20.6 CHARACTERISTICS AND THE EXISTENCE OF SOLUTIONS
20.6 Characteristics and the existence of solutionsSo far in this chapter we have discussed how to find general solutions to various
types of first- and second-order linear PDE. Moreover, given a set of boundary
conditions we have shown how to find the particular solution (or class of solutions)
that satisfies them. For first-order equations, for example, we found that if the
value ofu(x, y) is specified along some curve in thexy-plane then the solution to
the PDE is in general unique, but that ifu(x, y) is specified at only a single point
then the solution is not unique: there exists a class of particular solutions all of
which satisfy the boundary condition. In this section and the next we make more
rigorous the notion of the respective types of boundary condition that cause a
PDE to have a unique solution, a class of solutions, or no solution at all.
20.6.1 First-order equationsLet us consider the general first-order PDE (20.9) but now write it as
A(x, y)∂u
∂x+B(x, y)∂u
∂y=F(x, y, u). (20.39)Suppose we wish to solve this PDE subject to the boundary condition that
u(x, y)=φ(s) is specified along some curveCin thexy-plane that is described
parametrically by the equationsx=x(s)andy=y(s), wheresis the arc length
alongC. The variation ofualongCis therefore given by
du
ds=∂u
∂xdx
ds+∂u
∂ydy
ds=dφ
ds. (20.40)
We may then solve the two (inhomogeneous) simultaneous linear equations
(20.39) and (20.40) for∂u/∂xand∂u/∂y,unlessthe determinant of the coefficients
vanishes (see section 8.18), i.e. unless
∣
∣
∣
∣
dx/ds dy/ds
AB∣
∣
∣
∣=0.At each point in thexy-plane this equation determines a set of curves called
characteristic curves(or justcharacteristics), which thus satisfy
Bdx
ds−Ady
ds=0,or, multiplying through byds/dxand dividing through byA,
dy
dx=B(x, y)
A(x, y). (20.41)
However, we have already met (20.41) in subsection 20.3.1 on first-order PDEs,where solutions of the formu(x, y)=f(p), wherepis some combination ofxandy,