.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples_gallery/advanced_pdes/heterogeneous_bcs.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_gallery_advanced_pdes_heterogeneous_bcs.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_gallery_advanced_pdes_heterogeneous_bcs.py:


Heterogeneous boundary conditions
=================================

This example implements a `spatially coupled SIR model 
<https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology>`_ 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.

.. GENERATED FROM PYTHON SOURCE LINES 19-39



.. image-sg:: /examples_gallery/advanced_pdes/images/sphx_glr_heterogeneous_bcs_001.png
   :alt: heterogeneous bcs
   :srcset: /examples_gallery/advanced_pdes/images/sphx_glr_heterogeneous_bcs_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


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|

.. code-block:: Python


    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)


    # define and solve a simple diffusion equation
    eq = DiffusionPDE(bc={"*": {"derivative": 0}, "y+": {"value_expression": bc_value}})
    res = eq.solve(field, t_range=10, dt=0.01, backend="numpy")
    res.plot()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.766 seconds)


.. _sphx_glr_download_examples_gallery_advanced_pdes_heterogeneous_bcs.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: heterogeneous_bcs.ipynb <heterogeneous_bcs.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: heterogeneous_bcs.py <heterogeneous_bcs.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: heterogeneous_bcs.zip <heterogeneous_bcs.zip>`