Partial Differential Equation Canonical Form

A partial differential equation canonical form is a simplified form of a partial differential equation obtained by a coordinate transformation. For a second-order linear partial differential equation in two variables, the transformation simplifies the second-order terms according to the elliptic, hyperbolic, or parabolic type of the equation.

Consider

(1)

where the coefficients are smooth real functions of and , and , , and do not all vanish at any point of the neighborhood considered. Subscripts denote partial derivatives. Choose smooth real coordinate functions and , and write , with nonzero Jacobian

(2)

The chain rule gives the transformed equation

(3)

where

			(4)
			(5)
			(6)
			(7)
			(8)

Here and , with all right-hand sides expressed in the new coordinates (Pennell n.d., p. 1). In particular,

(9)

so the sign of , and hence the type of the equation, is unchanged.

In a neighborhood where the type is constant, suitable real coordinates and division by a nonvanishing factor give the following local canonical forms (Pennell n.d., pp. 2-3). Write , where , , , and are functions of and . For a hyperbolic partial differential equation, , and the canonical form is

(10)

The characteristic curves satisfy

(11)

Taking and constant along the two distinct families gives . For a parabolic partial differential equation, , and the canonical form is

(12)

Here is constant along the single family of characteristic curves, and is chosen to make . For an elliptic partial differential equation, , and the canonical form is

(13)

The coordinates satisfy and . These reductions simplify the second-order terms without generally eliminating the lower-order terms.

For example, the hyperbolic partial differential equation

(14)

for has characteristic curves satisfying . Choose and , so and . The transformed equation is

(15)

which gives the canonical form

(16)