A SOFTWARE TOOL FOR SOLVING A FRACTIONAL-ORDER DIFFERENTIAL EQUATION: A NUMERICAL EXPERIMENT BASED ON THE MITTAG-LEFFLER FUNCTION AND THE EULER METHOD

Abstract

A software tool has been developed for the simplest linear equation with a Caputo fractionalorder derivative, CDαy = −λy. In a window based on Python and Tkinter, the user enters the values of α, λ and T, and the program computes the exact solution via the Mittag-Leffler series and plots its graph. Together with the tool, the error of the fractional-order Euler method was measured as a function of the step h. For α = 0.5, 0.7 and 0.9 the method converges with first order; for α = 0.3 the order of the maximum error remains around 0.55, because the error accumulates near the initial point. If the step is large, the explicit scheme becomes unstable: for α = 0.5, starting from λhα/Γ(α+1) ≈ 1.4 the solution grew without bound. It is shown that summing the series in ordinary floating-point arithmetic is unreliable for large λtα and that the program warns about this. The computations were performed with Python 3.12, NumPy and mpmath; the codes are included in the article.