Non-autonomous linear ODEs show up in a lot of scientific problems where the differential equation lives on a manifold such as Lie Group. In these situations, specialized solvers can be utilized to enforce physical bounds on the solution and enhance the solving.
These algorithms require a Non-autonomous linear ODE of the form:
\[u^\prime = A(u,p,t)u\]
Where $A$ is an AbstractDiffEqOperator that is multiplied against $u$. Many algorithms specialize on the form of $A$, such as $A$ being a constant or $A$ being only time-dependent ($A(t)$).
Creating a non-autonomous linear ODE is the same as an
f is represented by an
AbstractDiffEqOperator (note: this means that any standard ODE solver can also be applied to problems written in this form). As an example:
function update_func(A,u,p,t) A[1,1] = cos(t) A[2,1] = sin(t) A[1,2] = -sin(t) A[2,2] = cos(t) end A = DiffEqArrayOperator(ones(2,2),update_func=update_func) prob = ODEProblem(A, ones(2), (10, 50.))
defines a quasi-linear ODE $u^\prime = A(t)u$ where the components of $A$ are the given functions. Using that formulation, we can see that the general form is $u^\prime = A(u,p,t)u$, for example:
function update_func(A,u,p,t) A[1,1] = 0 A[2,1] = 1 A[1,2] = -2*(1 - cos(u) - u*sin(u)) A[2,2] = 0 end
has a state-dependent linear operator. Note that many other
AbstractDiffEqOperators can be used and
DiffEqArrayOperator is just one version that represents
A via a matrix (other choices are matrix-free).
Note that if $A$ is a constant, then it is sufficient to supply $A$ directly without an
Note that the affine equation
\[u^\prime = A(u,p,t)u + g(u,p,t)\]
can be written as a linear form by extending the size of the system by one to have a constant term of 1. This is done by extending
A with a new row, containing only zeros, and giving this new state an initial value of 1. Then extend
A to have a new column containing the values of
g(u,p,t). In this way, these types of equations can be handled by these specialized integrators.