This paper is concerned with the approximate solution of second order differential equations where is a non zero constant and is a sufficiently smooth function, supplemented with linear integral boundary conditions of type: with positive constants and given smooth functions .
The proposed approach starts establishing an homotopy defined by family of differential equations , with the parameter so that for gives a linear equation such that with the boundary conditions has a unique solution easily computed and for the parameter value is the desired solution of the non linear problem. Now by using as a small parameter the solution of can be written as an asymptotic series where the successive can be computed recursively as a solution linear problems and then the solution for is approximated by the -sum . For solving each linear boundary value problem of the authors propose a reproducing kernel Hilbert space method. Two numerical experiments are presented to show the behaviour of the method depending on the terms of the series and the number of grid points in the interval