A new Abel inversion algorithm by using legendre polynomials

Abstract

Abel’s integral equation is frequently used in many areas of physics to reconstruct the radial physical quantity distribution from its projection data. In this paper, a new effective and accurate Abel inversion algorithm based on shifted Legendre polynomials is proposed and analyzed. The proposed method is derivative-free and singularity-free. Both the input projection data and the unknown solutions of the Abel’s integral equation are approximately expressed as Legendre expansions. A Legendre operational matrix of integral is constructed and then reduced to a discrete algebraic sum, which makes it easy and fast to compute the coefficients matrix of approximate solutions for the inverse Abel transform. Finally, the accuracy and stability are proved and then illustrated by some numerical experiments widely used in plasma diagnostics.