Problemi al contorno per equazioni del tipoy″=f(x,y) trattati come equazioni integrali con metodi alle differenze finite di elevata accuratezza

This work faces the problem of the numerical treatment on digital computers of boundary value problems of the type: $$y'' = f(x,y); y(a) = A,y(b) = B$$ through the reduction into the equivalent integral equation: $$y(x) = mathop smallint limits_a^b g_K (x,xi )[K^2 y(xi ) - f(xi ,y(xi ))]dxi $$ whereg _K (x,ξ) is the Green function associated to the differential operator (frac{{d^2 }}{{dx^2 }} - K^2 ) . I have extended to this problem a discrete analogue of higher accuracy introduced in [1] with reference to a boundary value problem analised under a differential point of view: this extention costitutes the original part of the work. The above problem is analysed with reference to the discretization error and the convergence of the discrete analogue solution algorithm; the actnal numerical treatment of a few systems follows.