APPROXIMATE SOLVING OF THE FREDHOLM INTEGRAL EQUATION OF SECOND KIND

The convergence order of the numerical solution, of Fredholm integral equations of the second kind, obtained by substitution kernel methods depends on the possibility of solving the substitute equation in a simple way. In this paper, we develop a modification which increases the quality of the approximating substitution kernels considerably by using blending-splines. In this way, we are led to procedures of extremely high order of convergence. In [1] a method was given which presents how to construct degenerate substitution kernels by interpolation. Thos method allowed the authors to build the explicit solution of the substitute equation. Error estimates for the eigenvalues approximations and for the eigenfunctions of the original integral equation in the homogeneous case could also be found using the classical results. The originality of the paper is that we developed the method presented in [1] in a more general framework by using two-dimensional splines, generated by forming tensor products of the common one-dimensional splines. Several generalizations were then than possible to be demonstrated too using proprieties related to the dimension of approximating spaces. The use of approximating instead of interpolating splines is important in calculating the order of convergence of the numerical solution. In addition, we can extend the error estimates to integral equations with kernels possessing only low differentiability properties. This plays a role, for example, in the case of kernels originated as Green's functions([2],[3]).