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DifferentialEquations.jl: Efficient Differential Equation Solving in Julia

This is a suite for numerically solving differential equations written in Julia and available for use in Julia, Python, and R. The purpose of this package is to supply efficient Julia implementations of solvers for various differential equations. Equations within the realm of this package include:

  • Discrete equations (function maps, discrete stochastic (Gillespie/Markov) simulations)
  • Ordinary differential equations (ODEs)
  • Split and Partitioned ODEs (Symplectic integrators, IMEX Methods)
  • Stochastic ordinary differential equations (SODEs or SDEs)
  • Stochastic differential-algebraic equations (SDAEs)
  • Random differential equations (RODEs or RDEs)
  • Differential algebraic equations (DAEs)
  • Delay differential equations (DDEs)
  • Neutral, retarded, and algebraic delay differential equations (NDDEs, RDDEs, and DDAEs)
  • Stochastic delay differential equations (SDDEs)
  • Experimental support for stochastic neutral, retarded, and algebraic delay differential equations (SNDDEs, SRDDEs, and SDDAEs)
  • Mixed discrete and continuous equations (Hybrid Equations, Jump Diffusions)
  • (Stochastic) partial differential equations ((S)PDEs) (with both finite difference and finite element methods)

The well-optimized DifferentialEquations solvers benchmark as some of the fastest implementations, using classic algorithms and ones from recent research which routinely outperform the “standard” C/Fortran methods, and include algorithms optimized for high-precision and HPC applications. At the same time, it wraps the classic C/Fortran methods, making it easy to switch over to them whenever necessary. Solving differential equations with different methods from different languages and packages can be done by changing one line of code, allowing for easy benchmarking to ensure you are using the fastest method possible.

DifferentialEquations.jl integrates with the Julia package sphere with:

  • GPU acceleration through CUDA.jl and DiffEqGPU.jl
  • Automated sparsity detection with Symbolics.jl
  • Automatic Jacobian coloring with SparseDiffTools.jl, allowing for fast solutions to problems with sparse or structured (Tridiagonal, Banded, BlockBanded, etc.) Jacobians
  • Allowing the specification of linear solvers for maximal efficiency with LinearSolve.jl
  • Progress meter integration with the Visual Studio Code IDE for estimated time to solution
  • Automatic plotting of time series and phase plots
  • Built-in interpolations
  • Wraps for common C/Fortran methods like Sundials and Hairer's radau
  • Arbitrary precision with BigFloats and ArbNumerics.jl
  • Arbitrary array types, allowing the definition of differential equations on matrices and distributed arrays
  • Unit checked arithmetic with Unitful

Additionally, DifferentialEquations.jl comes with built-in analysis features, including:

Contributing

Supporting and Citing

The software in this ecosystem was developed as part of academic research. If you would like to help support it, please star the repository, as such metrics may help us secure funding in the future. If you use SciML software as part of your research, teaching, or other activities, we would be grateful if you could cite our work as necessary for any use of DifferentialEquations.jl or the packages that are maintained as part of its suite (OrdinaryDiffEq.jl, Sundials.jl, DiffEqDevTools.jl, etc.).

@article{rackauckas2017differentialequations,
  title={Differential{E}quations.jl--a performant and feature-rich ecosystem for solving differential equations in {J}ulia},
  author={Rackauckas, Christopher and Nie, Qing},
  journal={Journal of Open Research Software},
  volume={5},
  number={1},
  year={2017},
  publisher={Ubiquity Press}
}

Additionally, many of the solvers utilize novel algorithms, and if these algorithms are used, we ask that you cite the methods. Please see our citation page for guidelines.

Getting Started: Installation And First Steps

Installing from Julia

To install the package, use the following command inside the Julia REPL:

import Pkg
Pkg.add("DifferentialEquations")

To load the package, use the command:

import DifferentialEquations as DE

This will add solvers and dependencies for all kinds of Differential Equations (e.g. ODEs or SDEs etc., see the Supported Equations section below). If you are interested in only one type of equation solver of DifferentialEquations.jl or simply want a more lightweight version, see the Reduced Compile Time and Low Dependency Usage page.

To understand the package in more detail, check out the following tutorials in this manual. It is highly recommended that new users start with the ODE tutorial. Example IJulia notebooks can also be found in SciMLTutorials.jl. If you find any example where there appears to be an error, please open an issue.

For the most up-to-date information on using the package, please join the Julia Zulip channel.

Using the bleeding edge for the latest features and development is only recommended for power users. Information on how to get to the bleeding edge is found in the developer documentation.

