Splines are an important mathematical tool in Applied and Theoretical Mechanics. Several
Problems in Mechanics are modeled with Differential Equations the solution of which demands Finite Elements
and Splines. In this paper, we consider the construction of computational schemes for the numerical solution of
integral equations of the second kind with a weak singularity. To construct the numerical schemes, local
polynomial quadratic spline approximations and second-order nonpolynomial spline approximations are used. The
results of the numerical experiments are given. This methodology has many applications in problems in Applied
and Theoretical Mechanics.