k-2}\,x^{k-2} \end{equation} The above polynomial providing the estimation of the second derivative can be found by a direct calculation of Lagrange polynomial.

You don't have to use Lagrange polynomial. As an alternative you can use splines. This gives you the estimation of the second derivative of a signal. To calculate the signal values you are still missing initial values.