OF CLASSIFICATION LINEAR PDEs OF SECOND ORDER
Second order linear PDEs with two independent variables x; y have the general form:
Au + Bu + Cu + Du + Eu + Fu = G
xx xy yy x y
where the coe cients A; B; C; D; E; F; and G are functions of x and y or they could be constants. The PDEs can be classi ed as either: 1: Hyperbolic if B 2 , 4AC 3: Elliptic if 0
2: Parabolic if B 2 , 4AC = 0
B 2 , 4AC 0
The PDEs are de ned to be homogeneous if G = 0, otherwise they are de ned to be nonhomogeneous. The PDEs are de ned to be nonlinear if they contain terms like
2 u @u = 1 @u @x 2 @x
EXAMPLES OF HYPERBOLIC, PARABOLIC, AND ELLIPTIC The PDEs
1. Di usion Heat Equation
u =u
xx
t
1
is a second-order linear PDE with coe cients:
A=1
Therefore
B=0
C=0 F =0
D=0 G=0
E = ,1
B 2 , 4AC = 0
for all x and t. The Di usion Equation is Parabolic. 2. The Wave Equation
u =u
xx
tt
is a second-order linear PDE with coe cients:
A=1
Therefore
B=0 E =0
C = ,1 F =0 G=0
D=0
B 2 , 4AC = 4 0
for al...