Solving system of nonlinear Abel's integral equations in frequency domain

In this paper, we solve both linear and nonlinear systems of Abel’s integral equations by combining the Laplace transform with Bernstein polynomials. In the proposed approach, the unknown functions are approximated using Bernstein polynomials. The Laplace transform is then applied to convert the equations from the time domain to the frequency domain. This results in a nonlinear system in terms of unknown coefficients, which is solved using Newtons iterative method. The unknown function is thus approximated. To illustrate the efficacy of the proposed method, we present detailed analytical and numerical examples.