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 we forgot something, or if a new package should be mentioned here, please let us know.

**Stochastic Differential Equations (SDEs)**

- 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 andGillespieSSA implement Gillespie's "exact" stochastic simulation algorithm (direct method) and several approximate methods. - The package
Sim.DiffProc provides functions for simulation of ItÃ´ and Stratonovitch stochastic differential equations.

**Ordinary Differential Equations (ODEs)**

- The "odesolve" package was the first to solve ordinary differential equations in R.
It contains two integration methods. It is not actively maintained and has been replaced by the package
deSolve . - The package
deSolve contains several solvers for solving ODEs. It can deal with stiff and nonstiff problems. - The package
deTestSet contains solvers designed for solving very stiff equations. - The package
odeintr generates and compiles C++ ODE solvers on the fly using Rcpp and Boost odeint.

**Delay Differential Equations (DDEs)**

- 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.

**Partial Differential Equations (PDEs)**

- 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. byReacTran ). - Solvers for 1-D time varying problems can also be found in the package
deTestSet . - 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.

**Differential Algebraic Equations (DAEs)**

- The package
deSolve provides two solvers, that can handle DAEs up to index 3. - Three more DAE solvers are in the package
deTestSet .

**Boundary Value Problems (BVPs)**

- Package
bvpSolve deals only with this type of equations. - The package
ReacTran can solve BVPs that belong to the class of reactive transport equations.

**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. - Package
scaRabee offers frameworks for simulation and optimization of Pharmacokinetic-Pharmacodynamic 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 mixed-effects modelling using differential equations. -
mkin provides routines for fitting kinetic models with one or more state variables to chemical degradation data. - 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
deTestSet package contains many test problems for differential equations. - 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. - 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). primer is a support package for the book of Stevens (2009).