Installing from Python

Use of DifferentialEquations.jl from the Python programming language is available through the diffeqpy module. To install diffeqpy, use pip:

pip install diffeqpy

Using diffeqpy requires that Julia is installed and in the path, along with DifferentialEquations.jl and PyCall.jl. To install Julia, download a generic binary from the JuliaLang site and add it to your path. To install Julia packages required for diffeqpy, open up Python interpreter then run:

>>> import diffeqpy
>>> diffeqpy.install()

and you're good! In addition, to improve the performance of your code, it is recommended that you use Numba to JIT compile your derivative functions. To install Numba, use:

pip install numba

diffeqpy supports the majority of DifferentialEquations.jl with very similar syntax, see the diffeqpy README for more details. One important point to note is that Numba is generally an order of magnitude slower than Julia in terms of the generated differential equation solver code, and thus it is recommended to use julia.Main.eval for Julia-side derivative function implementations for maximal efficiency. See this blog post for more information.

Installing from R

Use of DifferentialEquations.jl from the R programming language is available through the diffeqr module. diffeqr is registered into CRAN. Thus to add the package, use:

install.packages("diffeqr")

To install the master branch of the package (for developers), use:

devtools::install_github('SciML/diffeqr', build_vignettes=T)

You will need a working installation of Julia in your path. To install Julia, download a generic binary from the JuliaLang site and add it to your path. The download and installation of DifferentialEquations.jl will happen on the first invocation of diffeqr::diffeq_setup().

Currently, use from R supported a subset of DifferentialEquations.jl which is documented through CRAN.

Video Tutorial

Video Tutorial

Tutorials

The following tutorials will introduce you to the functionality of DifferentialEquations.jl. More examples can be found by checking out the IJulia notebooks in the examples folder.

Removing and Reducing Compile Times

In some situations, one may wish to decrease the compile time associated with DifferentialEquations.jl usage. If that's the case, there are two strategies to employ. One strategy is to use the low dependency usage. DifferentialEquations.jl is a metapackage composed of many smaller packages, and thus one could directly use a single component, such as OrdinaryDiffEq.jl for the pure Julia ODE solvers, and decrease the compile times by ignoring the rest (note: the interface is exactly the same, except using a solver apart from those in OrdinaryDiffEq.jl will error). We recommend that downstream packages rely solely on the packages they need.

The other strategy is to use PackageCompiler.jl to create a system image that precompiles the whole package. To do this, one simply does:

import PackageCompiler
PackageCompiler.create_sysimage([:DifferentialEquations, :Plots]; replace_default = true)

Note that there are some drawbacks to adding a package in your system image. For example, the package will never update until you manually rebuild the system image again. For more information on the consequences, see this portion of the PackageCompiler manual.

Basics

These pages introduce you to the core of DifferentialEquations.jl and the common interface. It explains the general workflow, options which are generally available, and the general tools for analysis.

Problem Types

These pages describe building the problem types to define differential equations for the solvers, and the special features of the different solution types.

Solver Algorithms

These pages describe the solvers and available algorithms in detail.

Additional Features

These sections discuss extra performance enhancements, event handling, and other in-depth features.

Extra Details

These are just assorted extra explanations for the curious.

Acknowledgements

Core Contributors

JuliaDiffEq and DifferentialEquations.jl has been a collaborative effort by many individuals. Significant contributions have been made by the following individuals:

  • Chris Rackauckas (@ChrisRackauckas) (lead developer)
  • Yingbo Ma (@YingboMa)
  • David Widmann (@devmotion)
  • Hendrik Ranocha (@ranocha)
  • Ethan Levien (@elevien)
  • Tom Short (@tshort)
  • @dextorious
  • Samuel Isaacson (@isaacsas)

Google Summer of Code Alumni

  • Yingbo Ma (@YingboMa)
  • Shivin Srivastava (@shivin9)
  • Ayush Pandey (@Ayush-iitkgp)
  • Xingjian Guo (@MSeeker1340)
  • Shubham Maddhashiya (@sipah00)
  • Vaibhav Kumar Dixit (@Vaibhavdixit02)

