A Numerical solution to the linear and nonlinear Fredholm integral equations using Legendre wavelet functions

Two methods for solving the Fredholm integral equation of the second kind in linear case, i.e. f ( x ) – λ ∫ a b K ( x,y ) f ( y ) dy = g ( x ), and nonlinear case, i.e., f ( x ) = g ( x ) + λ ∫ a b K ( x,y ) F ( f ( y )) dy , are proposed. In order to solve the linear equation, the kernel K ( x,y ) as well as the functions f and g are initially approximated through Legendre wavelet functions. This leads to a system of linear equations its solution culminates in a solution to the Fredholm integral equation. In nonlinear case only K ( x,y ) is approximated by Legendre wavelet base functions. This leads to a separable kernel and makes it possible to employ a number of earlier methods in solving nonlinear Fredholm integral equation with separable kernels. Another feature of the proposed method is that it finds the solution as a function instead of specific solution points, what is done by the majority of the existing methods. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)