# Heat Diffusion Equation

## $\LARGE\frac{{\partial}T}{{\partial}t}=\kappa\nabla^2 T+f$

### The Terms

$T$ - Temperature [Units: K, Kelvin]

$t$ - Time [Units: s]

$\kappa$ - Thermal diffusivity, material specific. $\kappa=k/c_p\rho$ (thermal conductivity divided by the volumetric heat capacity - the product of the density and the specific heat capacity [Units: m2 s-1]

$\nabla^2$ - Laplace operator, second order partial differential operator with respect to the three spatial directions, $\nabla^2T=\partial_{xx}^2T+\partial_{yy}^2T+\partial_{zz}^2T$

$f$ - Rate of heat input, known function varying in space and time

### What Does It Mean?

This is a partial differential equation describing the distribution of heat (or variation in temperature) in a particular body, over time. This equation has other important applications in mathematics, statistical mechanics, probability theory and financial mathematics.

### Further information at Warwick

The principles governing this equation are explored in the first year module "PX121 Thermal Physics I", the second year module "PX265 Thermal Physics II" and the third year module "PX366 Statistical Physics".