Partial Differential Equations with MATLAB

Characteristics 177

Then, supposing that the ODE (5.6) has general solutionh(x, y)=c,we
make the transformation

ξ=x

η=h(x, y).

In this case, we have

ux=uξ+uηhx

uy=uηhy

and
aux+buy=auξ+(ahx+bhy)uη.

But

dx

a

=

dy

b

⇒h(x, y)=c

⇒dh=0=hxdx+hydy

=dx

(

hx+hy

dy

dx

)

=dx

(

hx+hy

b

a

)

=

dx

a

(ahx+bhy)

and we have
aux+buy=auξ,

so the PDE has been reduced to an ODE.
We haven’t been very rigorous here. For example, what happens at points
wherea=0orb= 0? Theorem 5.1 can be extended somewhat to the case
of variable coefficients.†It turns out that a necessary condition for existence
and uniqueness of a solution throughout a neighborhood of a point is that
a=0orb= 0 at that point.

Example 4 Solve

ux+yuy=x,

u(1,y)=cosy.

The characteristics are given by

dx

1

=

dy

y

†See, e.g.,Introduction to Partial Differential Equations with Applicationsby E. C. Zach-
manoglou and Dale W. Thoe.