I am currently trying to solve a PDE using the Runge-Kutta method. The original equation is the Young-Laplace equation: $$ \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right) = \Delta P \label{yl} $$ where $\gamma$ is the interfacial tension, $R_1$ and $R_2$ are the principle radii of curvature, and $\Delta P$ is the change in pressure across a fluid interface. As I am solving the equation for a gravity distorted droplet shape the equation can be simplified down to a set of equations, these have been obtained from extensive literature on the topic: $$\frac{d\theta}{ds} = 2 - \beta z - \frac{\sin\theta}{x}$$ $$\frac{dx}{ds} = \cos\theta$$ $$\frac{dz}{ds} = \sin\theta$$ At $s=0$: $x=0$, $z=0$, and $\theta=0$
A mechanical test rig I have designed incorporates a camera which will give me a discrete shape of the droplet. Basically, I have known values for the function $x(z)$ and also for $z(x)$. I can also set practical boundaries for $s$, $\theta$, $x$, and $z$. The value that I would like to calculate is the parameter $\beta$ which represents the Bond number. This dimensionless number will allow me to calculate the interfacial tension I am looking for. Although the question does sound like a physics one, what I am really looking for is a way to implement the set of equations into a Runge-Kutta method so the PDE can be solved as a system of ODEs.