# Help solving coupled differential equations involving complex numbers
**URL:** https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893
**Category:** Modelling & Simulations
**Tags:** differentialequation
**Created:** [November 7, 2022, 4:24pm UTC](https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893 "2022-11-07T16:24:43Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@TheLateKronos](https://discourse.julialang.org/u/TheLateKronos)
#### Post date: [November 7, 2022, 4:24pm UTC](https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893/1 "2022-11-07T16:24:43Z")
Original title: “Some help solving set of differential equations”
Hi. I am stuck in doing some homework, and I was hoping I could get some help.
The task is to analytically or numerically solve some differential equations. I will provide the full task for completeness:
These are the DE’s to solve, in the following task:
“Real valued” Rabi frequencies means that \Omega is simply some real number, which reduces the complexity quite a bit. Below is my attempt at defining this, using ModelingToolkit.jl:
```julia
using ModelingToolkit
using DifferentialEquations
@variables t c̃₁(t) c̃₂(t) # independent and dependent variables
@parameters Ω Δ # parameters
∂ₜ = Differential(t) # define an operator for the differentiation w.r.t. time
```
Inspired by the docs, I tried to define a vector of equations:
```julia
@named eqs = ODESystem([∂ₜ(c̃₁) ~ Ω*im*c̃₂*cis(t*Δ),
∂ₜ(c̃₂) ~ Ω*im*c̃₁*cis(-t*Δ)
])
```
Resulting in
```julia
ERROR: type Array has no field lhs
Stacktrace:
```
So instead, I tried what I found inntuitive - defining two equations, and giving _those_ as a vector:
```julia
julia> @named eq1 = ODESystem(∂ₜ(c̃₁) ~ Ω*im*c̃₂*cis(t*Δ))
Model eq1 with 2 equations
States (2):
c̃₁(t)
c̃₂(t)
Parameters (2):
Ω
Δ
julia> @named eq2 = ODESystem(∂ₜ(c̃₂) ~ Ω*im*c̃₁*cis(-t*Δ))
Model eq2 with 2 equations
States (2):
c̃₂(t)
c̃₁(t)
Parameters (2):
Ω
Δ
```
So far so good I think. I do not get why both equations say “with 2 equations”.
I then attempt:
```julia
julia> using DifferentialEquations: solve
julia> initial_conds = [c̃₁ => 1.0, c̃₂ => 0.0];
julia> tspan = (0.0, 10.0);
julia> params = [Ω => 1.0, Δ => 1.0];
julia> prob = ODEProblem([eq1, eq2], initial_conds, tspan, params)
ERROR: No methods were found for the model function passed to the equation solver.
The function `f` needs to have dispatches, for example, for an ODEProblem
`f` must define either `f(u,p,t)` or `f(du,u,p,t)`. For more information
on how the model function `f` should be defined, consult the docstring for
the appropriate `AbstractSciMLFunction`.
Offending function: f
Stacktrace:
[1] isinplace(f::Vector{ODESystem}, inplace_param_number::Int64, fname::String, iip_preferred::Bool; has_two_dispatches::Bool, isoptimization::Bool)
@ SciMLBase ~/.julia/packages/SciMLBase/RAGXU/src/utils.jl:236
[2] isinplace (repeats 2 times)
@ ~/.julia/packages/SciMLBase/RAGXU/src/utils.jl:229 [inlined]
[3] ODEProblem(f::Vector{ODESystem}, u0::Vector{Pair{Num, Float64}}, tspan::Tuple{Float64, Float64}, p::Vector{Pair{Num, Float64}}; kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
@ SciMLBase ~/.julia/packages/SciMLBase/RAGXU/src/problems/ode_problems.jl:168
[4] ODEProblem(f::Vector{ODESystem}, u0::Vector{Pair{Num, Float64}}, tspan::Tuple{Float64, Float64}, p::Vector{Pair{Num, Float64}})
@ SciMLBase ~/.julia/packages/SciMLBase/RAGXU/src/problems/ode_problems.jl:167
[5] top-level scope
@ REPL[33]:1
```
Now I realize that I could be going about it the wrong way. Is it even an ODE system, or something else? I do however not know how to proceed, and would much appreciate some pointers.
Do anyone have the know-how and time to help me out?
