CRAN Task View: Differential Equations

Differential equations (DE) are mathematical equations that describe how a quantity changes as a function of one or several (independent) variables, often time or space. Differential equations play an important role in biology, chemistry, physics, engineering, economy and other disciplines.

Differential equations can be separated into stochastic versus deterministic DEs. Problems can be split into initial value problems versus boundary value problems. One also distinguishes ordinary differential equations from partial differential equations, differential algebraic equations and delay differential equations. All these types of DEs can be solved in R. DE problems can be classified to be either stiff or nonstiff; the former type of problems are much more difficult to solve.

The dynamic models SIG is a suitable mailing list for discussing the use of R for solving differential equation and other dynamic models such as individual-based or agent-based models.

This task view was created to provide an overview on the topic. If something is missing, or if a new package should be mentioned here, please e-mail the maintainers or submit an issue or pull request in the GitHub repository linked above.

Stochastic Differential Equations (SDEs)

In a stochastic differential equation, the unknown quantity is a stochastic process.

• The package sde provides functions for simulation and inference for stochastic differential equations. It is the accompanying package to the book by Iacus (2008).
• The package pomp contains functions for statistical inference for partially observed Markov processes.
• Packages adaptivetau and GillespieSSA implement Gillespie’s “exact” stochastic simulation algorithm (direct method) and several approximate methods.
• The package resde computes maximum likelihood parameter estimates for univariate reducible stochastic differential equation models.
• The package Sim.DiffProc provides functions for simulation of Itô and Stratonovitch stochastic differential equations.
• Package diffeqr can solve SDE problems using the DifferentialEquations.jl package from the Julia programming language.

Ordinary Differential Equations (ODEs)

In an ODE, the unknown quantity is a function of a single independent variable. Several packages offer to solve ODEs.

• The “odesolve” package was the first to solve ordinary differential equations in R. It contained two integration methods. It has been replaced by the package deSolve.
• The package deSolve contains several solvers for solving ODE, DAE, DDE and PDE. It can deal with stiff and nonstiff problems.
• Package r2sundials is a wrapper for the widely used SUNDIALS SUite of Nonlinear and DIfferential/ALgebraic Equation Solvers, and more precisely to its CVODES solver.
• The R package diffeqr provides a seamless interface to the DifferentialEquations.jl package from the Julia programming language. It has unique high performance methods for solving ODE, SDE, DDE, DAE and more. Models can be written in either R or Julia. It requires an installation of the Julia language.
• Package pracma implements several adaptive Runge-Kutta solvers such as ode23, ode23s, ode45, or the Burlisch-Stoer algorithm to obtain numerical solutions to ODEs with higher accuracy.
• Package rODE (inspired by the Java related book of Gould, Tobochnik and Christian, 2016) aims to show physics, math and engineering students how ODE solvers can be made with R’s S4 classes.
• The package mrgsolve compiles ODEs on the fly and allows shorthand prescription dosing.
• The package rxode2 is similar to mrgsolve, but has the added value of being the backend of the nonlinear mixed effects modeling R package nlmixr2.

Delay Differential Equations (DDEs)

In a DDE, the derivative at a certain time is a function of the variable value at a previous time.

• The dde package implements solvers for ordinary (ODE) and delay (DDE) differential equations, where the objective function is written in either R or C. Suitable only for non-stiff equations. Support is also included for iterating difference equations.
• The package PBSddesolve (originally published as “ddesolve”) includes a solver for non-stiff DDE problems.
• Functions in the package deSolve can solve both stiff and non-stiff DDE problems.
• Package diffeqr can solve DDE problems using the DifferentialEquations.jl package from the Julia programming language.

Partial Differential Equations (PDEs)

PDEs are differential equations in which the unknown quantity is a function of multiple independent variables. A common classification is into elliptic (time-independent), hyperbolic (time-dependent and wavelike), and parabolic (time-dependent and diffusive) equations. One way to solve them is to rewrite the PDEs as a set of coupled ODEs, and then use an efficient solver.

