# System of BVPs with complex dependent variables

**URL:** <https://discourse.julialang.org/t/system-of-bvps-with-complex-dependent-variables/79636>\
**Category:** New to Julia\
**Tags:** differentialequation\
**Created:** [April 18, 2022, 1:44pm UTC](https://discourse.julialang.org/t/system-of-bvps-with-complex-dependent-variables/79636 "2022-04-18T13:44:23Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Fishperson](https://avatars.discourse-cdn.com/v4/letter/f/ccd318/32.png) [@Fishperson](https://discourse.julialang.org/u/Fishperson)\
**Post date:** [April 18, 2022, 1:44pm UTC](https://discourse.julialang.org/t/system-of-bvps-with-complex-dependent-variables/79636/1 "2022-04-18T13:44:23Z")

</div>

Hi,  
I have the sort of thankless job of writing up a Julia-implementation for what I guess is a matrix-version of a [Riccati equation](https://en.wikipedia.org/wiki/Riccati_equation) using the DifferentialEquations.jl package. More spesifically this is a boundary value problem.

I think I ran into some problems regarding complex numbers, and type conversion - and could use some pointers on type conversion etc.

More spesifically, I think I need help type hinting the ComplexF64 type everywhere. I have a hunch it is the formatting of the initial guess that is off. The documentations says I could supply it as a function of the free variable, here x, but not how to do so. The way I have done it now, I have 16 coupled equations, but I mathematically speaking I only need 4 coupled equations for 2x2 matrices.

```julia
#Functions
equation!(dg, g, p, x) #This encodes the equation - with the independent variable being x and not t. 
bc!(res, sol, p, x) #Again, sorry, x is the independent variable.

#Calls
bvp1 = BVProblem(equation!, bc!, g_0, xspan)
sol1 = solve(bvp1, Shooting(Vern7()), dt = 0.03) #dt should be read dx

```

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [April 18, 2022, 3:04pm UTC](https://discourse.julialang.org/t/system-of-bvps-with-complex-dependent-variables/79636/2 "2022-04-18T15:04:35Z")

</div>

Can you share an example code which demonstrates your issue?

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<div class="post-metadata">

**Author:** ![Fishperson](https://avatars.discourse-cdn.com/v4/letter/f/ccd318/32.png) [@Fishperson](https://discourse.julialang.org/u/Fishperson)\
**Post date:** [April 18, 2022, 4:25pm UTC](https://discourse.julialang.org/t/system-of-bvps-with-complex-dependent-variables/79636/3 "2022-04-18T16:25:45Z")

</div>

Sure! I was a bit hesitant to do so - as I am probably breaking every best-practice imaginable. Here is a simpler version with scalar values. Chances are I created some numerical instabilities now.

EDIT: Having the initial guess as zeros produces a valid solution, and as ones a singularity - now it does not do that

```julia

function simple_eqn!(dg, g, p, x)
    #Here, I think I want dg and g to be Vector{ComplexF64}, but they look real
    γ1, γ2, dγ1, dγ2 = g[1:4]

    N = 1/(1 - γ1*γ2) #Readability

    dg[1] = dγ1
    dg[2] = dγ2

    dg[3] = - 2 * dγ1 * γ2 * N * dγ1 - 2im * E * γ1 
    dg[4] = - 2 * dγ2 * γ1 * N * dγ2 + 2im * (-E) * γ2

end

function simple_bc!(res, sol, p, x)

    res[1] = sol[1][3]
    res[2] = sol[1][4]
    res[2] = sol[end][3]
    res[2] = sol[end][4]

    #Chances are I did this wrong, but I want the derivatives, i.e. dγ1, dγ2 to be zero at the edges

end

E = 0.2 #Physical constant
g_0 = 0.5 * ones(ComplexF64, 4) 
xspan = (0.0, 1)

bvp1 = BVProblem(simple_eqn!, simple_bc!, g_0, xspan) 
sol1 = solve(bvp1, GeneralMIRK4(), dt = 0.03) #dt should be read dx

```