Fredholm Integral Equation and Splines of the Fifth Order of Approximation

Abstract

This paper considers the numerical solution of the Fredholm integral equation of the second kind using local polynomial splines of the fifth order of approximation and the fourth order of approximation (cubic splines). The basis splines in these cases occupy five and four adjacent grid intervals respectively. Different local spline approximations of the fifth (or fourth) order of approximation are used at the beginning of the integration interval, in the middle of the integration interval, and at the end of the integration interval. The construction of the calculation schemes for solving the Fredholm equation of the second kind with these splines is considered. The results of the numerical experiments on the approximation of functions and on the solution of the Fredholm integral equations are presented. The results of the solution of the integral equation which uses the polynomial splines of the fifth order of approximation are compared with ones obtained with cubic splines and with the application of the Simpson’s method. Note that in order to achieve a given error using the approximation with quadratic splines, a denser grid of nodes is required than when using the approximation with the cubic splines or splines of the fifth order of approximation.