# How to fit experimental data with by numerical integration?

folks.

I'm a begginer with Mathematica and all I know is just a few basics commands and i'm trying to improve it.

I have this list of experimental data taken from a paper which describes the magnetization of a sample (actually a rock) toward the equilibrium when subject to an external magnetic field (y axis is magnetization, x axis is time in log scale):

data={{0.995, 0.142}, {3.003, 0.2}, {5.908, 0.25}, {10.525, 0.36}, {13.617,
0.498}, {24.321, 0.616}, {33.917, 0.599}, {47.843, 0.7}, {64.172,
0.835}, {91.353, 1.102}, {126.745, 1.083}, {174.118,
1.225}, {225.059, 1.133}, {292.998, 1.165}, {369.133,
1.298}, {640.295, 1.365}, {828.169, 1.298}, {1255.39,
1.373}, {1496.61, 1.409}, {1942.79, 1.538}}


This data should be ploted using LogLinearPlot[data] which yields:

There's a integral equation that should fit this data:

$$m(t) = \int N V M_{eq} (1-e^{-t/\tau})f(V) dV$$

where $$N$$ is the total number of particle, $$t$$ is the elapsed time,$$V$$ the volume of the particles, $$M_{eq}$$ the magnetization at the equilibrium state,$$\tau$$ is a parameter that dependes on $$V$$ and $$f(V)$$ is the distribution of volume of the particles.

I'd like to know:

i)if it's possible (and if so, how to do it) to fit this data using this integral equation with, for instance, a log gaussian distribution $$f(V)$$.

ii) Given this equation, is there a way to create a routine which calculates the best-fitting of the data and, as an output, gives the $$f(V)$$ distribution which provides this best fitting?

• Hi, user64494. Thanks for your feedback. So let's suppose that I have a log-normal distribution $f(V)$. Is there a way to numerically solve the equation using the given distribution and plot this solution as a function of time $t$? – José Augusto Devienne May 2 '19 at 18:36