 To define the simplest kind of parabolic PDE, consider a real-valued function formula_1 of two independent real variables, formula_2 and formula_3.A second-order, linear, constant-coefficient PDE for formula_4 takes the form

and this PDE is classified as being "parabolic" if the coefficients satisfy the condition

Usually formula_2 represents one-dimensional position and formula_3 represents time,
and the PDE is solved subject to prescribed initial and boundary conditions.The name "parabolic" is used because the assumption on the coefficients is the same as the condition
for the analytic geometry equation formula_9
to define a planar parabola.