 The basic example of a parabolic PDE is the one-dimensional heat equation,

where formula_11 is the temperature at time formula_12 and at position formula_2 along a thin rod, and formula_14 is a positive constant (the "thermal diffusivity").The symbol formula_15 signifies the partial derivative of formula_4 with respect to the time variable formula_12, and similarly formula_18 is the second partial derivative with respect to formula_2.For this example, formula_12 plays the role of formula_3 in the general second-order linear PDE