Stable inversion of the Abel integral equation of the first kind by means of orthogonal polynomials

In this paper, we describe two stable methods for the inversion of the Abel integral operator of the first kind. The first method is based on the use of appropriate families of orthonormal polynomials of Jacobi type that constitute orthonormal bases of the L2([0, 1])-space. We prove that this first method for solving the ill-posed problem described by the first kind of Abel integral equation is stable. This stability is proved by using some properties of our orthogonal polynomials as well as using a convenient weighted subspace of L2([0, 1]). The second method is based on expanding the solution with respect to the basis of Legendre polynomials. More precisely, by using a special technique for the computation of the Legendre expansion coefficients of the solution, we prove that our second method is stable under weak conditions on the smoothness of the solution as well as on the data function. Also, we give a brief review of a modified Thikhonov method for solving the ill-posed Abel problem and we show how our second proposed method fits in the modified Tikhonov regularization scheme. Finally, we provide the reader with some numerical tests that illustrate the results of this work.