Unified scientific tool to investigate fractional derivatives of arbitrary variable order with time-memory and order-memory: The VOFD Python package

Fractional derivatives of arbitrary order may enhance several practical applications in science and engineering, ranging from physics, chemistry, economics, and botany to robotics, neural networks, data encryption, and internet of things (IoT). In such variable order (VO) fractional derivatives, the memory property changes as a function of time and order. However, despite a strong mathematical background, there are no software tools dedicated to exploiting the distinctive properties of VO derivatives. This is mainly due to the difficulty of capturing the complex properties of VO derivatives by a suitable computational method for numerical simulations. Therefore, this tutorial paper introduces a simple open-source Python library (VOFD Python package which can be downloaded from http://pypi.org/project/vofd/) that implements numerical integration schemes based on finite-difference approximations to solve Caputo VO derivatives (V1) and two convolution-based Caputo VO derivatives (V2 and V3). In addition, the proposed package includes a subroutine for generating bifurcation diagrams. Step-by-step examples for the Riccati equation and Chen system, along with error and convergence analyses, are provided to demonstrate the benefits of the proposed tool. The variable-order fractional derivative (VOFD) package provides high-performance numerical routines accelerated with the Numba JIT compiler, significantly reducing computation time for numerical solutions and large-scale bifurcation analysis, enabling efficient exploration of variable-order fractional models by both experts and practitioners.