Equation in homogeneous medium in one dimension
===============================================

Heat equation governs heat diffusion, as well as other diffusive processes. It describes
the evolution of heat transferred from hotter to colder environments in time and space.

**Notes:**

#. There is no straghtforward solution to this equation, and it depends on initial
   conditions as well.

**Conditions:**

#. There are no heat sources in the system, i.e. the heat distribution only depends on
   the initial conditions.
#. Thermal diffusivity :math:`\chi` does not depend on position.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Heat_equation#Heat_flow_in_a_uniform_rod>`__.

.. py:currentmodule:: symplyphysics.laws.thermodynamics.heat_transfer.equation_in_homogeneous_medium_in_one_dimension

.. py:data:: position

    :attr:`~symplyphysics.symbols.classical_mechanics.position`, or spatial variable.

Symbol:
    :code:`x`

Latex:
    :math:`x`

Dimension:
    :code:`length`


.. py:data:: time

    :attr:`~symplyphysics.symbols.basic.time`.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: temperature

    Temperature as a function of :attr:`~position` and :attr:`~time`.

Symbol:
    :code:`T(x, t)`

Latex:
    :math:`T{\left(x,t \right)}`

Dimension:
    :code:`temperature`


.. py:data:: thermal_diffusivity

    :attr:`~symplyphysics.symbols.thermodynamics.thermal_diffusivity`.

Symbol:
    :code:`alpha`

Latex:
    :math:`\alpha`

Dimension:
    :code:`area/time`


.. py:data:: law

    :code:`Derivative(T(x, t), t) = alpha * Derivative(T(x, t), (x, 2))`


    Latex:
        .. math::
            \frac{\partial}{\partial t} T{\left(x,t \right)} = \alpha \frac{\partial^{2}}{\partial x^{2}} T{\left(x,t \right)}