Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential Equations

First-order ordinary differential equations (ODEs) are widely used in various fields of science and engineering to model natural phenomena. This study proposes an extension of these equations using fractional derivatives, specifically the Grünwald-Letnikov definition, to explore their impact on the behavior of solution curves. The fractional ODE considered is discretized using the finite difference method and solved numerically for different values of the derivative order (α). Tests were conducted to verify mesh independence and the quality of the computational implementation of the method, through which the accuracy and the absence of implementation errors were confirmed. The behavior of the solution curves for different values of α was analyzed, revealing a sharp decrease near the initial point (t = 0) and an almost linear growth at higher values of t, within the considered domain. Additionally, in solving a specific initial value problem with a known analytical solution, it was discovered that the accuracy of the numerical solutions for higher values of α was more dependent on the mesh refinement than the solutions for lower values.