A partial differential equation of second-order, i.e., one of the form A u_(x x) + 2B u_(x y) + C u_(y y) + D u_x + E u_y + F = 0, is called parabolic if the matrix Z congruent [A | B B | C] satisfies det(Z) = 0. The heat conduction equation and other diffusion equations are examples.

boundary conditions | boundary value problem | elliptic partial differential equation | hyperbolic partial differential equation | initial value problem | partial differential equation