Steady State Problems

Defines an Defines a steady state ODE problem. Documentation Page: https://diffeq.sciml.ai/stable/types/steadystatetypes/

Mathematical Specification of a Steady State Problem

To define an Steady State Problem, you simply need to give the function $f$ which defines the ODE:

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

and an initial guess $u_0$ of where f(u,p,t)=0. f should be specified as f(u,p,t) (or in-place as f(du,u,p,t)), and u₀ should be an AbstractArray (or number) whose geometry matches the desired geometry of u. Note that we are not limited to numbers or vectors for u₀; one is allowed to provide u₀ as arbitrary matrices / higher dimension tensors as well.

Note that for the steady-state to be defined, we must have that f is autonomous, that is f is independent of t. But the form which matches the standard ODE solver should still be used. The steady state solvers interpret the f by fixing t=0.

Problem Type

Constructors

SteadyStateProblem(f::ODEFunction,u0,p=NullParameters();kwargs...)
SteadyStateProblem{isinplace}(f,u0,p=NullParameters();kwargs...)

isinplace optionally sets whether the function is inplace or not. This is determined automatically, but not inferred. Additionally, the constructor from ODEProblems is provided:

SteadyStateProblem(prob::ODEProblem)

Parameters are optional, and if not given then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

For specifying Jacobians and mass matrices, see the DiffEqFunctions page.

Fields

• f: The function in the ODE.
• u0: The initial guess for the steady state.
• p: The parameters for the problem. Defaults to NullParameters
• kwargs: The keyword arguments passed onto the solves.

Special Solution Fields

The SteadyStateSolution type is different from the other DiffEq solutions because it does not have temporal information.

struct NonlinearSolution{T, N, uType, R, P, A, O, uType2} <: SciMLBase.AbstractNonlinearSolution{T, N}

Representation of the solution to an nonlinear equation defined by an NonlinearProblem, or the steady state solution to a differential equation defined by a SteadyStateProblem.

Fields

• u: the representation of the nonlinear equation's solution.
• resid: the residual of the solution.
• prob: the original NonlinearProblem/SteadyStateProblem that was solved.
• alg: the algorithm type used by the solver.
• original: if the solver is wrapped from an alternative solver ecosystem, such as NLsolve.jl, then this is the original return from said solver library.
• retcode: the return code from the solver. Used to determine whether the solver solved successfully (sol.retcode === :Success), whether it terminated due to a user-defined callback (sol.retcode === :Terminated), or whether it exited due to an error. For more details, see the return code section of the DifferentialEquations.jl documentation.
• left: if the solver is bracketing method, this is the final left bracket value.
• right: if the solver is bracketing method, this is the final right bracket value.