# Unusual boundary conditions on Matlab

I'm trying to solve the following PDE by Matlab,

$$u_t-\Delta u = 0, \quad \text{in}\quad \Omega\times (0,T) \tag{1}$$ $$u_t-\Delta_\Gamma u + \partial_\nu u=0,\quad \text{on}\quad \Gamma\times(0,T) \tag{2}$$

where $$\Omega$$ is a bounded domain of $$\mathbb{R}^n$$, $$\Gamma =\partial \Omega$$ is the boundary of $$\Omega$$, $$\partial_\nu$$ is the normal derivative, and $$\nu$$ is the outer unit vector.

For simplicity, I considered the one dimensional case when $$\Omega$$ is the interval $$(0,1)$$, and discretized the equation following the Finite Difference method (we can neglect the Laplace-Beltrami operator $$\Delta_\Gamma$$ in this case), but I didn't find how to solve such PDE with the dynamic boundary conditions (2) in a simple way using pdepe or PDE toolbox. Usual Matlab functions use the usual boundary conditions (Dirichlet or Neumann), but the dynamic boundary conditions are not included. So I'm asking how to solve this PDE with Matlab in a simple way. For the 2D heat equation can we define $$\Delta_\Gamma$$ by Matlab tools.

• I think this is a question best asked on the Matlab forums... Jan 28, 2019 at 16:35