From an experiment, I have data for time t and a function of time $f(t)$. Data can be described by a Fredholm integral equation such that:
$$ F(t) = \int_{0}^{1}ke^{-kt}f(k) \,{\rm d} k $$
Here is the data:
| t | f(t) |
|---|---|
| 1.221 | 1.03114 |
| 3.34709 | 0.72101 |
| 5.81983 | 0.51572 |
| 8.06147 | 0.39997 |
| 10.37244 | 0.31578 |
| 12.9145 | 0.25919 |
| 15.64144 | 0.21551 |
| 18.02174 | 0.1878 |
| 20.26338 | 0.16526 |
| 22.80544 | 0.14566 |
| 24.95464 | 0.13047 |
| 27.51982 | 0.11763 |
| 30.15432 | 0.10588 |
| 32.69639 | 0.09439 |
| 35.00736 | 0.08663 |
| 37.57253 | 0.07672 |
| 40.1146 | 0.07076 |
| 42.84154 | 0.06369 |
| 44.99074 | 0.06018 |
| 47.55591 | 0.05296 |
| 49.93621 | 0.05004 |
| 52.57071 | 0.04563 |
| 54.78924 | 0.04277 |
| 57.51619 | 0.04015 |
| 59.91959 | 0.03812 |
| 62.39233 | 0.03415 |
| 64.86506 | 0.03317 |
| 67.66134 | 0.0299 |
| 69.87987 | 0.02858 |
| 72.19083 | 0.02619 |
| 74.59424 | 0.0247 |
| 77.22874 | 0.02443 |
| 79.95569 | 0.02177 |
My goal is to find $f(k)$ and $k$ that satisfies the equation.
I am not a mathematician and this type of maths is exotic to me!