## Inverse Problem

A very much improved version of the paper by the same authors is the following.
The problem herein may be regarded as the **inverse** of the above:
### Deficient Reasoning for Dark Matter in Galaxies

The mathematical model, as has been found in a previous section, is repeated here for convenience:
$$
\int_0^1 \left[ \int_0^\pi
\frac{\left[(r/R)-\cos(\phi)\right]\,d\phi}{\left[(r/R)^2-2(r/R)\cos(\phi)+1\right]^{3/2}}
\right] (r/R)\,\rho(R)\;dR = \frac{v^2}{2\,G}
$$
Upon discretization we get, with $\,i = 0,1,2,3,\cdots ,N$ :
$$
\sum_{j=0}^N \left[ \int_0^\pi
\frac{\left[(r_i/R_j)-\cos(\phi)\right]\,d\phi}{\left[(r_i/R_j)^2-2(r_i/R_j)\cos(\phi)+1\right]^{3/2}}
\right] (r_i/R_j)\,\rho(R_j)\;\Delta R_j = \frac{v_i^2}{2\,G}
$$
This has been used already in the calculations, of course. But now the interpretation will be different.
Define matrix elements:
$$
A_{ij} = \left[ \int_0^\pi
\frac{\left[(r_i/R_j)-\cos(\phi)\right]\,d\phi}{\left[(r_i/R_j)^2-2(r_i/R_j)\cos(\phi)+1\right]^{3/2}}
\right] (r_i/R_j) \;\Delta R_j
$$
Define the unknowns $\,x_j\,$, being the a discretization of the now supposed *unknown density profile*.

And define a right hand side $\,b_i$ , consisting of the velocity profile that is now supposed to be *well
known* instead of unknown:
$$
x_j = \rho(R_j) \quad ; \quad b_i = \frac{v_i^2}{2\,G}
$$
Then what we have is a linear system of equations, to be solved for the unknown density profile $\,x_i$ :
$$
\sum_{j=0}^N A_{ij}\,x_j = b_i
$$
I have not pursued this excercise any further myself, because James Q. Feng and C. F. Gallo have.