# Problem implementing perpendicular Neumann boundary conditions using MethodOfLines.jl

**URL:** <https://discourse.julialang.org/t/problem-implementing-perpendicular-neumann-boundary-conditions-using-methodoflines-jl/108925>\
**Category:** Numerics\
**Created:** [January 17, 2024, 7:37pm UTC](https://discourse.julialang.org/t/problem-implementing-perpendicular-neumann-boundary-conditions-using-methodoflines-jl/108925 "2024-01-17T19:37:21Z")\
**Posts on this page:** 1\
**Page:** 1

<div class="post-metadata">

**Author:** ![aharbick](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/aharbick/32/202619_2.png) [@aharbick](https://discourse.julialang.org/u/aharbick)\
**Post date:** [January 17, 2024, 7:37pm UTC](https://discourse.julialang.org/t/problem-implementing-perpendicular-neumann-boundary-conditions-using-methodoflines-jl/108925/1 "2024-01-17T19:37:21Z")

</div>

I am currently bug-fixing a more complicated set of equations I am trying to solve using MethodOfLines.jl and I have narrowed down the current issue I am facing to a particular term which involves a perpendicular Neumann boundary condition.

Here is a simple example using a diffusion equation which reproduces my issue:

```julia
using OrdinaryDiffEq, ModelingToolkit, MethodOfLines, DomainSets

@parameters t x y
@variables u(..)
Dt = Differential(t)
Dx = Differential(x)
Dxx = Differential(x)^2
Dy = Differential(y)
Dyy = Differential(y)^2

# 2D PDE and boundary conditions
eq = Dt(u(t, x, y)) ~ 0.1 * (Dxx(u(t, x, y)) + Dyy(u(t, x, y)))
bcs = [u(0, x, y) ~ 0.0,
    u(t, 0, y) ~ 0.0,
    u(t, 1, y) ~ 0.0,
    u(t, x, 0) ~ 0.0,
    Dx(u(t, x, 1)) ~ sin(2 * pi * x)] #<-- This term produces the issue

# Space and time domains
domains = [t ∈ Interval(0.0, 10.0),
    x ∈ Interval(0.0, 1.0),
    y ∈ Interval(0.0, 1.0)]

# PDE system
@named pdesys = PDESystem(eq, bcs, domains, [t, x, y], [u(t, x, y)])

# Method of lines discretization
discretization = MOLFiniteDifference([x => 0.1, y => 0.1], t, approx_order=2)

# Convert the PDE problem into an ODE problem
prob = discretize(pdesys, discretization)

```

This results in a large printout of the system ending by saying the Discretization failed and then gives the following error:

ERROR: LoadError: ArgumentError: Differential w.r.t. multiple variables Any[0.4, 0.7, 0.3, 0.5, 0.2, t, 0.9, 0.1, 0.8, 0.6] are not allowed.

Have I made a mistake in my implementation here, or is there otherwise a better way to use such a boundary condition? Thanks!