Last updated on 20210922
by Karline Soetaert, Thomas Petzoldt
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 individualbased or agentbased models.
This task view was created to provide an overview on the topic.
If we forgot something, or if a new package should be mentioned here, please let
us know.
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 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 It has been replaced by the package deSolve.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.
 The package odeintr generates and compiles C++ ODE solvers on the fly using Rcpp
and Boost odeint.

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 RungeKutta
solvers such as ode23, ode23s, ode45, or the BurlischStoer algorithm to obtain
numerical solutions to ODEs with higher accuracy.
 Package rODE (inspired from the 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.
 Package sundialr provides a way to call the 'CVODE' function from the 'SUNDIALS' C ODE solving library.
The package requires the ODE to be written as an 'R' or 'Rcpp' function.

The package mrgsolve compiles ODEs on the fly and allows shorthand prescription dosing.

The package RxODE is similar to mrgsolve, but has the added value of being the backend of the nonlinear mixed effects modeling R package nlmixr.
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 nonstiff equations.
Support is also included for iterating difference equations.
 The package PBSddesolve (originally published as "ddesolve")
includes a solver for nonstiff DDE problems.
 Functions in the package deSolve can solve both stiff and nonstiff 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 (timeindependent), hyperbolic (timedependent and wavelike),
and parabolic (timedependent 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 Rpackage 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 1D, 2D and
3D timevarying ODE problems as generated from PDEs (e.g. by ReacTran).
 The package rootSolve contains optimized solvers for 1D, 2D and
3D algebraic problems generated from (timeinvariant) 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.
 The package ReacTran can solve BVPs that belong to the
class of reactive transport equations.
 Package diffeqr can also solve BVPs using the DifferentialEquations.jl
package from the Julia programming language.
Population ODE modeling

The package nlmixr fits ODEbased nonlinear mixed effects models using RxODE.
Other
 The simecol package provides an interactive environment to
implement and simulate dynamic models.
Next to DE models, it also provides functions for gridoriented,
individualbased, and particle diffusion models.
 In the package FME are functions for inverse modelling (fitting to data),
sensitivity analysis, identifiability and Monte Carlo Analysis of DE models.
 The package nlmeODE has functions for
mixedeffects modelling using differential equations.
 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 collocationinference
for continuoustime and discretetime stochastic processes.
 Root finding, equilibrium and steadystate analysis of ODEs can be
done with the package rootSolve.
 The PBSmodelling package adds GUI functions to models.
 Package cOde supports the automatic creation of dynamically linked
code for packages deSolve (or a builtin implementation
of the sundials cvode solver) from inline C embedded in the R code.
 Package rodeo is an object oriented system and code generator that
creates and compiles efficient Fortran code for deSolve from models
defined in stoichiomatry matrix notation.
 Package ecolMod contains the figures, data sets and examples from a book
on ecological modelling (Soetaert and Herman, 2009).
 Task view: ChemPhys
 Task view: Econometrics
 Task view: Environmetrics
 Task view: Optimization
 Task view: Pharmacokinetics
 Wikipedia: Differential equation
 RForge website for package deSolve
 RForge website for package pomp
 Book: Iacus, SM. 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.
 Book: Soetaert, K., Cash, J. and Mazzia, F. 2012. Solving Differential Equations in R, Springer.