Stability and Convergence of a Nonlocal Explicit Finite-Difference Scheme for Numerical Solution of the Fractional Van der Pol--Airy Equation

Abstract

This paper presents a study of a nonlocal explicit finite-difference scheme for the numerical solution of the Cauchy problem for the fractional Van der Pol--Airy oscillator, which describes forced oscillations with memory effects. The study focuses on the stability and convergence of the proposed numerical scheme. These properties are rigorously established using the properties of the scheme's coefficients, the approximation of fractional derivative operators, and discrete Volterra inequalities. The conditional convergence and stability of the numerical scheme are proved. The scheme is shown to have first-order accuracy, provided that a condition on the time step of the computational grid is satisfied, which is also derived.