Determining $t(x)$ from $\frac{dx}{dt}$?

Question

I have a question that "feels" basic/stupid, but I've been really struggling with it. The basic question is: is there an easy way to determine $t(x)$ from $x(t)$ or $\frac{dx}{dt}$?

To explain further, I have a differential-algebraic system of equations I'm trying to solve. The system I'm solving is technically a 1D partial-differential equation, but all of the academic papers I've come across (at least 15) eliminate the spatial dimension by numerically integrating it, and this t(x) term shows up in the integral. However, none of them give any details on HOW they do that - they just say "numerical methods were used, and it was easily solved." See below for a representation of the full equation set:

$$\frac{d(F)}{dt} = C - E(t) $$ $$\frac{d(x)}{dt} = \sqrt{g*H(t)}$$ $$H(t) = \frac{F(t)}{\pi*x(t)^2}$$ $$E(t) = Const*\int\limits_0^{x(t)}\frac{2*\pi*x(t)}{t(x)} dx$$

I've managed to calculate t(x) via brute force with a coding algorithm, but I feel like there should be an easier way to find it either analytically or numerically. My approach is not very robust - especially for small values of x/t (x and t both technically start at 0, where everything goes to infinity - so I add a small offset to prevent divide-by-zero issues). All of the other equations/math are very straightforward and make sense to me - this one term is just tripping me up!

Answer 1

Numerically, the calculation of $x(t)$ is a table with two rows, $t$, and $x$, and it simultaneously calculates $t(x)$: calculating $t(x)$ from is simply a matter of reading the table inversely. Estimates on $dx/dt$ will allow you to estimate how large your table will need to be according to your accuracy needs. The integrand you have written is wrong; you should use $t_0$ instead of $t$, like so:

$$E(t_0) = 2\pi x(t_0)\int_0^{x(t_0)}\frac{dx}{t(x)}$$

In $E(t_0)$, the numerator of the integrand is constant. You're merely integrating $\int^{x(t_0)}_0\frac{dx}{t(x)}$, which, from the table, is approximated by the sum $\displaystyle\frac{x(t_0)}{N}\sum_{k=1}^N \frac{1}{t(x(t_0)\frac{k}{N})}$. The value for $N$ will depend on your estimates of $dx/dt$, where in general, small values of $dx/dt$ will require larger $N$ and/or splitting the integration interval in many intervals. See the inverse function rule on Wikipedia.