Algorithm XXXX: FdeSolver, a Julia Package for Solving Fractional Differential Equations

Abstract

We introduce FdeSolver, an open-source Julia package designed to solve fractional-order differential equations efficiently. The available solutions are based on product-integration rules, predictor–corrector algorithms, and the Newton-Raphson method. The package covers solutions for one-dimensional equations with orders of positive real numbers. For higher-dimensional systems, it supports orders up to one. Incommensurate derivatives are allowed and defined in the Caputo sense. Here, we summarize the implementation for a representative class of problems and compare it with available alternatives in Julia and MATLAB. Moreover, FdeSolver leverages the power and flexibility of the Julia environment to offer enhanced computational performance, and our development emphasizes adherence to the best practices of open research software. To highlight its practical utility, we demonstrate its capability in simulating microbial community dynamics and modeling the spread of COVID-19. This latter application involves fitting the order of derivatives grounded on real-world epidemiological data. Overall, these results highlight the efficiency, reliability, and practicality of the FdeSolver Julia package.