Classification and Examples - Main Theorem

The canonical form theorem shows that every second-order PDE can be locally transformed to a standard form that reveals its type. This fundamental result underlies both theoretical understanding and practical solution methods.

TheoremCanonical Form Theorem

Let $Lu = A\frac{\partial^2 u}{\partial x^2} + 2B\frac{\partial^2 u}{\partial x\partial y} + C\frac{\partial^2 u}{\partial y^2} + \text{lower order terms}$ be a second-order linear PDE with smooth coefficients in a domain $\Omega \subset \mathbb{R}^2$ .

Then in a neighborhood of any point $(x_0, y_0) \in \Omega$ , there exists a smooth change of variables $(\xi, \eta) = (\xi(x,y), \eta(x,y))$ that transforms $L$ to one of the following canonical forms:

Hyperbolic ( $B^2 - AC > 0$ ): $\frac{\partial^2 u}{\partial \xi \partial \eta} + \text{lower order terms}$
Parabolic ( $B^2 - AC = 0$ ): $\frac{\partial^2 u}{\partial \xi^2} + \text{lower order terms}$
Elliptic ( $B^2 - AC < 0$ ): $\frac{\partial^2 u}{\partial \xi^2} + \frac{\partial^2 u}{\partial \eta^2} + \text{lower order terms}$

This theorem is proved by finding appropriate characteristic coordinates. For hyperbolic equations, the transformation is constructed using the two families of real characteristics. For parabolic equations, one coordinate is along the characteristic, the other is arbitrary but transverse. For elliptic equations, complex characteristics lead to a transformation that symmetrizes the principal part.

Remark

The canonical form reveals the essential structure of the PDE. For instance, the hyperbolic canonical form $\frac{\partial^2 u}{\partial \xi \partial \eta} = F$ can be integrated twice, showing that hyperbolic equations have a fundamentally different solution character than elliptic equations.

The existence of canonical forms has profound implications. It allows us to understand general second-order PDEs by studying the three canonical types. Moreover, numerical methods often work by transforming a PDE to a form where standard techniques apply. The theorem guarantees that such transformations exist locally, though finding them explicitly may be challenging for variable coefficient equations.

ExampleTransformation Example

Consider $\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0$ (the wave equation in characteristic form).

The characteristics are $x \pm y = \text{const}$ . Setting $\xi = x + y$ , $\eta = x - y$ transforms this to: $\frac{\partial^2 u}{\partial \xi \partial \eta} = 0$ with general solution $u(\xi, \eta) = f(\xi) + g(\eta) = f(x+y) + g(x-y)$ .