A singularity free and derivative free approach for Abel integral equation in analyzing the laser-induced breakdown spectroscopy

Abstract In order to make a detailed analysis of the laser-induced breakdown spectroscopy (LIBS), we present a new approach which has advantages of singularity free and derivative free for solving the Abel integral equation. After presenting the model describing the LIBS, a collocation method is applied and two main steps are introduced. First, the Abel integral equation is transformed to a singularity free and derivative free integral equation. Then, by expanding the solution of the integral equation to a polynomial of degree n, the solution can be approximated by a linear system established by the collocation method. Our approach is suitable for dealing with LIBS problem for any form of the measured intensity. In particular, the method is insensitive to the noisy intensity data. A convergence analysis is also given to show the validity of the method and numerical examples are provided to show its efficiency and accuracy.