Solution of the interval equations of dynamics by using adaptive approximation

In this paper a new method for solution of the interval equations of dynamics will be presented. In this approach, in order to estimate upper (y) and lower bound (y) of the solution y = y(t, p) it is necessary to create approximation y^approx = y^approx(t, p₀, ..., p_n, p) (p₀, ..., p_n are some point values, and y ≈ y^approx). Then this approximation can be applied for calculation of y^approx and y^approx. It is also possible to get reliable inner estimation y^inner and y^inner of the solution. Using the differences y^approx - y^inner, y^approx - y^inner it is possible to control accuracy of the calculations. This method gives the possibility to calculate combinations of parameters (p_approx^min(t), p_approx^max (t), p_inner^min(t), p_inner^max (t)), which generate both interval solutions (i.e. y^approx = y(t, p_approx^min (t)), y^approx = y(t, p_approx^max (t)), y^inner = y(t, p_inner^min(t)), y^inner = y(t, p_inner^max (t))). This is very useful in applications of the interval methods.