Steady State Problems

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:

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

and an initial guess $u₀$ of where f(t,u)=0. f should be specified as f(t,u) (or in-place as f(t,u,du)), 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{isinplace}(f,u0,mass_matrix=I)

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)

Fields

f: The function in the ODE.
u0: The initial guess for the steady state.
mass_matrix: The mass-matrix. Defaults to I, the UniformScaling identity matrix.

Special Solution Fields

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