Nonstandard finite difference schemes for solving systems of two linear fractional differential equations with complex eigenvalues

Abstract

This paper presents a novel nonstandard finite difference (NSFD) scheme for solving a system of linear Caputo-type fractional differential equations (FDEs) characterized by complex eigenvalues. First, the analytical solutions for a two-dimensional linear FDE system are derived. The proposed numerical method is then constructed by employing complex Mittag-Leffler functions for establishing numerator and denominator functions within the NSFD framework. A rigorous convergence analysis is conducted, establishing the scheme’s consistency and stability. It is proven that the method is unconditionally stable provided the real parts of the system’s eigenvalues are negative. The applicability of the scheme is extended to fractional harmonic oscillators by reformulating them as equivalent systems of FDEs with complex eigenvalues. The efficacy and accuracy of the method are demonstrated through four numerical examples. The solutions exhibit the expected damped oscillatory behavior. The results show exceptional accuracy, with errors on the order of 10^-15 or less, even when using relatively large step sizes, demonstrating the robustness and computational efficiency of the proposed scheme.