Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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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)

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