The number of extrema of the error function of a class of methods for differential equations

We give sharp estimates for the number of extrema of the error of approximation of functions implicitly defined by linear and nonlinear differential equations, and by systems of differential equations when the latter are approximated using the Tau Method. Our estimates are deduced using the concept of length of a maximal monotone partition relative to a given function. This work is related to interesting previous work of Pittnauer [1] and to more recent work of El-Daou and Ortiz [2]. Our estimation techniques use explicitly analytical results from the Tau Method, relating the approximation error to the perturbation term. Recent work of El-Daou and Ortiz [3,4] and El-Daou, Ortiz and Samara [5], showed the possibility of simulating with the Tau Method, through a suitable choice of the perturbation term, of a variety of other, apparently diverse, numerical techniques. Among them Galerkinʹs method, polynomial expansion techniques, spectral methods, collocation and finite difference methods. Therefore, although our results are formulated in the language of the Tau Method, they are immediately applicable to a wider class of numerical methods for the approximate solution of differential equations.