Here is the code in Mathematica, The symmetric initial condition of v is just for simplification, otherwise the initial distribution of v does not have to be symmetric or Gaussian and in practice it should be a random distribution. Also the Neumann boundary condition in general will depend on the value of other variables which only exist on the boundary (here for simplification it is not the case). For example protein (variable) m could detach from the boundary and converts to protein (variable) v with a rate proportional to m.:
The variable v and m represent two proteins. Protein v diffuses freely inside the cytosol (inside of the cell, here represented as a disk). Protein m is a membrane-bound protein that is it attaches to the cell's membrane (here the boundary of the disk) and only can exist as a membrane-bound protein. The protein v diffuses freely inside the disk and reaches the membrane or the boundary. There it converts to protein m with a rate that is proportional to the value of protein v on the membrane. The created membrane-bound protein m then diffuses on the membrane. Protein m cannot detach from the membrane and thus it must not exist in the cytosol (inside the disk).
Edit
I added this explanation to the question:
The symmetric initial condition of v is just for simplification, otherwise the initial distribution of v does not have to be symmetric or Gaussian and in practice it should be a random distribution. Also the Neumann boundary condition in general will depend on the value of other variables which only exist on the boundary (here for simplification it is not the case). For example protein (variable) m could detach from the boundary and converts to protein (variable) v with a rate proportional to m.
