FractionalDiffEq.jl: High Performance Fractional Differential Equation Solver in Julia

TL;DR

FractionalDiffEq.jl is a high-performance, user-friendly Julia package for solving various fractional differential equations using advanced numerical algorithms, validated through extensive benchmarks and real-world applications.

Contribution

This paper introduces FractionalDiffEq.jl, a comprehensive Julia package that unifies multiple fractional differential equation solvers with high efficiency and scalability.

Findings

Outperforms existing implementations in benchmarks

Successfully applied to real-world scientific problems

Demonstrates robustness and flexibility in diverse scenarios

Abstract

We present FractionalDiffEq.jl, a comprehensive solver suite for solving fractional differential equations, featuring high-performance numerical algorithms in the Julia programming language. FractionalDiffEq.jl is designed to be user-friendly and scalable, tackling different types of fractional differential equations, encompassing powerful numerical algorithms including predictor-corrector methods, product-integral methods, and linear multistep methods, etc, and providing a unifying API to accommodate diverse solver features. This paper illustrates the convenient usage of FractionalDiffEq.jl in modeling various scientific problems, accompanied by detailed examples and applications. FractionalDiffEq.jl leverages best practices in Julia to ensure the high performance of numerical solvers. To validate the efficiency of FractionalDiffEq.jl , we conducted extensive benchmarks that prove the superiority of FractionalDiffEq.jl against other implementations on both stiff and non-stiff problems. We further demonstrate its capability on several challenging real-life scenarios including parameter estimation in fractional-order tequila fermentation processes, and harmonic oscillator problems, etc, emphasizing the robustness and flexibility of FractionalDiffEq.jl.