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FdeSolver: A Julia Package for Solving Fractional Differential Equations

Moein KhalighiGiulio BenedettiLeo Lahti
Dec 2022
摘要
Implementing and executing numerical algorithms to solve fractionaldifferential equations has been less straightforward than using theirinteger-order counterparts, posing challenges for practitioners who wish toincorporate fractional calculus in applied case studies. Hence, we created anopen-source Julia package, FdeSolver, that provides numerical solutions forfractional-order differential equations based on product-integration rules,predictor-corrector algorithms, and the Newton-Raphson method. The packagecovers solutions for one-dimensional equations with orders of positive realnumbers. For high-dimensional systems, the orders of positive real numbers arelimited to less than (and equal to) one. Incommensurate derivatives are allowedand defined in the Caputo sense. Here, we summarize the implementation for arepresentative class of problems, provide comparisons with availablealternatives in Julia and Matlab, describe our adherence to good practices inopen research software development, and demonstrate the practical performanceof the methods in two applications; we show how to simulate microbial communitydynamics and model the spread of Covid-19 by fitting the order of derivativesbased on epidemiological observations. Overall, these results highlight theefficiency, reliability, and practicality of the FdeSolver Julia package.
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