A collocation method for solving nonlinear Volterra integro-differential equations of the neutral type by sigmoidal functions

A numerical collocation method is developed for solving nonlinear Volterra integro-differential equations (VIDEs) of the neutral type, as well as other non-standard and classical VIDEs. A sigmoidal functions approximation is used to suitably represent the solutions. Special computational advantages are obtained using unit step functions, and important applications can be obtained also using other sigmoidal functions, such as logistic and Gompertz functions. The method allows to obtain a {\em simultaneous} approximation of the solution to a given VIDE and its first derivative, by means of an explicit formula. 'A priori' as well as 'posteriori' estimates are derived for the numerical errors, and numerical examples are given for the purpose of illustration. A comparison is made with the classical piecewise polynomial collocation method as for accuracy and CPU time.