Can I model a 2nd order elliptical PDE using the PDE toolbox that has no BC for one of its edges?

I am trying to use the PDE toolbox to solve a 2nd order, which is governed by the Laplace equation for stream function. I have already solved the problem using a hand-coded rectangular mesh finite difference method which produces the results I expected. Now I am trying to perform an examination of the capabilities of the PDE toolbox using this problem as the common denominator.

Most of the problem is pretty easy to set up using pde toolbox commands. This is a 2D rectangular diffusion channel that essentially has two "holes" on the top and bottom of the box. These are defined as Dirichlet boundary conditions that have a position-dependent component. I believe I have successfully modeled these using applyBoundaryCondition calls to custom made functions. However the diffusion channel, my box, has an open end where there are no Dirichlet BCs and I believe they are not Neumann BCs either. Any nodes on the left most edge (this is a rectangle whose horizontal length is greater than its vertical height and the flow is from left to right) simply have the value of the solution at that point.

In the FDM code, I solve the problem using ghost nodes. I can't figure out how to get this to work within the PDE Toolbox frame work though. Essentially I want to set up this edge's boundary condition such that I can generate any mesh size that I'd like and have the solvepde do it's magic to find the solution.

Is this possible? I've attached the simple drawing of the construct that defines the diffusion channel.

Thank you for the help! David

Answer 1

[Bjorn Gustavsson]

Maybe I get this all wrong but to me it seems that what you implement in the FDM version is a Neumann BC with the gradient of the flux kept to zero? So maybe that is what you should use for that boundary in your PDE-implementation.

HTH