Modelling: Derivation of the heat equation

In an object, heat will flow in the direction of decreasing temperature. In other words, heat flows from hot to cool. To derive the heat equation, we will consider the flow of heat along a metal rod. The rod will allow us to consider the temperature, $u(x,t)$, as one dimensional in $x$ but changing in time, $t$. The heat flow is proportional to the temperature gradient, i.e.

$$- K{\partial u \over \partial x},$$

$K$ is a constant of proportionality. Consider a small element of the rod between the positions $x$ and $x + \delta x$. The amount of heat in the element, at time $t$, is

$$H(t) = \sigma \rho u(x, t) \delta x,$$

where $\sigma$ is the specific heat of the rod and $\rho$ is the mass per unit length. At time $t + \delta t$, the amount of heat is

$$H(t + \delta t) = \sigma \rho u(x, t+\delta t) \delta x$$

Thus, the change in heat is simply

$$H(t+\delta t) - H(t) = \sigma \rho \left (u(x, t+\delta t) - u(x, t)\right ) \delta x.$$

This change of heat must equal the heat flowing in at $x$ minus the heat flowing out at $x + \delta x$ during the time interval $\delta t$. This may be expressed as

$$\left [- K\left ({\partial u \over \partial x}\right )_{x} ... ...ial u \over \partial x}\right )_{x+\delta x}\right ]\delta t$$

Equating these expressions and dividing by $\delta x$ and $\delta t$ gives,

$$\sigma \rho {u(x, t + \delta t) - u(x, t)\over \delta t} = ... ...t ({\partial U \over \partial x}\right )_{x}\over \delta x}.$$

Taking the limits of $\delta x$ and $\delta t$ tending to zero, we obtain the partial derivatives. Hence, the heat equation is

$${\partial u \over \partial t} = c^{2}{\partial^{2} u\over \partial x^{2}},$$ (2.47)

where $c^{2}= K/\sigma \rho$ is the constant thermal conductivity and $\partial ^{2}y/\partial x^{2}$ is the thermal conduction.

The heat equation has the same form as the equation describing diffusion. Both processes describe physical phenomena that are being smoothed in time. Thus, initial irregularities and spatial variations are smoothed due to the thermal conduction term.