Applied Numerical Mathematics 62 (2012) 567–579 Contents lists available at SciVerse ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum A stable numerical inversion of generalized Abel’s integral equation Sandeep Dixit, Om P. Singh ∗ , Sunil Kumar Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India article info abstract Article history: Received 25 April 2011 Received in revised form 12 November 2011 Accepted 30 December 2011 Available online 20 January 2012 Keywords: Abel inversion Bernstein polynomials Almost Bernstein operational matrix of integration Noise resistance A direct almost Bernstein operational matrix of integration is used to propose a stable al- gorithm for numerical inversion of the generalized Abel integral equation. The applicability of the earlier proposed methods was restricted to the numerical inversion of a part of the generalized Abel integral equation. The method is quite accurate and stable as illustrated by applying it to intensity data with and without random noise to invert and compare it with the known analytical inverse. Thus it is a good method for applying to experimental intensities distorted by noise.  2012 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction Recently, Chakrabarti [6] employed a direct function theoretic method to determine the closed form solution of the following generalized Abel integral equation a( y) y  α r γ −1 ε(r ) dr ( y γ − r γ ) μ + b( y) β  y r γ −1 ε(r ) dr (r γ − y γ ) μ = I ( y), 0 < μ < 1, α  y  β, (1) where the coefficients a( y) and b( y) do not vanish simultaneously. Earlier the generalized Abel equation (1) was examined in Gakhov’s book [10], under the special assumptions that the co- efficients a( y) and b( y) satisfy Holder’s condition in [α,β ], whereas the forcing term (data function) I ( y) and the unknown function (emissivity) ε( y) belong to those class of functions which admit representations of the form I ( y) =  ( y − α)(β − y)  δ I ∗ ( y), and ε( y) = ε ∗ ( y) [( y − α)(β − y)] 1−μ−δ y γ −1 , δ> 0, (2) where I ∗ ( y) possesses a Holder continuous derivative in [α,β ] and ε ∗ ( y) satisfies Holder’s condition in [α,β ]. The method of Gakhov has a particular disadvantage in the sense that while solving a singular equation that involves integrals only with weak singularity of the type (r − y) −μ (0 < μ < 1), occurrence of strongly singular integrals involving Cauchy type singularities of the type (r − y) −1 has to be permitted [6,10]. Chakrabarti [6] obtained the solution involving only weakly singular integrals of Abel type and thus Cauchy type singular integrals were avoided. But the numerical inver- sion is still needed for its application in physical models since the experimental data for the intensity I ( y) is available only at a discrete set of points and it may also be distorted by noise. * Corresponding author. E-mail address: singhom@gmail.com (O.P. Singh). 0168-9274/$36.00  2012 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2011.12.008