  We propose an efficient and flexible method for solving Abel integral
equation of the first kind, frequently appearing in many fields of
astrophysics, physics, chemistry, and applied sciences. This equation
represents an ill-posed problem, thus solving it requires some kind of
regularization. Our method is based on solving the equation on a so-called
compact set of functions and/or using Tikhonov's regularization. A priori
constraints on the unknown function, defining a compact set, are very loose and
can be set using simple physical considerations. Tikhonov's regularization on
itself does not require any explicit a priori constraints on the unknown
function and can be used independently of such constraints or in combination
with them. Various target degrees of smoothness of the unknown function may be
set, as required by the problem at hand. The advantage of the method, apart
from its flexibility, is that it gives uniform convergence of the approximate
solution to the exact solution, as the errors of input data tend to zero. The
method is illustrated on several simulated models with known solutions. An
example of astrophysical application of the method is also given.
