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 \(\chi\) does not depend on position.

Links:

Wikipedia.

position¶: position, or spatial variable.

Symbol:: x
Latex:: \(x\)
Dimension:: length

time¶: time.

Symbol:: t
Latex:: \(t\)
Dimension:: time

temperature¶: Temperature as a function of position and time.

Symbol:: T(x, t)
Latex:: \(T{\left(x,t \right)}\)
Dimension:: temperature

thermal_diffusivity¶: thermal_diffusivity.

Symbol:: alpha
Latex:: \(\alpha\)
Dimension:: area/time

law¶

Derivative(T(x, t), t) = alpha * Derivative(T(x, t), (x, 2))

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