Basics

In this chapter, we focus on second order linear PDEs in two independent variables $x$ and $y$. Generic form of such a equation is

$$\underbrace{au_{xx} + b u_{xy} + c u_{yy}}_{\mbox{\small The principal part}} + d u_x + e u_y + fu = g,$$ (2.1)

Where $a, b, c, d, e, f, g$ are constants or given functions of $x$ and $y$, $u(x,y)$ is the unknown function. We assume that the coefficients $a, b$ and $c$ do not vanish simultaneously.

The part $au_{xx}+bu_{xy}+cu_{yy}$ consisting of second order terms of the PDE is called the principal part ( the homogeneous part of given PDE).

We will see that many fundamental properties of the solutions of Eq. () are determined by its principle part (actually, the sign of the discriminant $\Delta := b^2 - 4ac$ of the equation). We classify the Eq. () according to the sign of its discriminant.

Definition 6 (Classification of second order PDEs)   We call PDE () is

1. hyperbolic at a point $(x,y)$ if $\Delta(x,y) = b^2(x,y) - 4a(x,y)c(x,y) >0,$
2. parabolic at a point $(x,y)$ if $\Delta(x,y) = b^2(x,y) - 4a(x,y)c(x,y) =0,$
3. elliptic at a point $(x,y)$ if $\Delta(x,y) = b^2(x,y) - 4a(x,y)c(x,y) <0.$

If this is true at all points in the domain $\Omega$, then Eq. () is said to be hyperbolic, parabolic or elliptic in that domain.

Fundamental PDEs of mathematical physics $\mapsto$ all are second order PDEs.

$$\left. \begin{array}{rrcll} \mbox{Wave equation} & u_{tt} &=&... ...elta u & \mbox{Elliptic PDE} \end{array}\right \} \mbox{Check.}$$

These three form of PDEs are called canonical form or normal form. If the number of independent variables is two or three, a transformation can always be found to reduce the given PDEs to one of the canonical form.