Local Splines of the Zero and First Degrees, and the Solution of Integral Equation

This paper discusses the features of using local splines of the first and second orders of approximation for solving Fredholm integral equations of the second kind. Note that basis splines with “narrow” support (i.e. if the length of the support of the basis spline is equal to one or two grid intervals) do not give a boundary layer when we construct the approximation on a finite interval. In addition, such splines are more convenient when building a non-uniform adaptive grid. The local splines of the first and second orders of approximation allow us to construct approximations and solve the integral equations on a non-uniform grid. The solution of the Fredholm integral equations of the second kind with a weak singularity is discussed too.