20.6 CHARACTERISTICS AND THE EXISTENCE OF SOLUTIONS
we may differentiate the two first derivatives∂u/∂xand∂u/∂yalong the boundary
to obtain the pair of equations
d
ds(
∂u
∂x)
=dx
ds∂^2 u
∂x^2+dy
ds∂^2 u
∂x∂y,d
ds(
∂u
∂y)
=dx
ds∂^2 u
∂x∂y+dy
ds∂^2 u
∂y^2.We may now solve these two equations, together with the original PDE (20.43),
for the second partial derivatives ofu,exceptwhere the determinant of their
coefficients equals zero,
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
ABC
dx
dsdy
ds00dx
dsdy
ds∣
∣
∣
∣
∣
∣
∣
∣
∣
∣=0.Expanding out the determinant,
A(
dy
ds) 2
−B(
dx
ds)(
dy
ds)
+C(
dx
ds) 2
=0.Multiplying through by (ds/dx)^2 we obtain
A(
dy
dx) 2
−Bdy
dx+C=0, (20.44)which is the ODE for the curves in thexy-plane along which the second partial
derivatives ofucannotbe found.
As for the first-order case, the curves satisfying (20.44) are called characteristicsof the original PDE. These characteristics have tangents at each point given by
(whenA=0)
dy
dx=B±√
B^2 − 4 AC
2 A. (20.45)
Clearly, when the original PDE is hyperbolic (B^2 > 4 AC), equation (20.45)
defines two families of real curves in thexy-plane; when the equation is parabolic
(B^2 =4AC) it defines one family of real curves; and when the equation is elliptic
(B^2 < 4 AC) it defines two families of complex curves. Furthermore, whenA,
BandCare constants, rather than functions ofxandy, the equations of the
characteristics will be of the formx+λy= constant, which is reminiscent of the
form of solution discussed in subsection 20.3.3.