.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples_gallery/simple_pdes/pde_1d_expression.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_simple_pdes_pde_1d_expression.py>`
        to download the full example code.

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

.. _sphx_glr_examples_gallery_simple_pdes_pde_1d_expression.py:


1D problem - Using `PDE` class
==============================

This example implements a PDE that is only defined in one dimension.
Here, we chose the `Korteweg-de Vries equation
<https://en.wikipedia.org/wiki/Korteweg–de_Vries_equation>`_, given by

.. math::
    \partial_t \phi = 6 \phi \partial_x \phi - \partial_x^3 \phi

which we implement using the :class:`~pde.pdes.pde.PDE`.

.. GENERATED FROM PYTHON SOURCE LINES 14-30



.. image-sg:: /examples_gallery/simple_pdes/images/sphx_glr_pde_1d_expression_001.png
   :alt: pde 1d expression
   :srcset: /examples_gallery/simple_pdes/images/sphx_glr_pde_1d_expression_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    from math import pi

    from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph

    # initialize the equation and the space
    eq = PDE({"φ": "6 * φ * d_dx(φ) - laplace(d_dx(φ))"})
    grid = CartesianGrid([[0, 2 * pi]], [32], periodic=True)
    state = ScalarField.from_expression(grid, "sin(x)")

    # solve the equation and store the trajectory
    storage = MemoryStorage()
    eq.solve(state, t_range=3, solver="scipy", tracker=storage.tracker(0.1))

    # plot the trajectory as a space-time plot
    plot_kymograph(storage)


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

   **Total running time of the script:** (0 minutes 9.978 seconds)


.. _sphx_glr_download_examples_gallery_simple_pdes_pde_1d_expression.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: pde_1d_expression.ipynb <pde_1d_expression.ipynb>`

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

      :download:`Download Python source code: pde_1d_expression.py <pde_1d_expression.py>`

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

      :download:`Download zipped: pde_1d_expression.zip <pde_1d_expression.zip>`