r"""
Heterogeneous boundary conditions
=================================
This example implements a `spatially coupled SIR model
`_ with
the following dynamics for the density of susceptible, infected, and recovered
individuals:
.. math::
\partial_t s &= D \nabla^2 s - \beta is \\
\partial_t i &= D \nabla^2 i + \beta is - \gamma i \\
\partial_t r &= D \nabla^2 r + \gamma i
Here, :math:`D` is the diffusivity, :math:`\beta` the infection rate, and
:math:`\gamma` the recovery rate.
"""
import numpy as np
from pde import CartesianGrid, DiffusionPDE, ScalarField
# define grid and an initial state
grid = CartesianGrid([[-5, 5], [-5, 5]], 32)
field = ScalarField(grid)
# define the boundary conditions, which here are calculated from a function
def bc_value(adjacent_value, dx, x, y, t):
"""return boundary value"""
return np.sign(x)
bc_x = "derivative"
bc_y = ["derivative", {"value_expression": bc_value}]
# define and solve a simple diffusion equation
eq = DiffusionPDE(bc=[bc_x, bc_y])
res = eq.solve(field, t_range=10, dt=0.01, backend="numpy")
res.plot()