ABSTRACT
If B AC2 4 0-< , the equation is said to be elliptic. If B AC2 4 0-= , the equation is said to be parabolic. If B AC2 4 0-> , the equation is said to be hyperbolic.
The steady-state heat conduction problem in two dimensions is an example of an elliptic PDE. Laplace’s PDE falls into this category. The parabolic PDE is also called the diffusion equation. The unsteady heat conduction problem is an example of a parabolic PDE. The hyperbolic PDE is also called the wave equation. Sound waves and vibration problems, such as the vibrating string, fall into this category. How a PDE is treated numerically depends into which category it falls. However, there are cases in all three categories where a closed-form solution can be obtained by a method called separation of variables. This solution method is discussed in the next section.
