Numerical analysis of the Cauchy problem for a wide class fractal oscillators

R. I. Parovik

Vestnik KRAUNC. Fiziko-matematičeskie nauki, no. 1 (2018), pp. 93-116

Résumé

The Cauchy problem for a wide class of fractal oscillators is considered in the paper and its numerical investigation is carried out using the theory of finite-difference schemes. Fractal oscillators characterize oscillatory processes with power memory or, in general, with heredity, and are described by means of integro-differential equations with difference kernels—memory functions. By choosing memory functions as power functions, integrodifferential equations are reduced to equations with derivatives of fractional orders. In this paper, using the approximation of the fractional derivatives of Gerasimov–Kaputo, a non-local explicit finite-difference scheme was developed, its stability and convergence are justified, estimates of the numerical accuracy of computational accuracy are presented. Examples of the work of the proposed explicit-finite scheme are given. It is shown that the order of computational accuracy tends to unity as the number of grid nodes increases and coincides with the order of approximation of the explicit finite difference scheme.

Détail

Keywords: Cauchy problem, fractal oscillators, hereditary, Gerasimov–Caputo operator, numerical scheme, stability, Runge rule, Cauchy problem, fractal oscillators, hereditary, Gerasimov–Caputo operator, numerical scheme, stability, Runge rule.
Mots-clés : convergence, convergence