A modified linear approximation for weakly singular integrals

In this work, we present a new approximation for a weakly singular integral, in particular Abel's integral. This approximation is based on the modification of the linear spline function, this one leads to eliminate the weak singularity. Noting that, it is clear that in the future we use this approximation for solving numerically all weakly singular integrals equations on an oriented smooth curve or on an interval.