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Overview of DifferentialEquations.jl

The general workflow for using the package is as follows:

  • Define a problem
  • Solve the problem
  • Analyze the output

Defining Problems

Problems are specified via a type interface. The problem types are designed to contain the necessary information to fully define their associated differential equation. Each problem type has a page explaining their problem type and the special features associated with them. For example, an ordinary differential equation is defined by

$$\frac{du}{dt} = f(u,p,t)$$

over some time interval tspan with some initial condition u0, and therefore the ODEProblem is defined by those components:

prob = ODEProblem(f, u0, tspan)
prob = ODEProblem(f, u0, tspan, p)

Note that the number types in the solution will match the types you designate in the problem. For example, if one uses Rational{BigInt} for specifying the timespan and BigFloat for specifying the initial condition, then the solution will solve using Rational{BigInt} for the timesteps and BigFloat for the independent variables. A wide variety of number types are compatible with the solvers such as complex numbers, unitful numbers (via Unitful.jl), decimals (via DecFP), dual numbers, and many more which may not have been tested yet (thanks to the power of multiple dispatch!). For information on type-compatibility, please see the solver pages for the specific problems.

Solving the Problems

Each type of differential equation has its own problem type which allow the solvers to dispatch to the right methods. The common interface for calling the solvers is:

sol = solve(prob, alg; kwargs)

Into the command, one passes the differential equation problem that they defined prob, optionally choose an algorithm alg (a default is given if not chosen), and change the properties of the solver using keyword arguments. The common arguments which are accepted by most methods is defined in [the common solver options manual page](@ref solver_options). The solver returns a solution object sol which hold all the details for the solution.

Analyzing the Solution

With the solution object, you do the analysis as you please! The solution type has a common interface, which makes handling the solution similar between the different types of differential equations. Tools such as interpolations are seamlessly built into the solution interface to make analysis easy. This interface is described in the [solution handling manual page](@ref solution).

Plotting functionality is provided by a recipe to Plots.jl. To use plot solutions, simply call the plot(sol) and the plotter will generate appropriate plots. If save_everystep was used, the plotters can generate animations of the solutions to evolution equations using the animate(sol) command. Plots can be customized using all the keyword arguments provided by Plots.jl. Please see Plots.jl's documentation for more information.

Add-on Tools

One of the most compelling features of DifferentialEquations.jl is that the common solver interface allows one to build tools which are “algorithm and problem agnostic”. For example, one of the provided tools allows for performing parameter estimation on ODEProblems. Since the solve interface is the same for the different algorithms, one can use any of the associated solving algorithms. This modular structure allows one to mix and match overarching analysis tools with specialized algorithms to one's problem, leading to high performance with a large feature base. Isn't that the promise of Julia just being fulfilled?

Development and Testing Tools

Lastly, one unique feature of DifferentialEquations.jl is the existence of algorithm development and testing functionality. This suite was designed by researchers in the field of numerical differential equations to both try out new ideas and distribute finalized results to large audiences. The tools for algorithm development allow for easy convergence testing, benchmarking, and higher order analysis (stability plotting, etc.). This is one of the reasons why DifferentialEquations.jl contains many algorithms which are unique and the results of recent publications! Please check out the developer documentation for more information on using the development tools.

Note that DifferentialEquations.jl allows for distributed development, meaning that algorithms which “plug-into the ecosystem” don't have to be a part of the major packages. If you are interested in adding your work to the ecosystem, checkout the developer documentation for more information.