The numerical method of successive interpolations for two-point boundary value problems with deviating argument

A new numerical method for two-point boundary value problems associated to differential equations with deviating argument is obtained. The method uses the fixed point technique, the trapezoidal quadrature rule, and the cubic spline interpolation procedure. The convergence of the method is proved without smoothness conditions, the kernel function being Lipschitzian in each argument. The interpolation procedure is used only on the points where the argument is modified. A practical stopping criterion of the algorithm is obtained and the accuracy of the method is illustrated on some numerical examples of the pantograph type.