Numerical Modeling Tools and S-derivatives

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