Bernstein Polynomials for Solving Some Classes of Abel’s Integral Equations with Weakly-Singular Kernels

Abstract

Most physical phenomena are described by differential equations, such equations are frequently hard to solve in a direct way, they are commonly converted into integral equations, which can then be tackled with simpler and more efficient methods. This paper aims to numerically obtain numerical solutions of the linear main generalized Abel’s integral equations (weakly-singular kernel) of both first and second kinds using the Bernstein polynomials where the unknown function is approximated in terms of such polynomials. Then relying on some properties of the Bernstein polynomials, the considered integral equation is converted into a linear system of algebraic equations, that can be easily solved to obtain the coefficients of expansion. To evaluate its performance, some examples were presented comparing the numerical solutions with the exact ones. These comparisons showed that, even for low-degrees Bernstein polynomials, the approximate solution agrees very well with the exact one. The obtained results confirm the method's effectiveness and dependability for solving integral equations with weakly-singular kernels. Additionally, the numerical findings ensure the analytical theorems concerning existence, uniqueness, and convergence of the numerical solution.