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juliasymbolic-mathmodelingtoolkit

Symbolics: Replacing symbolic equation with variables to solve

In Symbolics.jl, I can formulate a set of equations purely symbolically. I can for example define this differential equation using @syms:

using Symbolics

@syms α ρ[1:2, 1:2] dαdt dρdt[1:2, 1:2]

eqs = []
push!(eqs, dαdt == α*(ρ[1,1] +ρ[1,2] + ρ[2,1] + ρ[2,2]))
for i in 1:2, j in 1:2
    push!(eqs, dρdt[i,j] == α*ρ[j,i])
end

eqs then has the form:

5-element Vector{Any}:
 dαdt == (α*(ρ[1, 1] + ρ[1, 2] + ρ[2, 1] + ρ[2, 2]))
 dρdt[1, 1] == (α*ρ[1, 1])
 dρdt[1, 2] == (α*ρ[2, 1])
 dρdt[2, 1] == (α*ρ[1, 2])
 dρdt[2, 2] == (α*ρ[2, 2])

To solve the above equation using e.g. ModelingToolkit.jl, the symbolic parameters must be replaced by variables, e.g.

@variables t::Real, αvar(t)::Complex{Real}, ρvar(t)[1:2, 1:2]::Complex{Real}

Question: How can I transform the above symbolic equation to the correct variables, so that it can be solved using e.g. an ODEProblem?

Remark: A trivial solution is of course to just use the variables as defined above in the first place. This is however not the point of this question.

Solution

  • Something like this may work:

    using Symbolics

@syms τ α ρ[1:2, 1:2]
D = Differential(τ)

eqs = []
push!(eqs, D(α) ~ α*(ρ[1,1] +ρ[1,2] + ρ[2,1] + ρ[2,2]))
for i in 1:2, j in 1:2
    push!(eqs, D(ρ[i,j]) ~ α*ρ[j,i])
end

    and then

    using ModelingToolkit

@parameters t
@variables αvar(t), ρvar(t)[1:2, 1:2]

subs = Dict(τ => t, α => αvar, ρ => ρvar)
eqs_subbed = substitute.(eqs, Ref(subs))