Fractional Differential Equation Solver Benchmarks

This repository contains a cross-language benchmark suite for fractional ordinary differential equations (FODEs). It compares runtime and numerical error across Julia, MATLAB, and Python implementations on several representative problems:

• Single-term linear FODE
• Single-term nonlinear FODE
• Stiff FODE system
• Multi-term FODE

The benchmark outputs are stored as CSV files and then aggregated into publication-style performance profiles (time vs error) using plotting scripts.

What Is Benchmarked

Each benchmark sweeps the same step-size family:

• dt = 2^(-3), 2^(-4), 2^(-5), 2^(-6), 2^(-7)

For every solver and dt:

• Runtime is measured (wall-clock time)
• Error is computed against an analytic or manufactured reference solution
• One CSV file is generated per solver

Most CSV files follow the two-column convention:

• time: execution time in seconds
• error: numerical error norm

Repository Layout

Core benchmark scripts are under benchmarks/benchmark_fode:

• benchmark_scipt_Julia: Julia benchmark generators
• benchmark_scipt_MATLAB: MATLAB benchmark generators and solver functions
• benchmark_scipt_Python: Python benchmark generators (pycaputo)

Generated benchmark data is under:

• benchmarks/data

Cross-language plotting and comparison scripts are under:

• benchmarks/fode_wpd_*.jl

Detailed File Guide

1) Julia benchmark generators

Location: benchmarks/benchmark_fode/benchmark_scipt_Julia

• xue_benchmarks.jl
  • Noncommensurate 3D nonlinear system benchmark
  • Exports: FdeSolver_PECE, FractionalDiffEq_PECE, FractionalDiffEq_PITrap, FractionalDiffEq_PIRect
• linear_singleterm_benchmarks.jl
  • Single-term linear model with Mittag-Leffler reference
  • Exports: Linear_Single_* CSV series
• nonlinear_singleterm_benchmarks.jl
  • Single-term nonlinear manufactured-solution benchmark
  • Exports: Single_* CSV series
• stiff_benchmarks.jl
  • Stiff 3D coupled system with manufactured forcing and analytic solution
  • Exports: Stiff_* CSV series
• multiterms_benchmarks.jl
  • Multi-term Caputo benchmark (MTPECE/MTPIEX/MTPITrap/MTPIRect)
  • Exports: FractionalDiffEq_MT* CSV series
• large_benchmarks.jl, fdesovler_on_stiff.jl
  • Additional or exploratory benchmark scripts
• Project.toml
  • Present as local environment marker (currently empty)

2) MATLAB benchmark generators and local solvers

Location: benchmarks/benchmark_fode/benchmark_scipt_MATLAB

Main benchmark scripts:

• xue_benchmarks.m
• linear_singleterm_benchmarks.m
• nonlinear_singleterm_benchmarks.m
• stiff_benchmarks.m
• multiterms_benchmarks.m

Local method implementations used by the scripts include:

• fde_pi1_ex.m, fde_pi12_pc.m, fde_pi1_im.m, fde_pi2_im.m
• mt_fde_pi1_ex.m, mt_fde_pi12_pc.m, mt_fde_pi1_im.m, mt_fde_pi2_im.m
• flmm2.m, ml.m
• MFC2demo3c directory and data directory for supporting routines/data

3) Python benchmark generators

Location: benchmarks/benchmark_fode/benchmark_scipt_Python

• benchmark.py
  • Noncommensurate 3D nonlinear benchmark using pycaputo PECE
  • Exports: PYCAPUTO_PECE.csv
• linear_singleterm_benchmark.py
  • Linear single-term benchmark (uses FractionalDiffEq.mittleff via python-julia bridge)
  • Exports: Linear_Singleterm_PYCAPUTO_PECE.csv
• nonlinear_singleterm_benchmark.py
  • Nonlinear single-term manufactured benchmark
  • Exports: Singleterm_PYCAPUTO_PECE.csv

Plotting and Cross-Framework Comparison Scripts

Location: benchmarks

• fode_wpd_nonstiff.jl: non-stiff noncommensurate comparison
• fode_wpd_stiff.jl: stiff comparison
• fode_wpd_singleterm.jl: single-term comparison
• fode_wpd_linear_singleterm.jl: linear + nonlinear single-term comparison panels
• fode_wpd_mtnonstiff.jl: multi-term comparison
• fode_wpd_nonstiff_and_stiff.jl: combined non-stiff and stiff figure
• fdde_wpd_nonstiff.jl: additional non-stiff figure variant

These scripts read CSV files from benchmarks/data and produce SVG figures (for example: singleterm_fode_benchmarks.svg, stiff_fode_benchmarks.svg, fode_general_benchmarks.svg).

How To Run

Julia benchmarks

1. Start Julia in the repository root.
2. Install required packages if needed: FractionalDiffEq, FdeSolver, DataFrames, CSV, BenchmarkTools, SpecialFunctions.
3. Run a generator script, for example:
   • include("benchmarks/benchmark_fode/benchmark_scipt_Julia/stiff_benchmarks.jl")

MATLAB benchmarks

1. Add benchmarks/benchmark_fode/benchmark_scipt_MATLAB to the MATLAB path.
2. Run one script, for example:
   • stiff_benchmarks
3. CSV files are written to benchmarks/data (or current folder for some scripts, depending on path usage).

Python benchmarks

1. Use Python 3 with numpy, pandas, pycaputo, matplotlib installed.
2. For linear_singleterm_benchmark.py, ensure python-julia bridge is configured and Julia package FractionalDiffEq is available.
3. Run, for example:
   • python benchmarks/benchmark_fode/benchmark_scipt_Python/nonlinear_singleterm_benchmark.py

Data Naming Convention

CSV prefixes indicate source and scenario:

• Stiff_*: stiff problem benchmarks
• Single_* or Singleterm_*: nonlinear single-term benchmarks
• Linear_Single_*: linear single-term benchmarks
• FractionalDiffEq_MT* and MATLAB_MT*: multi-term benchmarks
• PYCAPUTO_*: Python pycaputo benchmarks
• MATLAB_*: MATLAB benchmarks

Suffixes indicate solver family (for example PECE, PIEX, PITrap, PIRect, BDF, NewtonGregory, Trapzoid).

Reproducibility Notes

• Step-size sweep is consistent across languages in the benchmark scripts.
• Error definitions are based on script-level analytic references and may differ in norm details between implementations.
• Runtime is sensitive to machine load and language startup overhead.
• Some scripts use absolute output paths; update paths if the repository is moved.

Suggested Workflow

1. Run benchmark generators in each language to refresh CSV data.
2. Run Julia plotting scripts in benchmarks to create comparative figures.
3. Use benchmarks/data as the canonical source for tables and plots in reports.