Application of the second-order spline approximations to solving integral equations of the second kind

Local interpolation splines are used to solve various problems such as interpolation, approximation, the solving of boundary value problems and the solving of integral equations. Polynomial piecewise linear splines have been known for a long time and are a special case of local continuous splines of maximum defect. Non-polynomial local splines were constructed and studied by the authors earlier. The approximation of a function with polynomial or non-polynomial splines is constructed on each grid interval separately as a sum of products of basis splines and values of the function at the grid nodes. We find the formulas of the basis splines in analytical form by solving a system of approximation identities. The paper discusses a numerical method for solving a Fredholm integral equation of the second kind. This method is based on the use of polynomial or non-polynomial splines of the second order of the approximation. The main idea of the proposed method is based on calculating integrals of the product of the kernel and the basis functions. The best result will be obtained if we assume that the function (that is the solution of the integral equation) is a twice differentiable function. When comparing the method under discussion with the well-known numerical methods (for example, methods based on the use of composite quadrature rules of trapezoids and midpoint rectangles) for solving integral equations, it can be noted, that the proposed method in this paper can give a significantly smaller calculation error. The calculation error can be reduced due to the fact that in many cases the integrals can be calculated exactly. When the integrals can't be calculated exactly then various quadrature formulas can be used to calculate these integrals. The advantages of using local polynomial and non-polynomial splines of the second order of approximation should also include the convenience of their application on a non-uniform grid of nodes. The results of the numerical solution of linear and nonlinear integral Fredholm equations are presented.