## 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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