Page 412 - Applied Numerical Methods Using MATLAB

9

PARTIAL DIFFERENTIAL

EQUATIONS

What is a partial differential equation (PDE)? It is a class of differential equations
involving more than one independent variable. In this chapter, we consider a gen-
eral second-order PDE in two independent variables x and y, which is written as

2
2
2
∂ u ∂ u ∂ u ∂u ∂u
A(x, y) + B(x, y) + C(x, y) = f x, y, u, , (9.0.1)
∂x 2 ∂x∂y ∂y 2 ∂x ∂y
for x 0 ≤ x ≤ x f ,y 0 ≤ y ≤ y f
with the boundary conditions given by

u(x, y 0 ) = b y0 (x), u(x, y f ) = b yf (x),
(9.0.2)
u(x 0 ,y) = b x0 (y), and u(x f ,y) = b xf (y)
These PDEs are classified into three groups:

2
Elliptic PDE: if B − 4AC < 0
2
Parabolic PDE: if B − 4AC = 0
2
Hyperbolic PDE: if B − 4AC > 0
These three types of PDE are associated with equilibrium states, diffusion states,
and oscillating systems, respectively. We will study some numerical methods for
solving these PDEs, since their analytical solutions are usually difficult to find.


Applied Numerical Methods Using MATLAB , by Yang, Cao, Chung, and Morris
Copyright  2005 John Wiley & Sons, I nc., ISBN 0-471-69833-4

401