Instabilities in numerical solutions to Fredholm and Volterra integral equations of the first kind. Resolution by Tchebycheff polynomials. Application to photonuclear cross-sections

It is well known, if not well explained, that photo cross-sections curves depend on numerical resolution; as well as many other physical solutions from integral equations of the first kind, they are oscillating. In the first part of this report, a typical example points out how oscillations are growing. In the second part, a new method is explained yielding a smooth resolution. From experimental data on equidistant intervals, we build functions expanded in Tchebycheff polynomials; the solution is of this kind. Then, the third part points out that semi-analytical resolutions of this problem are illusive. (author)