On the maximal order of convergence of Green's function method for solving two-point boundary value problems with deviating argument

The Green's function method is applied to second, third, and fourth order two-point boundary value problems with deviating argument. At each iterative step, the Picard-Green's method is combined with suitable quadrature rule and interpolation procedures. The Hermite type cubic spline is applied to second and third order boundary value problems, while complete cubic splines are used at fourth order boundary value problems. The corrected trapezoidal quadrature rule is involved at third and fourth order boundary value problems with appropriate error bounds. The convergence of the method was proved, and the maximal order of convergence is obtained for each problem. The theoretical results are tested on some numerical examples.