Analytical-approximate solution of Abel integral equations

It is known that Abel integral equation has a solution in a closed form, with a removable singularity. The presence of Volterra inte-grals with weak singularity is not always integrable for continuous dif-ferentiable class of functions. In this work we propose an analyti-cal approximate method for the solution of Abel integral equations. We showed that the proposed method is exact for the known func-tion in the cases of polynomials and irrational function of the form f(t) = tα+1(a0+ a1t+ · · ·+ antn), 0 < α < 1. For the derivation of the proposed method we expand the known function to the Taylor series around a singular point s. Substituting this expansion into the solution of Abel equation we could remove the singularity. All evaluations of the integrals are calculated analytically. The obtained solution is a series that is uniformly convergence to the exact solution.

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