.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/heat/plot_01_fourier_heat_series.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_pde_heat_plot_01_fourier_heat_series.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_pde_heat_plot_01_fourier_heat_series.py:


Fourier's heat equation: a sine series whose modes decay
========================================================

Joseph Fourier solved :math:`u_t = \alpha u_{xx}` on a rod with ice-cold
ends by writing the initial temperature as a sine series. Each mode
:math:`\sin(k\pi x)` then evolves on its own, decaying like
:math:`e^{-\alpha (k\pi)^2 t}`: sharp features (high :math:`k`) vanish
almost instantly, leaving the smooth fundamental mode. This script
computes the sine coefficients of a square-wave start, watches the
series smooth out, and checks it against a finite-difference solve.

.. GENERATED FROM PYTHON SOURCE LINES 15-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import HeatEquation1D, fourier_sine_coefficients, heat_series_solution


    def square(x):
        return np.where((x > 0.25) & (x < 0.75), 1.0, 0.0)









.. GENERATED FROM PYTHON SOURCE LINES 26-28

The initial temperature as a sine series
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-43

.. code-block:: Python


    b = fourier_sine_coefficients(square, 60)
    for k in range(1, 8):
        print(f"b_{k} = {b[k - 1]:+.4f}")

    x = np.linspace(0.0, 1.0, 400)
    fig1, ax1 = plt.subplots(figsize=(7, 4))
    ax1.plot(x, square(x), color="black", lw=2, label="initial temperature")
    for n_terms, color in ((1, "goldenrod"), (5, "darkorange"), (60, "firebrick")):
        ax1.plot(x, heat_series_solution(b[:n_terms], x, t=0.0), color=color, label=f"{n_terms} sine terms")
    ax1.set_xlabel("$x$")
    ax1.set_title("A square wave as a sum of sine modes")
    ax1.legend(fontsize=8)
    fig1.tight_layout()




.. image-sg:: /api/gallery/pde/heat/images/sphx_glr_plot_01_fourier_heat_series_001.png
   :alt: A square wave as a sum of sine modes
   :srcset: /api/gallery/pde/heat/images/sphx_glr_plot_01_fourier_heat_series_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    b_1 = +0.9003
    b_2 = +0.0000
    b_3 = -0.3001
    b_4 = -0.0000
    b_5 = -0.1801
    b_6 = +0.0000
    b_7 = +0.1286




.. GENERATED FROM PYTHON SOURCE LINES 44-48

High modes die first
--------------------
The :math:`k`-th mode's decay rate grows like :math:`k^2`, so after
a short time only the first mode is visible.

.. GENERATED FROM PYTHON SOURCE LINES 48-58

.. code-block:: Python


    fig2, ax2 = plt.subplots(figsize=(7, 4))
    for t, color in zip((0.0, 0.002, 0.01, 0.05, 0.2), plt.get_cmap("viridis")(np.linspace(0, 0.9, 5)), strict=False):
        ax2.plot(x, heat_series_solution(b, x, t), color=color, label=f"t = {t}")
    ax2.set_xlabel("$x$")
    ax2.set_ylabel("$u$")
    ax2.set_title("Fourier's series solution smooths the square wave")
    ax2.legend(fontsize=8)
    fig2.tight_layout()




.. image-sg:: /api/gallery/pde/heat/images/sphx_glr_plot_01_fourier_heat_series_002.png
   :alt: Fourier's series solution smooths the square wave
   :srcset: /api/gallery/pde/heat/images/sphx_glr_plot_01_fourier_heat_series_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 59-61

The series agrees with a finite-difference solve
------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 61-68

.. code-block:: Python


    heat = HeatEquation1D(square, n=201)
    sol = heat.solve_theta(0.05, dt=1e-4, theta=0.5)
    series = heat_series_solution(b, sol.x, 0.05)
    print(f"max |finite difference - Fourier series| at t = 0.05: {np.max(np.abs(sol.final - series)):.2e}")

    plt.show()




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

 .. code-block:: none

    max |finite difference - Fourier series| at t = 0.05: 4.24e-03





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

   **Total running time of the script:** (0 minutes 0.260 seconds)


.. _sphx_glr_download_api_gallery_pde_heat_plot_01_fourier_heat_series.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/pde/heat/plot_01_fourier_heat_series.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_fourier_heat_series.ipynb <plot_01_fourier_heat_series.ipynb>`

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

      :download:`Download Python source code: plot_01_fourier_heat_series.py <plot_01_fourier_heat_series.py>`

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

      :download:`Download zipped: plot_01_fourier_heat_series.zip <plot_01_fourier_heat_series.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_