Stability and Convergence Issues of a Nonlocal Explicit Finite-Difference Scheme for the Numerical Analysis of the Fractional Zeeman Oscillator
DOI:
https://doi.org/10.56143/ujmcs.v2i2.3/Keywords:
fractional oscillator; finite-difference scheme; stability; convergence; numerical simulation; fractional calculusAbstract
This paper investigates the stability and convergence of a nonlocal explicit finite-difference scheme designed for the numerical solution of the Cauchy problem for the fractional Zeeman oscillator. This model is a generalization of the classical Zeeman model, which describes the contraction of heart muscle fibers, and employs the apparatus of fractional calculus to account for hereditary effects and memory. Fractional derivatives are understood in the Gerasimov--Caputo sense, and an explicit L1-type scheme is used for their approximation.
During the theoretical analysis, stability conditions for the proposed scheme were obtained, its first-order convergence in time was proved, and an explicit restriction on the time step ensuring computational stability was established. To confirm the theoretical conclusions, a computer program FracDynZe was developed in Python, allowing for the visualization of oscillograms and phase trajectories of the system. Numerical experiments based on the Runge rule confirm the first-order accuracy of the algorithm. The simulation results demonstrate that the fractional Zeeman oscillator can reproduce various dynamic modes, including self-oscillations with convergence to a stable limit cycle, which may be useful for analyzing cardiac pathologies such as arrhythmia and tachycardia.
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