Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.6 CHARACTERISTICS AND THE EXISTENCE OF SOLUTIONS

1

1

2

y

x

− 1

x=1

c=1

y=c/x^2

Figure 20.2 The characteristics of equation (20.42). The shaded region shows

where the solution to the equation is defined, given the imposed boundary

condition atx= 1 betweeny=0andy= 1, shown as a bold vertical line.

P

y

Q

C

R

x

Figure 20.3 A boundary curveCthat crosses characteristics more than once.

20.6.2 Second-order equations

The concept of characteristics can be extended naturally to second- (and higher-)

order equations. In this case let us write the general second-order linear PDE

(20.19) as

A(x, y)

∂^2 u

∂x^2

+B(x, y)

∂^2 u

∂x∂y

+C(x, y)

∂^2 u

∂y^2

=F

(

x, y, u,

∂u

∂x

,

∂u

∂y

)

. (20.43)