## 11

## Partial Differential Equations

Partial differential equations appear in the description of physical processes in applied sciences and engineering. A differential equation that involves more than one independent variable is called a partial differential equation. We restrict ourselves to second order partial differential equations. The general second order linear partial differential equation is of the form

*Au*_{xx} + *Bu*_{xy} + *Cu*_{yy} + *Du*_{x} + *Eu*_{y}+ *Fu* = *G*,

where *A*, *B*, *C*, *D*, *E*, *F*, *G* are all functions of *x* and *y*. Equations of the above form can be classified into three types:

- If
*B*^{2} – 4*AC* <; 0 at a point in the (*x*, *y*) plane, then the equation is called elliptic. For example, the equation *u*_{xx} + *u*_{yy} = 0, known as Laplace equation, is elliptic.
- If
*B*