Classification of Second-Order PDEs

Classification of PDEs is an important concept because the general theory and methods
of solution usually apply only to a given class of equations. Let us first discuss the
classification of PDEs involving two or more independent variables.

1) Classification with two independent variables

Consider the following general second order linear PDE in two independent variables:

\(A \frac{ \partial ^2u}{ \partial x^2}+ B\frac { \partial^2u}{ \partial x\partial y} + C\frac{ \partial^2u}{\partial y^2} + D\frac{ \partial u}{\partial x}+ E\frac{ \partial u}{\partial y} + F u + G = 0\)

where \(A, B, C, D, E, F\) and \(G\) are functions of the independent variables \(x\) and \(y\). This equation may be written in the form.



\(u_x=\frac{ \partial u}{\partial x}, u_y=\frac {\partial u}{\partial y}, u_{xx}= \frac{ \partial ^2u}{ \partial x^2}, u_{xy}=\frac { \partial^2u}{ \partial x\partial y}, u_{yy}=\frac{ \partial^2u}{\partial y^2}\)

Assume that \(A, B\) and \(C\) are continuous functions of \(x\) and \(y\) possessing continuous partial derivatives of as high order as necessary.

The classification of PDE is motivated by the classification of second order algebraic equations in two-variables

\(ax^2 + bxy + cy^2 + dx + ey + f = 0\)

We know that the nature of the curves will be decided by the principal part \(ax^2+bxy+cy^2\) i.e., the term containing highest degree. Depending on the sign of the discrimination \(b^2-4ac\), we classify the curve as follows:

  • If \(b^2-4ac>0\) then the curve traces hyperbola
  • If \(b^2-4ac=0\) then the curve traces parabola
  • If \(b^2-4ac<0\) then the curve traces ellipse.

With suitable transformation, we can transform into the following normal form:

  • For \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) –> hyperbola
  • For \(x^=y\) –> parabola
  • For \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) –> ellipse

Conclusion: There are three types of second order linear PDEs, each of which is invariant under changes in variables. A discriminant’s sign determines its type. The slope of the characteristic curves corresponds exactly to the different cases of the quadratic equation. The hyperbolic equations have two distinct families of (real) characteristic curves, the parabolic equations have one, and the elliptic equations have none. Canonical forms can be found for all three types of equations. In the leading terms, hyperbolic equations are equivalent to the wave equation, parabolic equations to the heat equation, and elliptic equations to Laplace’s equation. For all second order constant coefficient PDEs, the wave, heat, and Laplace’s equations serve as canonical models. Laplace’s equations, wave equations, and heat equations will be studied throughout the remainder of the quarter.