Numerical Solutions of the Abel Integral Equation

Abel inversion applied to plasma spectroscopy: a new interactive method

A new method has been developed to perform an Abel inversion. As smoothing is an essential step when one deals with experimental data, a controlled smoothing procedure using spline functions is used. This gives better results for test functions used by previous authors on data with or without scattering.

Spatial resolution of the volume emission coefficient in strongly self-absorbing sources of cylindrical symmetry

It is shown that the equations relating the radial profiles of the volume emission and absorption coefficients to the transmission and emitted intensity profiles in self-absorbing cylindrically symmetric sources, can be written in such a way that the problem of spatially resolving the volume emission coefficient gives rise to a Volterra integral equation of the second kind in a standard form. The theory of equations of this type is invoked to show the formal convergence of an iterative solution to the problem, subject only to a finite transmission and bounded slope to the absorption coefficient. A prescription for applying this iterative procedure is given that involves a series of numerical integrations and Abel inversions, and the convergence of some numerical solutions is demonstrated.