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 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.

Au_{xx}+Bu_{xy}+Cu_{yy}+f(x,y,u_x,u_y,u)=0,

where

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
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