New stable numerical solutions of singular integral equations of Abel type by using normalized Bernstein polynomials

Abstract

A new numerical method, based on the normalized Bernstein polynomials for solving singular integral equations of Abel type is presented here in this paper. We construct an othonormal family {bin}i=0n of polynomials of degree n from the nth degree Bernstein polynomials Bin and use them as a basis to approximate the known and unknown functions f(x) and (x) respectively in the Abel's integral equations. Then orthogonality is used to reduce the integral equation to a system of algebraic equations which can be solved easily. The method is quite accurate and stable even when the approximations are performed by orthonormal Bernstein polynomials bin of degree as low as 5, as illustrated by the given numerical examples with varying degree of noise terms ε added to f(x).