• The R-package ReacTran provides functions for converting the PDEs into a set of ODEs. Its main target is in the field of ‘’reactive transport’’ modelling, but it can be used to solve PDEs of the three main types. It provides functions for discretising PDEs on cartesian, polar, cylindrical and spherical grids.
• The package deSolve contains dedicated solvers for 1-D, 2-D and 3-D time-varying ODE problems as generated from PDEs (e.g. by ReacTran).
• The package rootSolve contains optimized solvers for 1-D, 2-D and 3-D algebraic problems generated from (time-invariant) PDEs. It can thus be used for solving elliptic equations.

Note that, to date, PDEs in R can only be solved using finite differences. At some point, we hope that finite element and spectral methods will become available.

Differential Algebraic Equations (DAEs)

Differential algebraic equations comprise both differential and algebraic terms. An important feature of a DAE is its differentiation index; the higher this index, the more difficult to solve the DAE.

• The package deSolve provides two solvers, that can handle DAEs up to index 3.
• Package diffeqr can solve DAE problems using the DifferentialEquations.jl package from the Julia programming language.

Boundary Value Problems (BVPs)

BVPs have solutions and/or derivative conditions specified at the boundaries of the independent variable.

• Package diffeqr can solve BVPs using the DifferentialEquations.jl package from the Julia programming language.
• The package ReacTran can solve BVPs that belong to the class of reactive transport equations.

Model Analysis and Calibration

• Package phaseR applys phase plane methods to one- and two-dimensional autonomous ODEs.
• In the package FME are functions for inverse modelling (fitting to data), sensitivity analysis, identifiability and Monte Carlo Analysis of DE models.
• Package fitode contains tools for fitting ODEs. Sensity equations are used to compute the gradients of ODE trajectories for more stable fitting. An MCMC method is also available.
• Package magi implements parameter estimation of dynamic systems from noisy and sparse data within a Bayesian framework, without the need for numerical integration.
  
• Package ODEsensitivity performs sensitivity analysis of ODE models. It utilizes theinterface from deSolve and connects it with the sensitivity analysis from sensitivity.
• Package deFit uses numerical optimization to fit ODEs to time series data to examine the dynamic relationships between variables.

Compiled Code

• Package odin implements a high-level language for describing and implementing ordinary differential equations in R. It provides a “domain specific language” (DSL) which looks like R but is compiled directly to C.
  
• Package rodeo is an object oriented system and code generator that creates and compiles efficient Fortran code for deSolve from models defined in stoichiometry matrix notation.
• Package cOde supports the automatic creation of dynamically linked code for packages deSolve (or a built-in implementation of the sundials cvode solver) from inline C embedded in the R code.

Population ODE modeling

• The package nlmixr2 fits ODE-based nonlinear mixed effects models using rxode2.

Other

• The simecol package provides an interactive environment to implement and simulate dynamic models. Next to DE models, it also provides functions for grid-oriented, individual-based, and particle diffusion models.
• mkin provides routines for fitting kinetic models with one or more state variables to chemical degradation data.
• Package dMod provides functions to generate ODEs of reaction networks, parameter transformations, observation functions, residual functions, etc. It follows the paradigm that derivative information should be used for optimization whenever possible.
• The package CollocInfer implements collocation-inference for continuous-time and discrete-time stochastic processes.
• Root finding, equilibrium and steady-state analysis of ODEs can be done with the package rootSolve.
• The PBSmodelling package adds GUI functions to models.
• Package ecolMod contains the figures, data sets and examples from a book on ecological modelling (Soetaert and Herman, 2009).

CRAN packages

Related links

• Wikipedia: Differential equation
• R-Forge website: deSolve (differential equation solvers)
• Github website: pomp (partially observed Markov process)
• Book: Iacus, S.M. 2008. Simulation and Inference for Stochastic Differential Equations: with R examples, Springer
• Book: Soetaert, K. and P.M.J. Herman, 2009. A Practical Guide to Ecological Modelling, using R as a simulation Platform, Springer.
• Book: Stevens, H, 2009. A Primer of Ecology with R, Springer and the 2021 online edition on Github.
• Book: Soetaert, K., Cash, J. and Mazzia, F. 2012. Solving Differential Equations in R, Springer.
• Book: Griffiths, G.W., 2016. Numerical Analysis Using R. Solutions to ODEs and PDEs. Cambridge University Press.