Reproducibility

The documentation of this SciML package was built using these direct dependencies,
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and using this machine and Julia version.
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Build Info:
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  [92933f4c] ProgressMeter v1.10.4
  [43287f4e] PtrArrays v1.3.0
  [1fd47b50] QuadGK v2.11.2
  [74087812] Random123 v1.7.1
  [e6cf234a] RandomNumbers v1.6.0
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [731186ca] RecursiveArrayTools v3.35.0
  [189a3867] Reexport v1.2.2
  [2792f1a3] RegistryInstances v0.1.0
  [05181044] RelocatableFolders v1.0.1
  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.1.1
  [79098fc4] Rmath v0.8.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.15
  [9dfe8606] SCCNonlinearSolve v1.4.0
  [c72e72a9] SDEProblemLibrary v0.1.7
  [94e857df] SIMDTypes v0.1.0
  [0bca4576] SciMLBase v2.104.0
  [19f34311] SciMLJacobianOperators v0.1.8
  [c0aeaf25] SciMLOperators v1.3.1
  [53ae85a6] SciMLStructures v1.7.0
  [7e506255] ScopedValues v1.4.0
  [6c6a2e73] Scratch v1.3.0
  [91c51154] SentinelArrays v1.4.8
  [efcf1570] Setfield v1.1.2
  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.2.0
  [727e6d20] SimpleNonlinearSolve v2.7.0
  [699a6c99] SimpleTraits v0.9.4
  [ce78b400] SimpleUnPack v1.1.0
  [a2af1166] SortingAlgorithms v1.2.1
  [9f842d2f] SparseConnectivityTracer v1.0.0
  [0a514795] SparseMatrixColorings v0.4.21
  [276daf66] SpecialFunctions v2.5.1
  [860ef19b] StableRNGs v1.0.3
  [aedffcd0] Static v1.2.0
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.14
  [1e83bf80] StaticArraysCore v1.4.3
  [10745b16] Statistics v1.11.1
  [82ae8749] StatsAPI v1.7.1
  [2913bbd2] StatsBase v0.34.5
  [4c63d2b9] StatsFuns v1.5.0
  [9672c7b4] SteadyStateDiffEq v2.5.0
  [789caeaf] StochasticDiffEq v6.80.0
  [7792a7ef] StrideArraysCore v0.5.7
  [892a3eda] StringManipulation v0.4.1
  [c3572dad] Sundials v4.28.0
  [2efcf032] SymbolicIndexingInterface v0.3.42
  [19f23fe9] SymbolicLimits v0.2.2
  [d1185830] SymbolicUtils v3.29.0
  [0c5d862f] Symbolics v6.45.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.1
  [ed4db957] TaskLocalValues v0.1.3
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [5d786b92] TerminalLoggers v0.1.7
  [1c621080] TestItems v1.0.0
  [8290d209] ThreadingUtilities v0.5.5
  [a759f4b9] TimerOutputs v0.5.29
  [0796e94c] Tokenize v0.5.29
  [3bb67fe8] TranscodingStreams v0.11.3
  [410a4b4d] Tricks v0.1.10
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.6.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.23.1
  [45397f5d] UnitfulLatexify v1.7.0
  [a7c27f48] Unityper v0.1.6
  [41fe7b60] Unzip v0.2.0
  [ea10d353] WeakRefStrings v1.4.2
  [897b6980] WeakValueDicts v0.1.0
  [76eceee3] WorkerUtilities v1.6.1
  [6e34b625] Bzip2_jll v1.0.9+0
  [83423d85] Cairo_jll v1.18.5+0
  [655fdf9c] DASKR_jll v1.0.0+0
  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.6.5+0
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
  [a3f928ae] Fontconfig_jll v2.16.0+0
  [d7e528f0] FreeType2_jll v2.13.4+0
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.0+2
  [d2c73de3] GR_jll v0.73.17+0
  [b0724c58] GettextRuntime_jll v0.22.4+0
  [f8c6e375] Git_jll v2.50.1+0
  [7746bdde] Glib_jll v2.84.3+0
  [3b182d85] Graphite2_jll v1.3.15+0
  [2e76f6c2] HarfBuzz_jll v8.5.1+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.1.1+0
  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.0.1+0
  [1d63c593] LLVMOpenMP_jll v18.1.8+0
  [dd4b983a] LZO_jll v2.10.3+0
  [81d17ec3] L_BFGS_B_jll v3.0.1+0
  [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.41.0+0
  [89763e89] Libtiff_jll v4.7.1+0
  [38a345b3] Libuuid_jll v2.41.0+0
  [856f044c] MKL_jll v2025.2.0+0
  [079eb43e] NLopt_jll v2.10.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [9bd350c2] OpenSSH_jll v10.0.1+0
  [458c3c95] OpenSSL_jll v3.5.1+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.5.2+0
  [36c8627f] Pango_jll v1.56.3+0
⌅ [30392449] Pixman_jll v0.44.2+0
  [c0090381] Qt6Base_jll v6.8.2+1
  [629bc702] Qt6Declarative_jll v6.8.2+1
  [ce943373] Qt6ShaderTools_jll v6.8.2+1
  [e99dba38] Qt6Wayland_jll v6.8.2+1
  [f50d1b31] Rmath_jll v0.5.1+0
⌅ [fb77eaff] Sundials_jll v5.2.3+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.1+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.12+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.7+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.1+0
  [a51aa0fd] Xorg_libXi_jll v1.8.3+0
  [d1454406] Xorg_libXinerama_jll v1.1.6+0
  [ec84b674] Xorg_libXrandr_jll v1.5.5+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.3+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.5+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
  [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.12.1+0
⌅ [0ac62f75] libass_jll v0.15.2+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.50+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.0.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.9.2+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [1a1011a3] SharedArrays v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.