An stable approach to a kind of problems involving the inversion of a Volterra integral equation of the first kind: application to x-ray fluorescence analysis

Abstract

In some physical problems it is necessary to obtain a function coming from the inversion of an unstable problem, and use it to calculate some global quantities by integrating it weighted by the appropriate weighting functions. When the desired function comes from a first kind Volterra integral equation, the explicit inversion can be avoided by integrating by parts in the integrals in which the above mentioned function appears. That is the case of the fundamental parameters method of x-ray fluorescence analysis. To obtain the concentrations of chemical elements in the sample which is analyzed it is necessary to calculate some integrals of the spectral distribution of the fluorescence exciting x-ray beam multiplied by a weighting function which depends on the concrete analysis to be done. The spectral distribution of the fluorescence exciting beam is related to the experimental measurements of the fluorescence excited on a set of targets made up of pure elements by a Volterra integral equation of the first kind, and it can be obtained by inverting the Volterra equation. By integrating by parts in the integrals in which the spectral distribution appears we avoid the unstable reconstruction of the spectrum of the fluorescence exciting x-ray beam and the concentrations can be calculated in a stable fashion.