---
### Author:  [@contradict](https://discourse.julialang.org/u/contradict)
#### Post date: [November 7, 2022, 9:11pm UTC](https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893/2 "2022-11-07T21:11:15Z")
ModelingToolkit variables are all real, as written you get
```julia
julia> eqs = [∂ₜ(c̃₁) ~ Ω*im*c̃₂*cis(t*Δ),
∂ₜ(c̃₂) ~ Ω*im*c̃₁*cis(-t*Δ)
]
2-element Vector{Vector{Equation}}:
[Differential(t)(c̃₁(t)) ~ -Ω*c̃₂(t)*sin(t*Δ), 0 ~ Ω*c̃₂(t)*cos(t*Δ)]
[Differential(t)(c̃₂(t)) ~ -Ω*c̃₁(t)*sin(-t*Δ), 0 ~ Ω*c̃₁(t)*cos(-t*Δ)]
```
Which probably isn’t what you mean. Try defining c̃₁ and c̃₂ with explicit real and imaginary parts.
```julia
@variables c̃r₁ c̃i₁ c̃₁ c̃r₂ c̃i₂ c̃₂
c̃₁ = c̃r₁ + im*c̃i₁
c̃₂ = c̃r₂ + im*c̃i₂
```
---
### Author:  [@TheLateKronos](https://discourse.julialang.org/u/TheLateKronos)
#### Post date: [November 8, 2022, 7:38am UTC](https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893/3 "2022-11-08T07:38:15Z")
> [@contradict](#):
>
> ```julia
>
> ```
Thank you for your suggestion.
With the following definitions:
```julia
@variables t c̃r₁ c̃i₁ c̃₁ c̃r₂ c̃i₂ c̃₂ # independent and dependent variables
@parameters Ω Δ # parameters
c̃₁ = c̃r₁ + im*c̃i₁
c̃₂ = c̃r₂ + im*c̃i₂
∂ₜ = Differential(t) # define an operator for the differentiation w.r.t. time
```
I get the following error:
```julia
julia> @named eqs = ODESystem([∂ₜ(c̃₁) ~ Ω*im*c̃₂*cis(t*Δ),
∂ₜ(c̃₂) ~ Ω*im*c̃₁*cis(-t*Δ)
])
ERROR: type ComplexTerm has no field metadata
Stacktrace:
[1] getproperty(x::Symbolics.ComplexTerm{Real}, f::Symbol)
...
```
I get the same error when defining a single equation:
```julia
julia> @named eq1 = ODESystem(∂ₜ(c̃₁) ~ Ω*im*c̃₂*cis(t*Δ))
ERROR: type ComplexTerm has no field metadata
```
Should I define seperate equations for the real and imaginary parts, giving me 4 equations? If yes, I am unsure how to even go about that. Should I use the `real` and `imag` functions in defining the equations?
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [November 8, 2022, 11:25am UTC](https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893/4 "2022-11-08T11:25:45Z")
Open an issue. This is a very specific question to ModelingToolkit with complex numbers and it would be helpful if you title it as such.
---
### Author:  [@TheLateKronos](https://discourse.julialang.org/u/TheLateKronos)
#### Post date: [November 8, 2022, 8:44pm UTC](https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893/5 "2022-11-08T20:44:04Z")
Done. Are you saying this could be solved without ModelingToolkit? Any quick pointers on how?
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [November 9, 2022, 10:27am UTC](https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893/6 "2022-11-09T10:27:55Z")
Just use complex numbers in DiffEq directly.
---
### Author:  [@TheLateKronos](https://discourse.julialang.org/u/TheLateKronos)
#### Post date: [November 9, 2022, 11:13am UTC](https://discourse.julialang.org/t/help-solving-coupled-differential-equations-involving-complex-numbers/89893/7 "2022-11-09T11:13:36Z")
Thanks guys. This ended up doing the trick:
```julia
using DifferentialEquations
using ParameterizedFunctions
my_ode = @ode_def begin
dc̃₁ = Ω*im*c̃₂*exp( im*t*Δ)
dc̃₂ = Ω*im*c̃₁*exp(-im*t*Δ)
end Ω Δ
u₀ = Complex{Float64}[1, 0]
tspan = (0.0, 10.0)
p = [1.0, 1.0]
prob = ODEProblem(my_ode, u₀, tspan, p)
sol = solve(prob)
```