Spline Collocation for Nonlinear Fredholm Integral‎ ‎Equations

The collocation method based on cubic B-spline‎, ‎is‎ ‎developed to approximate the solution of second kind nonlinear‎ ‎Fredholm integral equations‎. ‎First of all‎, ‎we collocate the‎ ‎solution by B-spline collocation method then the Newton-Cotes‎ ‎formula use to approximate the integrand‎. ‎Convergence analysis‎ ‎has been investigated and proved that the quadrature rule is‎ ‎third order convergent‎. ‎The presented method is tested with four‎ ‎examples‎, ‎and the errors in the solution are compared with the‎ ‎existing methods [1‎, ‎2‎, ‎3‎, ‎4] to verify the accuracy and‎ ‎convergent nature of proposed methods‎. ‎The RMS errors in the‎ ‎solutions are tabulated in table 3 which shows that our method‎ ‎can be applied for large values of n‎, ‎but the maximum $n$ which‎ ‎has been used by the existing methods are only n=10, ‎moreover‎ ‎our method is accurate and stable for different values of n.