Numerical integration of a derivative dataset

Question

I have an experimentally measured derivative data ($\frac {dy}{dx}$) at a range of times i.e $\frac {dy}{dx}$ for $0 \le t \le tf$.

Integrating for $t=0$ is fine, since $y(x,0)$ is known. However, for each further timestep, I don't know how to compute the definite integral (just the indefinite relative integral result).

So, in summary, what is known is $\frac {dy}{dx}$ for all $t$ and $x$, $y$ for $t=0$ and all $x$.

what I need to extract is y for all $t$ and $x$.

If I need to clarify, just let me know.

Answer 1

Your data is governed by the diffusion equation:

$$\frac{\partial^2 y}{\partial x^2} = c \frac{\partial y}{\partial t}$$

Here, you know $\partial y/\partial x$ for all $t$ and $x$ of interest, as well as $y(x,0)$ for all $x$ of interest. I will assume you know $c$, or at least have an estimate.

You can approximate $\partial y/\partial t$ from your derivative data, but taking the numerical derivative of that data in $x$, and dividing by $c$. From this, you may extract an approximation for $y$ at the next time step $t$. This is admittedly crude, but it is a first step.

Answer 2

If I understand you correctly you know $\frac{dy}{dx}$ in some discrete points $t_i$. And you know $y_0$ and $x_0$ at $t_0$. So why do not you approximate the derivative like this $$\frac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}} = \frac{dy}{dx}_{(i+1)}$$ and simply solve this equation. But to do this some additional information on $x$ or $y$ are required. Otherwise it seems the lack of information to find a solution.