Numerical Modeling Tools and S-derivatives

Abstract

Numerical study of various processes leads to the need of clarification (extensions) of the limits of applicability of computational constructs and modeling tools. For dynamical systems, this question may be related with a generalization of the concept of a derivative, which keeps the used constructions relevant. In this article we introduce the concept of weak local differentiability in a space of Lebesgue integrable functions and consider the consistency of this concept with such fundamental computational constructions as the Taylor expansion and finite differences, as well as properties of functions with a given type of differentiability on a segment. The function f from L₁[a; b] is called S-differentiable at the point x₀ from (a; b), if there are coefficients c and q, for which fx₀x₀+h (f (x) - c - q·(x-x₀)) dx = o(h²). Formulas are found for calculating the coefficients c and q, coefficients c and q, which are conveniently denoted fₛ(x₀) and fₛ ˊ(x₀) respectively. It is shown that if the function f belongs to W₁ⁿ⁻¹[a; b], n is greater than 1, and the function f⁽ⁿ⁻¹⁾ is S-differentiable at the point xₒ from (a; b), then f is approximated by a Taylor polynomial with accuracy o((x-xₒ)ⁿ), and the ratio of Δⁿₕ(f, xₒ) to hⁿ tends to fₛ⁽ⁿ⁾(xₒ) as h tends to 0. Based on the quotient Δⁿₕ (f, ·) and hⁿ, a sequence is built {Ʌₘⁿ [f]} piecewise constant functions subordinate to partitions segment [a; b] into m equal parts. It is shown that for the function f from W₁ⁿ⁻¹[a; b], for which the value is defined f ₛ⁽ⁿ⁾(xₒ), { Ʌₘⁿ [f] (xₒ)} converges to f ⁽ⁿ⁾(xₒ) as m tends to infinity, and for f from Wₚⁿ[a; b] the sequence { Ʌₘⁿ [f] } converges to f⁽ⁿ⁾ in the norm of the space Lₚ [I]. The place of S-differentiability in practical and theoretical terms is determined by its bilateral relations with ordinary differentiability. It is proved that if f belongs to W₁ⁿ⁻¹[I] and the function f⁽ⁿ⁻¹⁾ is uniformly S-differentiable on I, then f belongs to Cⁿ[f]. The constructions are algorithmic in nature and can be applied in numerically computer research of various relevant models.